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Degree 1 Guzmán–Neilan (first kind) on a tetrahedron
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- \(R\) is the reference tetrahedron. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{\sqrt{3} z \left(60 x^{2} + 180 x y + 132 x z - 96 x - 330 y z - 430 z^{2} + 192 z + 9\right)}{180}\\\displaystyle \tfrac{\sqrt{3} z \left(180 x y - 330 x z + 60 y^{2} + 132 y z - 96 y - 430 z^{2} + 192 z + 9\right)}{180}\\\displaystyle \tfrac{\sqrt{3} z \left(- 150 x z - 150 y z - 88 z^{2} + 96 z + 9\right)}{180}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{\sqrt{3} y \left(60 x^{2} + 132 x y + 180 x z - 96 x - 430 y^{2} - 330 y z + 192 y + 9\right)}{180}\\\displaystyle \tfrac{\sqrt{3} y \left(- 150 x y - 88 y^{2} - 150 y z + 96 y + 9\right)}{180}\\\displaystyle \tfrac{\sqrt{3} y \left(- 330 x y + 180 x z - 430 y^{2} + 132 y z + 192 y + 60 z^{2} - 96 z + 9\right)}{180}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{\sqrt{3} x \left(- 88 x^{2} - 150 x y - 150 x z + 96 x + 9\right)}{180}\\\displaystyle \tfrac{\sqrt{3} x \left(- 430 x^{2} + 132 x y - 330 x z + 192 x + 60 y^{2} + 180 y z - 96 y + 9\right)}{180}\\\displaystyle \tfrac{\sqrt{3} x \left(- 430 x^{2} - 330 x y + 132 x z + 192 x + 180 y z + 60 z^{2} - 96 z + 9\right)}{180}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{\sqrt{3} \left(112 x^{3} + 54 x^{2} y + 54 x^{2} z - 156 x^{2} - 168 x y^{2} - 216 x y z + 156 x y - 168 x z^{2} + 156 x z + 3 x - 110 y^{3} - 270 y^{2} z + 252 y^{2} - 270 y z^{2} + 444 y z - 183 y - 110 z^{3} + 252 z^{2} - 183 z + 41\right)}{180}\\\displaystyle \tfrac{\sqrt{3} \left(- 110 x^{3} - 168 x^{2} y - 270 x^{2} z + 252 x^{2} + 54 x y^{2} - 216 x y z + 156 x y - 270 x z^{2} + 444 x z - 183 x + 112 y^{3} + 54 y^{2} z - 156 y^{2} - 168 y z^{2} + 156 y z + 3 y - 110 z^{3} + 252 z^{2} - 183 z + 41\right)}{180}\\\displaystyle \tfrac{\sqrt{3} \left(- 110 x^{3} - 270 x^{2} y - 168 x^{2} z + 252 x^{2} - 270 x y^{2} - 216 x y z + 444 x y + 54 x z^{2} + 156 x z - 183 x - 110 y^{3} - 168 y^{2} z + 252 y^{2} + 54 y z^{2} + 156 y z - 183 y + 112 z^{3} - 156 z^{2} + 3 z + 41\right)}{180}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(- 90 x y + 216 x z + 12 x - 60 y^{2} + 105 y z + 60 y - 55 z^{2} - 54 z - 3\right)}{60}\\\displaystyle \tfrac{z \left(- 30 y^{2} - 219 y z + 42 y + 325 z^{2} - 69 z - 3\right)}{60}\\\displaystyle \tfrac{z \left(75 y z + z^{2} - 27 z - 3\right)}{60}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(216 x y - 90 x z + 12 x - 55 y^{2} + 105 y z - 54 y - 60 z^{2} + 60 z - 3\right)}{60}\\\displaystyle \tfrac{y \left(y^{2} + 75 y z - 27 y - 3\right)}{60}\\\displaystyle \tfrac{y \left(325 y^{2} - 219 y z - 69 y - 30 z^{2} + 42 z - 3\right)}{60}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{43 x^{3}}{30} + \tfrac{5 x^{2} y}{4} + \tfrac{5 x^{2} z}{4} - \tfrac{7 x^{2}}{10} - \tfrac{x}{20} - y^{2} z - y z^{2} + y z\\\displaystyle \tfrac{x \left(160 x^{2} - 129 x y + 165 x z - 69 x - 30 y^{2} - 90 y z + 42 y - 3\right)}{60}\\\displaystyle \tfrac{x \left(160 x^{2} + 165 x y - 129 x z - 69 x - 90 y z - 30 z^{2} + 42 z - 3\right)}{60}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{203 x^{3}}{30} + \tfrac{369 x^{2} y}{20} + \tfrac{369 x^{2} z}{20} - \tfrac{173 x^{2}}{10} + \tfrac{161 x y^{2}}{10} + \tfrac{151 x y z}{5} - \tfrac{151 x y}{5} + \tfrac{161 x z^{2}}{10} - \tfrac{151 x z}{5} + \tfrac{283 x}{20} + \tfrac{53 y^{3}}{12} + \tfrac{45 y^{2} z}{4} - \tfrac{62 y^{2}}{5} + \tfrac{45 y z^{2}}{4} - \tfrac{114 y z}{5} + \tfrac{58 y}{5} + \tfrac{53 z^{3}}{12} - \tfrac{62 z^{2}}{5} + \tfrac{58 z}{5} - \tfrac{217}{60}\\\displaystyle - \tfrac{8 x^{3}}{3} - \tfrac{203 x^{2} y}{20} - \tfrac{21 x^{2} z}{4} + \tfrac{137 x^{2}}{20} - \tfrac{59 x y^{2}}{5} - \tfrac{133 x y z}{10} + \tfrac{173 x y}{10} - \tfrac{5 x z^{2}}{2} + \tfrac{41 x z}{5} - \tfrac{113 x}{20} - \tfrac{259 y^{3}}{60} - \tfrac{151 y^{2} z}{20} + \tfrac{199 y^{2}}{20} - \tfrac{63 y z^{2}}{20} + \tfrac{103 y z}{10} - \tfrac{71 y}{10} + \tfrac{z^{3}}{12} + \tfrac{27 z^{2}}{20} - \tfrac{29 z}{10} + \tfrac{22}{15}\\\displaystyle - \tfrac{8 x^{3}}{3} - \tfrac{21 x^{2} y}{4} - \tfrac{203 x^{2} z}{20} + \tfrac{137 x^{2}}{20} - \tfrac{5 x y^{2}}{2} - \tfrac{133 x y z}{10} + \tfrac{41 x y}{5} - \tfrac{59 x z^{2}}{5} + \tfrac{173 x z}{10} - \tfrac{113 x}{20} + \tfrac{y^{3}}{12} - \tfrac{63 y^{2} z}{20} + \tfrac{27 y^{2}}{20} - \tfrac{151 y z^{2}}{20} + \tfrac{103 y z}{10} - \tfrac{29 y}{10} - \tfrac{259 z^{3}}{60} + \tfrac{199 z^{2}}{20} - \tfrac{71 z}{10} + \tfrac{22}{15}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(30 x^{2} + 219 x z - 42 x - 325 z^{2} + 69 z + 3\right)}{60}\\\displaystyle \tfrac{z \left(60 x^{2} + 90 x y - 105 x z - 60 x - 216 y z - 12 y + 55 z^{2} + 54 z + 3\right)}{60}\\\displaystyle \tfrac{z \left(- 75 x z - z^{2} + 27 z + 3\right)}{60}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(30 x^{2} + 129 x y + 90 x z - 42 x - 160 y^{2} - 165 y z + 69 y + 3\right)}{60}\\\displaystyle x^{2} z - \tfrac{5 x y^{2}}{4} + x z^{2} - x z - \tfrac{43 y^{3}}{30} - \tfrac{5 y^{2} z}{4} + \tfrac{7 y^{2}}{10} + \tfrac{y}{20}\\\displaystyle \tfrac{y \left(- 165 x y + 90 x z - 160 y^{2} + 129 y z + 69 y + 30 z^{2} - 42 z + 3\right)}{60}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(- x^{2} - 75 x z + 27 x + 3\right)}{60}\\\displaystyle \tfrac{x \left(55 x^{2} - 216 x y - 105 x z + 54 x + 90 y z - 12 y + 60 z^{2} - 60 z + 3\right)}{60}\\\displaystyle \tfrac{x \left(- 325 x^{2} + 219 x z + 69 x + 30 z^{2} - 42 z + 3\right)}{60}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{259 x^{3}}{60} + \tfrac{59 x^{2} y}{5} + \tfrac{151 x^{2} z}{20} - \tfrac{199 x^{2}}{20} + \tfrac{203 x y^{2}}{20} + \tfrac{133 x y z}{10} - \tfrac{173 x y}{10} + \tfrac{63 x z^{2}}{20} - \tfrac{103 x z}{10} + \tfrac{71 x}{10} + \tfrac{8 y^{3}}{3} + \tfrac{21 y^{2} z}{4} - \tfrac{137 y^{2}}{20} + \tfrac{5 y z^{2}}{2} - \tfrac{41 y z}{5} + \tfrac{113 y}{20} - \tfrac{z^{3}}{12} - \tfrac{27 z^{2}}{20} + \tfrac{29 z}{10} - \tfrac{22}{15}\\\displaystyle - \tfrac{53 x^{3}}{12} - \tfrac{161 x^{2} y}{10} - \tfrac{45 x^{2} z}{4} + \tfrac{62 x^{2}}{5} - \tfrac{369 x y^{2}}{20} - \tfrac{151 x y z}{5} + \tfrac{151 x y}{5} - \tfrac{45 x z^{2}}{4} + \tfrac{114 x z}{5} - \tfrac{58 x}{5} - \tfrac{203 y^{3}}{30} - \tfrac{369 y^{2} z}{20} + \tfrac{173 y^{2}}{10} - \tfrac{161 y z^{2}}{10} + \tfrac{151 y z}{5} - \tfrac{283 y}{20} - \tfrac{53 z^{3}}{12} + \tfrac{62 z^{2}}{5} - \tfrac{58 z}{5} + \tfrac{217}{60}\\\displaystyle - \tfrac{x^{3}}{12} + \tfrac{5 x^{2} y}{2} + \tfrac{63 x^{2} z}{20} - \tfrac{27 x^{2}}{20} + \tfrac{21 x y^{2}}{4} + \tfrac{133 x y z}{10} - \tfrac{41 x y}{5} + \tfrac{151 x z^{2}}{20} - \tfrac{103 x z}{10} + \tfrac{29 x}{10} + \tfrac{8 y^{3}}{3} + \tfrac{203 y^{2} z}{20} - \tfrac{137 y^{2}}{20} + \tfrac{59 y z^{2}}{5} - \tfrac{173 y z}{10} + \tfrac{113 y}{20} + \tfrac{259 z^{3}}{60} - \tfrac{199 z^{2}}{20} + \tfrac{71 z}{10} - \tfrac{22}{15}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(- 30 x^{2} - 90 x y - 129 x z + 42 x + 165 y z + 160 z^{2} - 69 z - 3\right)}{60}\\\displaystyle \tfrac{z \left(- 90 x y + 165 x z - 30 y^{2} - 129 y z + 42 y + 160 z^{2} - 69 z - 3\right)}{60}\\\displaystyle - x^{2} y - x y^{2} + x y + \tfrac{5 x z^{2}}{4} + \tfrac{5 y z^{2}}{4} + \tfrac{43 z^{3}}{30} - \tfrac{7 z^{2}}{10} - \tfrac{z}{20}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(- 30 x^{2} - 219 x y + 42 x + 325 y^{2} - 69 y - 3\right)}{60}\\\displaystyle \tfrac{y \left(75 x y + y^{2} - 27 y - 3\right)}{60}\\\displaystyle \tfrac{y \left(- 60 x^{2} + 105 x y - 90 x z + 60 x - 55 y^{2} + 216 y z - 54 y + 12 z - 3\right)}{60}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(x^{2} + 75 x y - 27 x - 3\right)}{60}\\\displaystyle \tfrac{x \left(325 x^{2} - 219 x y - 69 x - 30 y^{2} + 42 y - 3\right)}{60}\\\displaystyle \tfrac{x \left(- 55 x^{2} + 105 x y + 216 x z - 54 x - 60 y^{2} - 90 y z + 60 y + 12 z - 3\right)}{60}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{259 x^{3}}{60} - \tfrac{151 x^{2} y}{20} - \tfrac{59 x^{2} z}{5} + \tfrac{199 x^{2}}{20} - \tfrac{63 x y^{2}}{20} - \tfrac{133 x y z}{10} + \tfrac{103 x y}{10} - \tfrac{203 x z^{2}}{20} + \tfrac{173 x z}{10} - \tfrac{71 x}{10} + \tfrac{y^{3}}{12} - \tfrac{5 y^{2} z}{2} + \tfrac{27 y^{2}}{20} - \tfrac{21 y z^{2}}{4} + \tfrac{41 y z}{5} - \tfrac{29 y}{10} - \tfrac{8 z^{3}}{3} + \tfrac{137 z^{2}}{20} - \tfrac{113 z}{20} + \tfrac{22}{15}\\\displaystyle \tfrac{x^{3}}{12} - \tfrac{63 x^{2} y}{20} - \tfrac{5 x^{2} z}{2} + \tfrac{27 x^{2}}{20} - \tfrac{151 x y^{2}}{20} - \tfrac{133 x y z}{10} + \tfrac{103 x y}{10} - \tfrac{21 x z^{2}}{4} + \tfrac{41 x z}{5} - \tfrac{29 x}{10} - \tfrac{259 y^{3}}{60} - \tfrac{59 y^{2} z}{5} + \tfrac{199 y^{2}}{20} - \tfrac{203 y z^{2}}{20} + \tfrac{173 y z}{10} - \tfrac{71 y}{10} - \tfrac{8 z^{3}}{3} + \tfrac{137 z^{2}}{20} - \tfrac{113 z}{20} + \tfrac{22}{15}\\\displaystyle \tfrac{53 x^{3}}{12} + \tfrac{45 x^{2} y}{4} + \tfrac{161 x^{2} z}{10} - \tfrac{62 x^{2}}{5} + \tfrac{45 x y^{2}}{4} + \tfrac{151 x y z}{5} - \tfrac{114 x y}{5} + \tfrac{369 x z^{2}}{20} - \tfrac{151 x z}{5} + \tfrac{58 x}{5} + \tfrac{53 y^{3}}{12} + \tfrac{161 y^{2} z}{10} - \tfrac{62 y^{2}}{5} + \tfrac{369 y z^{2}}{20} - \tfrac{151 y z}{5} + \tfrac{58 y}{5} + \tfrac{203 z^{3}}{30} - \tfrac{173 z^{2}}{10} + \tfrac{283 z}{20} - \tfrac{217}{60}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
- \(\mathcal{L}=\{l_0,...,l_{15}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{0} = \begin{cases} \left(\begin{array}{c}\displaystyle 30 x y z - 72 x z^{2} - 4 x z - x + 20 y^{2} z - 35 y z^{2} - 20 y z - y + \tfrac{55 z^{3}}{3} + 18 z^{2} + 1\\\displaystyle \tfrac{z \left(30 y^{2} + 219 y z - 42 y - 325 z^{2} + 69 z + 3\right)}{3}\\\displaystyle \tfrac{z \left(- 75 y z - z^{2} + 27 z + 3\right)}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 72 x y^{2} + 30 x y z - 4 x y - x + \tfrac{55 y^{3}}{3} - 35 y^{2} z + 18 y^{2} + 20 y z^{2} - 20 y z - z + 1\\\displaystyle \tfrac{y \left(- y^{2} - 75 y z + 27 y + 3\right)}{3}\\\displaystyle \tfrac{y \left(- 325 y^{2} + 219 y z + 69 y + 30 z^{2} - 42 z + 3\right)}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{86 x^{3}}{3} - 25 x^{2} y - 25 x^{2} z + 14 x^{2} + 20 y^{2} z + 20 y z^{2} - 20 y z - y - z + 1\\\displaystyle \tfrac{x \left(- 160 x^{2} + 129 x y - 165 x z + 69 x + 30 y^{2} + 90 y z - 42 y + 3\right)}{3}\\\displaystyle \tfrac{x \left(- 160 x^{2} - 165 x y + 129 x z + 69 x + 90 y z + 30 z^{2} - 42 z + 3\right)}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{406 x^{3}}{3} - 369 x^{2} y - 369 x^{2} z + 346 x^{2} - 322 x y^{2} - 604 x y z + 604 x y - 322 x z^{2} + 604 x z - 284 x - \tfrac{265 y^{3}}{3} - 225 y^{2} z + 248 y^{2} - 225 y z^{2} + 456 y z - 233 y - \tfrac{265 z^{3}}{3} + 248 z^{2} - 233 z + \tfrac{220}{3}\\\displaystyle \tfrac{160 x^{3}}{3} + 203 x^{2} y + 105 x^{2} z - 137 x^{2} + 236 x y^{2} + 266 x y z - 346 x y + 50 x z^{2} - 164 x z + 113 x + \tfrac{259 y^{3}}{3} + 151 y^{2} z - 199 y^{2} + 63 y z^{2} - 206 y z + 142 y - \tfrac{5 z^{3}}{3} - 27 z^{2} + 58 z - \tfrac{88}{3}\\\displaystyle \tfrac{160 x^{3}}{3} + 105 x^{2} y + 203 x^{2} z - 137 x^{2} + 50 x y^{2} + 266 x y z - 164 x y + 236 x z^{2} - 346 x z + 113 x - \tfrac{5 y^{3}}{3} + 63 y^{2} z - 27 y^{2} + 151 y z^{2} - 206 y z + 58 y + \tfrac{259 z^{3}}{3} - 199 z^{2} + 142 z - \tfrac{88}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{0} = \begin{cases} \left(\begin{array}{c}\displaystyle 30 x y z - 72 x z^{2} - 4 x z - x + 20 y^{2} z - 35 y z^{2} - 20 y z - y + \tfrac{55 z^{3}}{3} + 18 z^{2} + 1\\\displaystyle \tfrac{z \left(30 y^{2} + 219 y z - 42 y - 325 z^{2} + 69 z + 3\right)}{3}\\\displaystyle \tfrac{z \left(- 75 y z - z^{2} + 27 z + 3\right)}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 72 x y^{2} + 30 x y z - 4 x y - x + \tfrac{55 y^{3}}{3} - 35 y^{2} z + 18 y^{2} + 20 y z^{2} - 20 y z - z + 1\\\displaystyle \tfrac{y \left(- y^{2} - 75 y z + 27 y + 3\right)}{3}\\\displaystyle \tfrac{y \left(- 325 y^{2} + 219 y z + 69 y + 30 z^{2} - 42 z + 3\right)}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{86 x^{3}}{3} - 25 x^{2} y - 25 x^{2} z + 14 x^{2} + 20 y^{2} z + 20 y z^{2} - 20 y z - y - z + 1\\\displaystyle \tfrac{x \left(- 160 x^{2} + 129 x y - 165 x z + 69 x + 30 y^{2} + 90 y z - 42 y + 3\right)}{3}\\\displaystyle \tfrac{x \left(- 160 x^{2} - 165 x y + 129 x z + 69 x + 90 y z + 30 z^{2} - 42 z + 3\right)}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{406 x^{3}}{3} - 369 x^{2} y - 369 x^{2} z + 346 x^{2} - 322 x y^{2} - 604 x y z + 604 x y - 322 x z^{2} + 604 x z - 284 x - \tfrac{265 y^{3}}{3} - 225 y^{2} z + 248 y^{2} - 225 y z^{2} + 456 y z - 233 y - \tfrac{265 z^{3}}{3} + 248 z^{2} - 233 z + \tfrac{220}{3}\\\displaystyle \tfrac{160 x^{3}}{3} + 203 x^{2} y + 105 x^{2} z - 137 x^{2} + 236 x y^{2} + 266 x y z - 346 x y + 50 x z^{2} - 164 x z + 113 x + \tfrac{259 y^{3}}{3} + 151 y^{2} z - 199 y^{2} + 63 y z^{2} - 206 y z + 142 y - \tfrac{5 z^{3}}{3} - 27 z^{2} + 58 z - \tfrac{88}{3}\\\displaystyle \tfrac{160 x^{3}}{3} + 105 x^{2} y + 203 x^{2} z - 137 x^{2} + 50 x y^{2} + 266 x y z - 164 x y + 236 x z^{2} - 346 x z + 113 x - \tfrac{5 y^{3}}{3} + 63 y^{2} z - 27 y^{2} + 151 y z^{2} - 206 y z + 58 y + \tfrac{259 z^{3}}{3} - 199 z^{2} + 142 z - \tfrac{88}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{1} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(30 x^{2} + 219 x z - 42 x - 325 z^{2} + 69 z + 3\right)}{3}\\\displaystyle 20 x^{2} z + 30 x y z - 35 x z^{2} - 20 x z - x - 72 y z^{2} - 4 y z - y + \tfrac{55 z^{3}}{3} + 18 z^{2} + 1\\\displaystyle \tfrac{z \left(- 75 x z - z^{2} + 27 z + 3\right)}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(30 x^{2} + 129 x y + 90 x z - 42 x - 160 y^{2} - 165 y z + 69 y + 3\right)}{3}\\\displaystyle 20 x^{2} z - 25 x y^{2} + 20 x z^{2} - 20 x z - x - \tfrac{86 y^{3}}{3} - 25 y^{2} z + 14 y^{2} - z + 1\\\displaystyle \tfrac{y \left(- 165 x y + 90 x z - 160 y^{2} + 129 y z + 69 y + 30 z^{2} - 42 z + 3\right)}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(- x^{2} - 75 x z + 27 x + 3\right)}{3}\\\displaystyle \tfrac{55 x^{3}}{3} - 72 x^{2} y - 35 x^{2} z + 18 x^{2} + 30 x y z - 4 x y + 20 x z^{2} - 20 x z - y - z + 1\\\displaystyle \tfrac{x \left(- 325 x^{2} + 219 x z + 69 x + 30 z^{2} - 42 z + 3\right)}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{259 x^{3}}{3} + 236 x^{2} y + 151 x^{2} z - 199 x^{2} + 203 x y^{2} + 266 x y z - 346 x y + 63 x z^{2} - 206 x z + 142 x + \tfrac{160 y^{3}}{3} + 105 y^{2} z - 137 y^{2} + 50 y z^{2} - 164 y z + 113 y - \tfrac{5 z^{3}}{3} - 27 z^{2} + 58 z - \tfrac{88}{3}\\\displaystyle - \tfrac{265 x^{3}}{3} - 322 x^{2} y - 225 x^{2} z + 248 x^{2} - 369 x y^{2} - 604 x y z + 604 x y - 225 x z^{2} + 456 x z - 233 x - \tfrac{406 y^{3}}{3} - 369 y^{2} z + 346 y^{2} - 322 y z^{2} + 604 y z - 284 y - \tfrac{265 z^{3}}{3} + 248 z^{2} - 233 z + \tfrac{220}{3}\\\displaystyle - \tfrac{5 x^{3}}{3} + 50 x^{2} y + 63 x^{2} z - 27 x^{2} + 105 x y^{2} + 266 x y z - 164 x y + 151 x z^{2} - 206 x z + 58 x + \tfrac{160 y^{3}}{3} + 203 y^{2} z - 137 y^{2} + 236 y z^{2} - 346 y z + 113 y + \tfrac{259 z^{3}}{3} - 199 z^{2} + 142 z - \tfrac{88}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{1} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(30 x^{2} + 219 x z - 42 x - 325 z^{2} + 69 z + 3\right)}{3}\\\displaystyle 20 x^{2} z + 30 x y z - 35 x z^{2} - 20 x z - x - 72 y z^{2} - 4 y z - y + \tfrac{55 z^{3}}{3} + 18 z^{2} + 1\\\displaystyle \tfrac{z \left(- 75 x z - z^{2} + 27 z + 3\right)}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(30 x^{2} + 129 x y + 90 x z - 42 x - 160 y^{2} - 165 y z + 69 y + 3\right)}{3}\\\displaystyle 20 x^{2} z - 25 x y^{2} + 20 x z^{2} - 20 x z - x - \tfrac{86 y^{3}}{3} - 25 y^{2} z + 14 y^{2} - z + 1\\\displaystyle \tfrac{y \left(- 165 x y + 90 x z - 160 y^{2} + 129 y z + 69 y + 30 z^{2} - 42 z + 3\right)}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(- x^{2} - 75 x z + 27 x + 3\right)}{3}\\\displaystyle \tfrac{55 x^{3}}{3} - 72 x^{2} y - 35 x^{2} z + 18 x^{2} + 30 x y z - 4 x y + 20 x z^{2} - 20 x z - y - z + 1\\\displaystyle \tfrac{x \left(- 325 x^{2} + 219 x z + 69 x + 30 z^{2} - 42 z + 3\right)}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{259 x^{3}}{3} + 236 x^{2} y + 151 x^{2} z - 199 x^{2} + 203 x y^{2} + 266 x y z - 346 x y + 63 x z^{2} - 206 x z + 142 x + \tfrac{160 y^{3}}{3} + 105 y^{2} z - 137 y^{2} + 50 y z^{2} - 164 y z + 113 y - \tfrac{5 z^{3}}{3} - 27 z^{2} + 58 z - \tfrac{88}{3}\\\displaystyle - \tfrac{265 x^{3}}{3} - 322 x^{2} y - 225 x^{2} z + 248 x^{2} - 369 x y^{2} - 604 x y z + 604 x y - 225 x z^{2} + 456 x z - 233 x - \tfrac{406 y^{3}}{3} - 369 y^{2} z + 346 y^{2} - 322 y z^{2} + 604 y z - 284 y - \tfrac{265 z^{3}}{3} + 248 z^{2} - 233 z + \tfrac{220}{3}\\\displaystyle - \tfrac{5 x^{3}}{3} + 50 x^{2} y + 63 x^{2} z - 27 x^{2} + 105 x y^{2} + 266 x y z - 164 x y + 151 x z^{2} - 206 x z + 58 x + \tfrac{160 y^{3}}{3} + 203 y^{2} z - 137 y^{2} + 236 y z^{2} - 346 y z + 113 y + \tfrac{259 z^{3}}{3} - 199 z^{2} + 142 z - \tfrac{88}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{2} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(30 x^{2} + 90 x y + 129 x z - 42 x - 165 y z - 160 z^{2} + 69 z + 3\right)}{3}\\\displaystyle \tfrac{z \left(90 x y - 165 x z + 30 y^{2} + 129 y z - 42 y - 160 z^{2} + 69 z + 3\right)}{3}\\\displaystyle 20 x^{2} y + 20 x y^{2} - 20 x y - 25 x z^{2} - x - 25 y z^{2} - y - \tfrac{86 z^{3}}{3} + 14 z^{2} + 1\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(30 x^{2} + 219 x y - 42 x - 325 y^{2} + 69 y + 3\right)}{3}\\\displaystyle \tfrac{y \left(- 75 x y - y^{2} + 27 y + 3\right)}{3}\\\displaystyle 20 x^{2} y - 35 x y^{2} + 30 x y z - 20 x y - x + \tfrac{55 y^{3}}{3} - 72 y^{2} z + 18 y^{2} - 4 y z - z + 1\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(- x^{2} - 75 x y + 27 x + 3\right)}{3}\\\displaystyle \tfrac{x \left(- 325 x^{2} + 219 x y + 69 x + 30 y^{2} - 42 y + 3\right)}{3}\\\displaystyle \tfrac{55 x^{3}}{3} - 35 x^{2} y - 72 x^{2} z + 18 x^{2} + 20 x y^{2} + 30 x y z - 20 x y - 4 x z - y - z + 1\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{259 x^{3}}{3} + 151 x^{2} y + 236 x^{2} z - 199 x^{2} + 63 x y^{2} + 266 x y z - 206 x y + 203 x z^{2} - 346 x z + 142 x - \tfrac{5 y^{3}}{3} + 50 y^{2} z - 27 y^{2} + 105 y z^{2} - 164 y z + 58 y + \tfrac{160 z^{3}}{3} - 137 z^{2} + 113 z - \tfrac{88}{3}\\\displaystyle - \tfrac{5 x^{3}}{3} + 63 x^{2} y + 50 x^{2} z - 27 x^{2} + 151 x y^{2} + 266 x y z - 206 x y + 105 x z^{2} - 164 x z + 58 x + \tfrac{259 y^{3}}{3} + 236 y^{2} z - 199 y^{2} + 203 y z^{2} - 346 y z + 142 y + \tfrac{160 z^{3}}{3} - 137 z^{2} + 113 z - \tfrac{88}{3}\\\displaystyle - \tfrac{265 x^{3}}{3} - 225 x^{2} y - 322 x^{2} z + 248 x^{2} - 225 x y^{2} - 604 x y z + 456 x y - 369 x z^{2} + 604 x z - 233 x - \tfrac{265 y^{3}}{3} - 322 y^{2} z + 248 y^{2} - 369 y z^{2} + 604 y z - 233 y - \tfrac{406 z^{3}}{3} + 346 z^{2} - 284 z + \tfrac{220}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{2} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(30 x^{2} + 90 x y + 129 x z - 42 x - 165 y z - 160 z^{2} + 69 z + 3\right)}{3}\\\displaystyle \tfrac{z \left(90 x y - 165 x z + 30 y^{2} + 129 y z - 42 y - 160 z^{2} + 69 z + 3\right)}{3}\\\displaystyle 20 x^{2} y + 20 x y^{2} - 20 x y - 25 x z^{2} - x - 25 y z^{2} - y - \tfrac{86 z^{3}}{3} + 14 z^{2} + 1\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(30 x^{2} + 219 x y - 42 x - 325 y^{2} + 69 y + 3\right)}{3}\\\displaystyle \tfrac{y \left(- 75 x y - y^{2} + 27 y + 3\right)}{3}\\\displaystyle 20 x^{2} y - 35 x y^{2} + 30 x y z - 20 x y - x + \tfrac{55 y^{3}}{3} - 72 y^{2} z + 18 y^{2} - 4 y z - z + 1\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(- x^{2} - 75 x y + 27 x + 3\right)}{3}\\\displaystyle \tfrac{x \left(- 325 x^{2} + 219 x y + 69 x + 30 y^{2} - 42 y + 3\right)}{3}\\\displaystyle \tfrac{55 x^{3}}{3} - 35 x^{2} y - 72 x^{2} z + 18 x^{2} + 20 x y^{2} + 30 x y z - 20 x y - 4 x z - y - z + 1\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{259 x^{3}}{3} + 151 x^{2} y + 236 x^{2} z - 199 x^{2} + 63 x y^{2} + 266 x y z - 206 x y + 203 x z^{2} - 346 x z + 142 x - \tfrac{5 y^{3}}{3} + 50 y^{2} z - 27 y^{2} + 105 y z^{2} - 164 y z + 58 y + \tfrac{160 z^{3}}{3} - 137 z^{2} + 113 z - \tfrac{88}{3}\\\displaystyle - \tfrac{5 x^{3}}{3} + 63 x^{2} y + 50 x^{2} z - 27 x^{2} + 151 x y^{2} + 266 x y z - 206 x y + 105 x z^{2} - 164 x z + 58 x + \tfrac{259 y^{3}}{3} + 236 y^{2} z - 199 y^{2} + 203 y z^{2} - 346 y z + 142 y + \tfrac{160 z^{3}}{3} - 137 z^{2} + 113 z - \tfrac{88}{3}\\\displaystyle - \tfrac{265 x^{3}}{3} - 225 x^{2} y - 322 x^{2} z + 248 x^{2} - 225 x y^{2} - 604 x y z + 456 x y - 369 x z^{2} + 604 x z - 233 x - \tfrac{265 y^{3}}{3} - 322 y^{2} z + 248 y^{2} - 369 y z^{2} + 604 y z - 233 y - \tfrac{406 z^{3}}{3} + 346 z^{2} - 284 z + \tfrac{220}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{3} = \begin{cases} \left(\begin{array}{c}\displaystyle - \tfrac{20 x^{2} z}{3} - 20 x y z - \tfrac{44 x z^{2}}{3} + \tfrac{32 x z}{3} + x + \tfrac{110 y z^{2}}{3} + \tfrac{430 z^{3}}{9} - \tfrac{64 z^{2}}{3} - z\\\displaystyle \tfrac{z \left(- 180 x y + 330 x z - 60 y^{2} - 132 y z + 96 y + 430 z^{2} - 192 z - 9\right)}{9}\\\displaystyle \tfrac{z \left(150 x z + 150 y z + 88 z^{2} - 96 z - 9\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{20 x^{2} y}{3} - \tfrac{44 x y^{2}}{3} - 20 x y z + \tfrac{32 x y}{3} + x + \tfrac{430 y^{3}}{9} + \tfrac{110 y^{2} z}{3} - \tfrac{64 y^{2}}{3} - y\\\displaystyle \tfrac{y \left(150 x y + 88 y^{2} + 150 y z - 96 y - 9\right)}{9}\\\displaystyle \tfrac{y \left(330 x y - 180 x z + 430 y^{2} - 132 y z - 192 y - 60 z^{2} + 96 z - 9\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{2 x^{2} \left(44 x + 75 y + 75 z - 48\right)}{9}\\\displaystyle \tfrac{x \left(430 x^{2} - 132 x y + 330 x z - 192 x - 60 y^{2} - 180 y z + 96 y - 9\right)}{9}\\\displaystyle \tfrac{x \left(430 x^{2} + 330 x y - 132 x z - 192 x - 180 y z - 60 z^{2} + 96 z - 9\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{112 x^{3}}{9} - 6 x^{2} y - 6 x^{2} z + \tfrac{52 x^{2}}{3} + \tfrac{56 x y^{2}}{3} + 24 x y z - \tfrac{52 x y}{3} + \tfrac{56 x z^{2}}{3} - \tfrac{52 x z}{3} + \tfrac{2 x}{3} + \tfrac{110 y^{3}}{9} + 30 y^{2} z - 28 y^{2} + 30 y z^{2} - \tfrac{148 y z}{3} + \tfrac{61 y}{3} + \tfrac{110 z^{3}}{9} - 28 z^{2} + \tfrac{61 z}{3} - \tfrac{41}{9}\\\displaystyle \tfrac{110 x^{3}}{9} + \tfrac{56 x^{2} y}{3} + 30 x^{2} z - 28 x^{2} - 6 x y^{2} + 24 x y z - \tfrac{52 x y}{3} + 30 x z^{2} - \tfrac{148 x z}{3} + \tfrac{61 x}{3} - \tfrac{112 y^{3}}{9} - 6 y^{2} z + \tfrac{52 y^{2}}{3} + \tfrac{56 y z^{2}}{3} - \tfrac{52 y z}{3} - \tfrac{y}{3} + \tfrac{110 z^{3}}{9} - 28 z^{2} + \tfrac{61 z}{3} - \tfrac{41}{9}\\\displaystyle \tfrac{110 x^{3}}{9} + 30 x^{2} y + \tfrac{56 x^{2} z}{3} - 28 x^{2} + 30 x y^{2} + 24 x y z - \tfrac{148 x y}{3} - 6 x z^{2} - \tfrac{52 x z}{3} + \tfrac{61 x}{3} + \tfrac{110 y^{3}}{9} + \tfrac{56 y^{2} z}{3} - 28 y^{2} - 6 y z^{2} - \tfrac{52 y z}{3} + \tfrac{61 y}{3} - \tfrac{112 z^{3}}{9} + \tfrac{52 z^{2}}{3} - \tfrac{z}{3} - \tfrac{41}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{3} = \begin{cases} \left(\begin{array}{c}\displaystyle - \tfrac{20 x^{2} z}{3} - 20 x y z - \tfrac{44 x z^{2}}{3} + \tfrac{32 x z}{3} + x + \tfrac{110 y z^{2}}{3} + \tfrac{430 z^{3}}{9} - \tfrac{64 z^{2}}{3} - z\\\displaystyle \tfrac{z \left(- 180 x y + 330 x z - 60 y^{2} - 132 y z + 96 y + 430 z^{2} - 192 z - 9\right)}{9}\\\displaystyle \tfrac{z \left(150 x z + 150 y z + 88 z^{2} - 96 z - 9\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{20 x^{2} y}{3} - \tfrac{44 x y^{2}}{3} - 20 x y z + \tfrac{32 x y}{3} + x + \tfrac{430 y^{3}}{9} + \tfrac{110 y^{2} z}{3} - \tfrac{64 y^{2}}{3} - y\\\displaystyle \tfrac{y \left(150 x y + 88 y^{2} + 150 y z - 96 y - 9\right)}{9}\\\displaystyle \tfrac{y \left(330 x y - 180 x z + 430 y^{2} - 132 y z - 192 y - 60 z^{2} + 96 z - 9\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{2 x^{2} \left(44 x + 75 y + 75 z - 48\right)}{9}\\\displaystyle \tfrac{x \left(430 x^{2} - 132 x y + 330 x z - 192 x - 60 y^{2} - 180 y z + 96 y - 9\right)}{9}\\\displaystyle \tfrac{x \left(430 x^{2} + 330 x y - 132 x z - 192 x - 180 y z - 60 z^{2} + 96 z - 9\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{112 x^{3}}{9} - 6 x^{2} y - 6 x^{2} z + \tfrac{52 x^{2}}{3} + \tfrac{56 x y^{2}}{3} + 24 x y z - \tfrac{52 x y}{3} + \tfrac{56 x z^{2}}{3} - \tfrac{52 x z}{3} + \tfrac{2 x}{3} + \tfrac{110 y^{3}}{9} + 30 y^{2} z - 28 y^{2} + 30 y z^{2} - \tfrac{148 y z}{3} + \tfrac{61 y}{3} + \tfrac{110 z^{3}}{9} - 28 z^{2} + \tfrac{61 z}{3} - \tfrac{41}{9}\\\displaystyle \tfrac{110 x^{3}}{9} + \tfrac{56 x^{2} y}{3} + 30 x^{2} z - 28 x^{2} - 6 x y^{2} + 24 x y z - \tfrac{52 x y}{3} + 30 x z^{2} - \tfrac{148 x z}{3} + \tfrac{61 x}{3} - \tfrac{112 y^{3}}{9} - 6 y^{2} z + \tfrac{52 y^{2}}{3} + \tfrac{56 y z^{2}}{3} - \tfrac{52 y z}{3} - \tfrac{y}{3} + \tfrac{110 z^{3}}{9} - 28 z^{2} + \tfrac{61 z}{3} - \tfrac{41}{9}\\\displaystyle \tfrac{110 x^{3}}{9} + 30 x^{2} y + \tfrac{56 x^{2} z}{3} - 28 x^{2} + 30 x y^{2} + 24 x y z - \tfrac{148 x y}{3} - 6 x z^{2} - \tfrac{52 x z}{3} + \tfrac{61 x}{3} + \tfrac{110 y^{3}}{9} + \tfrac{56 y^{2} z}{3} - 28 y^{2} - 6 y z^{2} - \tfrac{52 y z}{3} + \tfrac{61 y}{3} - \tfrac{112 z^{3}}{9} + \tfrac{52 z^{2}}{3} - \tfrac{z}{3} - \tfrac{41}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{4} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{5 z \left(6 x^{2} - 36 x y + 105 x z - 6 x + 66 y z - 109 z^{2} + 3 z\right)}{9}\\\displaystyle 20 x^{2} z + 10 x y z + \tfrac{5 x z^{2}}{3} - 20 x z + x - \tfrac{20 y^{2} z}{3} - \tfrac{260 y z^{2}}{3} + \tfrac{20 y z}{3} + \tfrac{595 z^{3}}{9} - \tfrac{10 z^{2}}{3}\\\displaystyle \tfrac{5 z^{2} \left(- 15 x + 30 y + 17 z - 3\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{5 y \left(6 x^{2} + 51 x y + 18 x z - 6 x - 10 y^{2} - 33 y z + 3 y\right)}{9}\\\displaystyle 20 x^{2} z - \tfrac{25 x y^{2}}{3} + 20 x z^{2} - 20 x z + x - \tfrac{170 y^{3}}{9} - \tfrac{25 y^{2} z}{3} + \tfrac{10 y^{2}}{3}\\\displaystyle \tfrac{5 y \left(- 33 x y + 18 x z - 10 y^{2} + 51 y z + 3 y + 6 z^{2} - 6 z\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{5 x^{2} \left(17 x + 30 y - 15 z - 3\right)}{9}\\\displaystyle \tfrac{x \left(595 x^{2} - 780 x y + 15 x z - 30 x - 60 y^{2} + 90 y z + 60 y + 180 z^{2} - 180 z + 9\right)}{9}\\\displaystyle \tfrac{5 x \left(- 109 x^{2} + 66 x y + 105 x z + 3 x - 36 y z + 6 z^{2} - 6 z\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{665 x^{3}}{9} + 230 x^{2} y + 145 x^{2} z - \tfrac{545 x^{2}}{3} + \tfrac{665 x y^{2}}{3} + 290 x y z - \tfrac{1090 x y}{3} + \tfrac{245 x z^{2}}{3} - \tfrac{670 x z}{3} + \tfrac{425 x}{3} + \tfrac{590 y^{3}}{9} + 135 y^{2} z - 165 y^{2} + 80 y z^{2} - \tfrac{640 y z}{3} + \tfrac{400 y}{3} + \tfrac{95 z^{3}}{9} - 55 z^{2} + \tfrac{235 z}{3} - \tfrac{305}{9}\\\displaystyle - \tfrac{685 x^{3}}{9} - \tfrac{910 x^{2} y}{3} - 195 x^{2} z + 220 x^{2} - 375 x y^{2} - 580 x y z + \tfrac{1760 x y}{3} - 195 x z^{2} + \tfrac{1220 x z}{3} - \tfrac{632 x}{3} - \tfrac{1330 y^{3}}{9} - 375 y^{2} z + \tfrac{1090 y^{2}}{3} - \tfrac{910 y z^{2}}{3} + \tfrac{1760 y z}{3} - \tfrac{850 y}{3} - \tfrac{685 z^{3}}{9} + 220 z^{2} - \tfrac{635 z}{3} + \tfrac{610}{9}\\\displaystyle \tfrac{95 x^{3}}{9} + 80 x^{2} y + \tfrac{245 x^{2} z}{3} - 55 x^{2} + 135 x y^{2} + 290 x y z - \tfrac{640 x y}{3} + 145 x z^{2} - \tfrac{670 x z}{3} + \tfrac{235 x}{3} + \tfrac{590 y^{3}}{9} + \tfrac{665 y^{2} z}{3} - 165 y^{2} + 230 y z^{2} - \tfrac{1090 y z}{3} + \tfrac{400 y}{3} + \tfrac{665 z^{3}}{9} - \tfrac{545 z^{2}}{3} + \tfrac{425 z}{3} - \tfrac{305}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{4} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{5 z \left(6 x^{2} - 36 x y + 105 x z - 6 x + 66 y z - 109 z^{2} + 3 z\right)}{9}\\\displaystyle 20 x^{2} z + 10 x y z + \tfrac{5 x z^{2}}{3} - 20 x z + x - \tfrac{20 y^{2} z}{3} - \tfrac{260 y z^{2}}{3} + \tfrac{20 y z}{3} + \tfrac{595 z^{3}}{9} - \tfrac{10 z^{2}}{3}\\\displaystyle \tfrac{5 z^{2} \left(- 15 x + 30 y + 17 z - 3\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{5 y \left(6 x^{2} + 51 x y + 18 x z - 6 x - 10 y^{2} - 33 y z + 3 y\right)}{9}\\\displaystyle 20 x^{2} z - \tfrac{25 x y^{2}}{3} + 20 x z^{2} - 20 x z + x - \tfrac{170 y^{3}}{9} - \tfrac{25 y^{2} z}{3} + \tfrac{10 y^{2}}{3}\\\displaystyle \tfrac{5 y \left(- 33 x y + 18 x z - 10 y^{2} + 51 y z + 3 y + 6 z^{2} - 6 z\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{5 x^{2} \left(17 x + 30 y - 15 z - 3\right)}{9}\\\displaystyle \tfrac{x \left(595 x^{2} - 780 x y + 15 x z - 30 x - 60 y^{2} + 90 y z + 60 y + 180 z^{2} - 180 z + 9\right)}{9}\\\displaystyle \tfrac{5 x \left(- 109 x^{2} + 66 x y + 105 x z + 3 x - 36 y z + 6 z^{2} - 6 z\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{665 x^{3}}{9} + 230 x^{2} y + 145 x^{2} z - \tfrac{545 x^{2}}{3} + \tfrac{665 x y^{2}}{3} + 290 x y z - \tfrac{1090 x y}{3} + \tfrac{245 x z^{2}}{3} - \tfrac{670 x z}{3} + \tfrac{425 x}{3} + \tfrac{590 y^{3}}{9} + 135 y^{2} z - 165 y^{2} + 80 y z^{2} - \tfrac{640 y z}{3} + \tfrac{400 y}{3} + \tfrac{95 z^{3}}{9} - 55 z^{2} + \tfrac{235 z}{3} - \tfrac{305}{9}\\\displaystyle - \tfrac{685 x^{3}}{9} - \tfrac{910 x^{2} y}{3} - 195 x^{2} z + 220 x^{2} - 375 x y^{2} - 580 x y z + \tfrac{1760 x y}{3} - 195 x z^{2} + \tfrac{1220 x z}{3} - \tfrac{632 x}{3} - \tfrac{1330 y^{3}}{9} - 375 y^{2} z + \tfrac{1090 y^{2}}{3} - \tfrac{910 y z^{2}}{3} + \tfrac{1760 y z}{3} - \tfrac{850 y}{3} - \tfrac{685 z^{3}}{9} + 220 z^{2} - \tfrac{635 z}{3} + \tfrac{610}{9}\\\displaystyle \tfrac{95 x^{3}}{9} + 80 x^{2} y + \tfrac{245 x^{2} z}{3} - 55 x^{2} + 135 x y^{2} + 290 x y z - \tfrac{640 x y}{3} + 145 x z^{2} - \tfrac{670 x z}{3} + \tfrac{235 x}{3} + \tfrac{590 y^{3}}{9} + \tfrac{665 y^{2} z}{3} - 165 y^{2} + 230 y z^{2} - \tfrac{1090 y z}{3} + \tfrac{400 y}{3} + \tfrac{665 z^{3}}{9} - \tfrac{545 z^{2}}{3} + \tfrac{425 z}{3} - \tfrac{305}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{5} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{5 z \left(6 x^{2} + 18 x y + 51 x z - 6 x - 33 y z - 10 z^{2} + 3 z\right)}{9}\\\displaystyle \tfrac{5 z \left(18 x y - 33 x z + 6 y^{2} + 51 y z - 6 y - 10 z^{2} + 3 z\right)}{9}\\\displaystyle 20 x^{2} y + 20 x y^{2} - 20 x y - \tfrac{25 x z^{2}}{3} + x - \tfrac{25 y z^{2}}{3} - \tfrac{170 z^{3}}{9} + \tfrac{10 z^{2}}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{5 y \left(6 x^{2} + 105 x y - 36 x z - 6 x - 109 y^{2} + 66 y z + 3 y\right)}{9}\\\displaystyle \tfrac{5 y^{2} \left(- 15 x + 17 y + 30 z - 3\right)}{9}\\\displaystyle 20 x^{2} y + \tfrac{5 x y^{2}}{3} + 10 x y z - 20 x y + x + \tfrac{595 y^{3}}{9} - \tfrac{260 y^{2} z}{3} - \tfrac{10 y^{2}}{3} - \tfrac{20 y z^{2}}{3} + \tfrac{20 y z}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{5 x^{2} \left(17 x - 15 y + 30 z - 3\right)}{9}\\\displaystyle \tfrac{5 x \left(- 109 x^{2} + 105 x y + 66 x z + 3 x + 6 y^{2} - 36 y z - 6 y\right)}{9}\\\displaystyle \tfrac{x \left(595 x^{2} + 15 x y - 780 x z - 30 x + 180 y^{2} + 90 y z - 180 y - 60 z^{2} + 60 z + 9\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{665 x^{3}}{9} + 145 x^{2} y + 230 x^{2} z - \tfrac{545 x^{2}}{3} + \tfrac{245 x y^{2}}{3} + 290 x y z - \tfrac{670 x y}{3} + \tfrac{665 x z^{2}}{3} - \tfrac{1090 x z}{3} + \tfrac{425 x}{3} + \tfrac{95 y^{3}}{9} + 80 y^{2} z - 55 y^{2} + 135 y z^{2} - \tfrac{640 y z}{3} + \tfrac{235 y}{3} + \tfrac{590 z^{3}}{9} - 165 z^{2} + \tfrac{400 z}{3} - \tfrac{305}{9}\\\displaystyle \tfrac{95 x^{3}}{9} + \tfrac{245 x^{2} y}{3} + 80 x^{2} z - 55 x^{2} + 145 x y^{2} + 290 x y z - \tfrac{670 x y}{3} + 135 x z^{2} - \tfrac{640 x z}{3} + \tfrac{235 x}{3} + \tfrac{665 y^{3}}{9} + 230 y^{2} z - \tfrac{545 y^{2}}{3} + \tfrac{665 y z^{2}}{3} - \tfrac{1090 y z}{3} + \tfrac{425 y}{3} + \tfrac{590 z^{3}}{9} - 165 z^{2} + \tfrac{400 z}{3} - \tfrac{305}{9}\\\displaystyle - \tfrac{685 x^{3}}{9} - 195 x^{2} y - \tfrac{910 x^{2} z}{3} + 220 x^{2} - 195 x y^{2} - 580 x y z + \tfrac{1220 x y}{3} - 375 x z^{2} + \tfrac{1760 x z}{3} - \tfrac{632 x}{3} - \tfrac{685 y^{3}}{9} - \tfrac{910 y^{2} z}{3} + 220 y^{2} - 375 y z^{2} + \tfrac{1760 y z}{3} - \tfrac{635 y}{3} - \tfrac{1330 z^{3}}{9} + \tfrac{1090 z^{2}}{3} - \tfrac{850 z}{3} + \tfrac{610}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{5} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{5 z \left(6 x^{2} + 18 x y + 51 x z - 6 x - 33 y z - 10 z^{2} + 3 z\right)}{9}\\\displaystyle \tfrac{5 z \left(18 x y - 33 x z + 6 y^{2} + 51 y z - 6 y - 10 z^{2} + 3 z\right)}{9}\\\displaystyle 20 x^{2} y + 20 x y^{2} - 20 x y - \tfrac{25 x z^{2}}{3} + x - \tfrac{25 y z^{2}}{3} - \tfrac{170 z^{3}}{9} + \tfrac{10 z^{2}}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{5 y \left(6 x^{2} + 105 x y - 36 x z - 6 x - 109 y^{2} + 66 y z + 3 y\right)}{9}\\\displaystyle \tfrac{5 y^{2} \left(- 15 x + 17 y + 30 z - 3\right)}{9}\\\displaystyle 20 x^{2} y + \tfrac{5 x y^{2}}{3} + 10 x y z - 20 x y + x + \tfrac{595 y^{3}}{9} - \tfrac{260 y^{2} z}{3} - \tfrac{10 y^{2}}{3} - \tfrac{20 y z^{2}}{3} + \tfrac{20 y z}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{5 x^{2} \left(17 x - 15 y + 30 z - 3\right)}{9}\\\displaystyle \tfrac{5 x \left(- 109 x^{2} + 105 x y + 66 x z + 3 x + 6 y^{2} - 36 y z - 6 y\right)}{9}\\\displaystyle \tfrac{x \left(595 x^{2} + 15 x y - 780 x z - 30 x + 180 y^{2} + 90 y z - 180 y - 60 z^{2} + 60 z + 9\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{665 x^{3}}{9} + 145 x^{2} y + 230 x^{2} z - \tfrac{545 x^{2}}{3} + \tfrac{245 x y^{2}}{3} + 290 x y z - \tfrac{670 x y}{3} + \tfrac{665 x z^{2}}{3} - \tfrac{1090 x z}{3} + \tfrac{425 x}{3} + \tfrac{95 y^{3}}{9} + 80 y^{2} z - 55 y^{2} + 135 y z^{2} - \tfrac{640 y z}{3} + \tfrac{235 y}{3} + \tfrac{590 z^{3}}{9} - 165 z^{2} + \tfrac{400 z}{3} - \tfrac{305}{9}\\\displaystyle \tfrac{95 x^{3}}{9} + \tfrac{245 x^{2} y}{3} + 80 x^{2} z - 55 x^{2} + 145 x y^{2} + 290 x y z - \tfrac{670 x y}{3} + 135 x z^{2} - \tfrac{640 x z}{3} + \tfrac{235 x}{3} + \tfrac{665 y^{3}}{9} + 230 y^{2} z - \tfrac{545 y^{2}}{3} + \tfrac{665 y z^{2}}{3} - \tfrac{1090 y z}{3} + \tfrac{425 y}{3} + \tfrac{590 z^{3}}{9} - 165 z^{2} + \tfrac{400 z}{3} - \tfrac{305}{9}\\\displaystyle - \tfrac{685 x^{3}}{9} - 195 x^{2} y - \tfrac{910 x^{2} z}{3} + 220 x^{2} - 195 x y^{2} - 580 x y z + \tfrac{1220 x y}{3} - 375 x z^{2} + \tfrac{1760 x z}{3} - \tfrac{632 x}{3} - \tfrac{685 y^{3}}{9} - \tfrac{910 y^{2} z}{3} + 220 y^{2} - 375 y z^{2} + \tfrac{1760 y z}{3} - \tfrac{635 y}{3} - \tfrac{1330 z^{3}}{9} + \tfrac{1090 z^{2}}{3} - \tfrac{850 z}{3} + \tfrac{610}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{6} = \begin{cases} \left(\begin{array}{c}\displaystyle - \tfrac{20 x^{2} z}{3} + 10 x y z - \tfrac{260 x z^{2}}{3} + \tfrac{20 x z}{3} + 20 y^{2} z + \tfrac{5 y z^{2}}{3} - 20 y z + y + \tfrac{595 z^{3}}{9} - \tfrac{10 z^{2}}{3}\\\displaystyle \tfrac{5 z \left(- 36 x y + 66 x z + 6 y^{2} + 105 y z - 6 y - 109 z^{2} + 3 z\right)}{9}\\\displaystyle \tfrac{5 z^{2} \left(30 x - 15 y + 17 z - 3\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(- 60 x^{2} - 780 x y + 90 x z + 60 x + 595 y^{2} + 15 y z - 30 y + 180 z^{2} - 180 z + 9\right)}{9}\\\displaystyle \tfrac{5 y^{2} \left(30 x + 17 y - 15 z - 3\right)}{9}\\\displaystyle \tfrac{5 y \left(66 x y - 36 x z - 109 y^{2} + 105 y z + 3 y + 6 z^{2} - 6 z\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{170 x^{3}}{9} - \tfrac{25 x^{2} y}{3} - \tfrac{25 x^{2} z}{3} + \tfrac{10 x^{2}}{3} + 20 y^{2} z + 20 y z^{2} - 20 y z + y\\\displaystyle \tfrac{5 x \left(- 10 x^{2} + 51 x y - 33 x z + 3 x + 6 y^{2} + 18 y z - 6 y\right)}{9}\\\displaystyle \tfrac{5 x \left(- 10 x^{2} - 33 x y + 51 x z + 3 x + 18 y z + 6 z^{2} - 6 z\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{1330 x^{3}}{9} - 375 x^{2} y - 375 x^{2} z + \tfrac{1090 x^{2}}{3} - \tfrac{910 x y^{2}}{3} - 580 x y z + \tfrac{1760 x y}{3} - \tfrac{910 x z^{2}}{3} + \tfrac{1760 x z}{3} - \tfrac{850 x}{3} - \tfrac{685 y^{3}}{9} - 195 y^{2} z + 220 y^{2} - 195 y z^{2} + \tfrac{1220 y z}{3} - \tfrac{632 y}{3} - \tfrac{685 z^{3}}{9} + 220 z^{2} - \tfrac{635 z}{3} + \tfrac{610}{9}\\\displaystyle \tfrac{590 x^{3}}{9} + \tfrac{665 x^{2} y}{3} + 135 x^{2} z - 165 x^{2} + 230 x y^{2} + 290 x y z - \tfrac{1090 x y}{3} + 80 x z^{2} - \tfrac{640 x z}{3} + \tfrac{400 x}{3} + \tfrac{665 y^{3}}{9} + 145 y^{2} z - \tfrac{545 y^{2}}{3} + \tfrac{245 y z^{2}}{3} - \tfrac{670 y z}{3} + \tfrac{425 y}{3} + \tfrac{95 z^{3}}{9} - 55 z^{2} + \tfrac{235 z}{3} - \tfrac{305}{9}\\\displaystyle \tfrac{590 x^{3}}{9} + 135 x^{2} y + \tfrac{665 x^{2} z}{3} - 165 x^{2} + 80 x y^{2} + 290 x y z - \tfrac{640 x y}{3} + 230 x z^{2} - \tfrac{1090 x z}{3} + \tfrac{400 x}{3} + \tfrac{95 y^{3}}{9} + \tfrac{245 y^{2} z}{3} - 55 y^{2} + 145 y z^{2} - \tfrac{670 y z}{3} + \tfrac{235 y}{3} + \tfrac{665 z^{3}}{9} - \tfrac{545 z^{2}}{3} + \tfrac{425 z}{3} - \tfrac{305}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{6} = \begin{cases} \left(\begin{array}{c}\displaystyle - \tfrac{20 x^{2} z}{3} + 10 x y z - \tfrac{260 x z^{2}}{3} + \tfrac{20 x z}{3} + 20 y^{2} z + \tfrac{5 y z^{2}}{3} - 20 y z + y + \tfrac{595 z^{3}}{9} - \tfrac{10 z^{2}}{3}\\\displaystyle \tfrac{5 z \left(- 36 x y + 66 x z + 6 y^{2} + 105 y z - 6 y - 109 z^{2} + 3 z\right)}{9}\\\displaystyle \tfrac{5 z^{2} \left(30 x - 15 y + 17 z - 3\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(- 60 x^{2} - 780 x y + 90 x z + 60 x + 595 y^{2} + 15 y z - 30 y + 180 z^{2} - 180 z + 9\right)}{9}\\\displaystyle \tfrac{5 y^{2} \left(30 x + 17 y - 15 z - 3\right)}{9}\\\displaystyle \tfrac{5 y \left(66 x y - 36 x z - 109 y^{2} + 105 y z + 3 y + 6 z^{2} - 6 z\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{170 x^{3}}{9} - \tfrac{25 x^{2} y}{3} - \tfrac{25 x^{2} z}{3} + \tfrac{10 x^{2}}{3} + 20 y^{2} z + 20 y z^{2} - 20 y z + y\\\displaystyle \tfrac{5 x \left(- 10 x^{2} + 51 x y - 33 x z + 3 x + 6 y^{2} + 18 y z - 6 y\right)}{9}\\\displaystyle \tfrac{5 x \left(- 10 x^{2} - 33 x y + 51 x z + 3 x + 18 y z + 6 z^{2} - 6 z\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{1330 x^{3}}{9} - 375 x^{2} y - 375 x^{2} z + \tfrac{1090 x^{2}}{3} - \tfrac{910 x y^{2}}{3} - 580 x y z + \tfrac{1760 x y}{3} - \tfrac{910 x z^{2}}{3} + \tfrac{1760 x z}{3} - \tfrac{850 x}{3} - \tfrac{685 y^{3}}{9} - 195 y^{2} z + 220 y^{2} - 195 y z^{2} + \tfrac{1220 y z}{3} - \tfrac{632 y}{3} - \tfrac{685 z^{3}}{9} + 220 z^{2} - \tfrac{635 z}{3} + \tfrac{610}{9}\\\displaystyle \tfrac{590 x^{3}}{9} + \tfrac{665 x^{2} y}{3} + 135 x^{2} z - 165 x^{2} + 230 x y^{2} + 290 x y z - \tfrac{1090 x y}{3} + 80 x z^{2} - \tfrac{640 x z}{3} + \tfrac{400 x}{3} + \tfrac{665 y^{3}}{9} + 145 y^{2} z - \tfrac{545 y^{2}}{3} + \tfrac{245 y z^{2}}{3} - \tfrac{670 y z}{3} + \tfrac{425 y}{3} + \tfrac{95 z^{3}}{9} - 55 z^{2} + \tfrac{235 z}{3} - \tfrac{305}{9}\\\displaystyle \tfrac{590 x^{3}}{9} + 135 x^{2} y + \tfrac{665 x^{2} z}{3} - 165 x^{2} + 80 x y^{2} + 290 x y z - \tfrac{640 x y}{3} + 230 x z^{2} - \tfrac{1090 x z}{3} + \tfrac{400 x}{3} + \tfrac{95 y^{3}}{9} + \tfrac{245 y^{2} z}{3} - 55 y^{2} + 145 y z^{2} - \tfrac{670 y z}{3} + \tfrac{235 y}{3} + \tfrac{665 z^{3}}{9} - \tfrac{545 z^{2}}{3} + \tfrac{425 z}{3} - \tfrac{305}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{7} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(- 60 x^{2} - 180 x y - 132 x z + 96 x + 330 y z + 430 z^{2} - 192 z - 9\right)}{9}\\\displaystyle - 20 x y z + \tfrac{110 x z^{2}}{3} - \tfrac{20 y^{2} z}{3} - \tfrac{44 y z^{2}}{3} + \tfrac{32 y z}{3} + y + \tfrac{430 z^{3}}{9} - \tfrac{64 z^{2}}{3} - z\\\displaystyle \tfrac{z \left(150 x z + 150 y z + 88 z^{2} - 96 z - 9\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(- 60 x^{2} - 132 x y - 180 x z + 96 x + 430 y^{2} + 330 y z - 192 y - 9\right)}{9}\\\displaystyle \tfrac{2 y^{2} \left(75 x + 44 y + 75 z - 48\right)}{9}\\\displaystyle \tfrac{y \left(330 x y - 180 x z + 430 y^{2} - 132 y z - 192 y - 60 z^{2} + 96 z - 9\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(88 x^{2} + 150 x y + 150 x z - 96 x - 9\right)}{9}\\\displaystyle \tfrac{430 x^{3}}{9} - \tfrac{44 x^{2} y}{3} + \tfrac{110 x^{2} z}{3} - \tfrac{64 x^{2}}{3} - \tfrac{20 x y^{2}}{3} - 20 x y z + \tfrac{32 x y}{3} - x + y\\\displaystyle \tfrac{x \left(430 x^{2} + 330 x y - 132 x z - 192 x - 180 y z - 60 z^{2} + 96 z - 9\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{112 x^{3}}{9} - 6 x^{2} y - 6 x^{2} z + \tfrac{52 x^{2}}{3} + \tfrac{56 x y^{2}}{3} + 24 x y z - \tfrac{52 x y}{3} + \tfrac{56 x z^{2}}{3} - \tfrac{52 x z}{3} - \tfrac{x}{3} + \tfrac{110 y^{3}}{9} + 30 y^{2} z - 28 y^{2} + 30 y z^{2} - \tfrac{148 y z}{3} + \tfrac{61 y}{3} + \tfrac{110 z^{3}}{9} - 28 z^{2} + \tfrac{61 z}{3} - \tfrac{41}{9}\\\displaystyle \tfrac{110 x^{3}}{9} + \tfrac{56 x^{2} y}{3} + 30 x^{2} z - 28 x^{2} - 6 x y^{2} + 24 x y z - \tfrac{52 x y}{3} + 30 x z^{2} - \tfrac{148 x z}{3} + \tfrac{61 x}{3} - \tfrac{112 y^{3}}{9} - 6 y^{2} z + \tfrac{52 y^{2}}{3} + \tfrac{56 y z^{2}}{3} - \tfrac{52 y z}{3} + \tfrac{2 y}{3} + \tfrac{110 z^{3}}{9} - 28 z^{2} + \tfrac{61 z}{3} - \tfrac{41}{9}\\\displaystyle \tfrac{110 x^{3}}{9} + 30 x^{2} y + \tfrac{56 x^{2} z}{3} - 28 x^{2} + 30 x y^{2} + 24 x y z - \tfrac{148 x y}{3} - 6 x z^{2} - \tfrac{52 x z}{3} + \tfrac{61 x}{3} + \tfrac{110 y^{3}}{9} + \tfrac{56 y^{2} z}{3} - 28 y^{2} - 6 y z^{2} - \tfrac{52 y z}{3} + \tfrac{61 y}{3} - \tfrac{112 z^{3}}{9} + \tfrac{52 z^{2}}{3} - \tfrac{z}{3} - \tfrac{41}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{7} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(- 60 x^{2} - 180 x y - 132 x z + 96 x + 330 y z + 430 z^{2} - 192 z - 9\right)}{9}\\\displaystyle - 20 x y z + \tfrac{110 x z^{2}}{3} - \tfrac{20 y^{2} z}{3} - \tfrac{44 y z^{2}}{3} + \tfrac{32 y z}{3} + y + \tfrac{430 z^{3}}{9} - \tfrac{64 z^{2}}{3} - z\\\displaystyle \tfrac{z \left(150 x z + 150 y z + 88 z^{2} - 96 z - 9\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(- 60 x^{2} - 132 x y - 180 x z + 96 x + 430 y^{2} + 330 y z - 192 y - 9\right)}{9}\\\displaystyle \tfrac{2 y^{2} \left(75 x + 44 y + 75 z - 48\right)}{9}\\\displaystyle \tfrac{y \left(330 x y - 180 x z + 430 y^{2} - 132 y z - 192 y - 60 z^{2} + 96 z - 9\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(88 x^{2} + 150 x y + 150 x z - 96 x - 9\right)}{9}\\\displaystyle \tfrac{430 x^{3}}{9} - \tfrac{44 x^{2} y}{3} + \tfrac{110 x^{2} z}{3} - \tfrac{64 x^{2}}{3} - \tfrac{20 x y^{2}}{3} - 20 x y z + \tfrac{32 x y}{3} - x + y\\\displaystyle \tfrac{x \left(430 x^{2} + 330 x y - 132 x z - 192 x - 180 y z - 60 z^{2} + 96 z - 9\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{112 x^{3}}{9} - 6 x^{2} y - 6 x^{2} z + \tfrac{52 x^{2}}{3} + \tfrac{56 x y^{2}}{3} + 24 x y z - \tfrac{52 x y}{3} + \tfrac{56 x z^{2}}{3} - \tfrac{52 x z}{3} - \tfrac{x}{3} + \tfrac{110 y^{3}}{9} + 30 y^{2} z - 28 y^{2} + 30 y z^{2} - \tfrac{148 y z}{3} + \tfrac{61 y}{3} + \tfrac{110 z^{3}}{9} - 28 z^{2} + \tfrac{61 z}{3} - \tfrac{41}{9}\\\displaystyle \tfrac{110 x^{3}}{9} + \tfrac{56 x^{2} y}{3} + 30 x^{2} z - 28 x^{2} - 6 x y^{2} + 24 x y z - \tfrac{52 x y}{3} + 30 x z^{2} - \tfrac{148 x z}{3} + \tfrac{61 x}{3} - \tfrac{112 y^{3}}{9} - 6 y^{2} z + \tfrac{52 y^{2}}{3} + \tfrac{56 y z^{2}}{3} - \tfrac{52 y z}{3} + \tfrac{2 y}{3} + \tfrac{110 z^{3}}{9} - 28 z^{2} + \tfrac{61 z}{3} - \tfrac{41}{9}\\\displaystyle \tfrac{110 x^{3}}{9} + 30 x^{2} y + \tfrac{56 x^{2} z}{3} - 28 x^{2} + 30 x y^{2} + 24 x y z - \tfrac{148 x y}{3} - 6 x z^{2} - \tfrac{52 x z}{3} + \tfrac{61 x}{3} + \tfrac{110 y^{3}}{9} + \tfrac{56 y^{2} z}{3} - 28 y^{2} - 6 y z^{2} - \tfrac{52 y z}{3} + \tfrac{61 y}{3} - \tfrac{112 z^{3}}{9} + \tfrac{52 z^{2}}{3} - \tfrac{z}{3} - \tfrac{41}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{8} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{5 z \left(6 x^{2} + 18 x y + 51 x z - 6 x - 33 y z - 10 z^{2} + 3 z\right)}{9}\\\displaystyle \tfrac{5 z \left(18 x y - 33 x z + 6 y^{2} + 51 y z - 6 y - 10 z^{2} + 3 z\right)}{9}\\\displaystyle 20 x^{2} y + 20 x y^{2} - 20 x y - \tfrac{25 x z^{2}}{3} - \tfrac{25 y z^{2}}{3} + y - \tfrac{170 z^{3}}{9} + \tfrac{10 z^{2}}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{5 y \left(6 x^{2} + 105 x y - 36 x z - 6 x - 109 y^{2} + 66 y z + 3 y\right)}{9}\\\displaystyle \tfrac{5 y^{2} \left(- 15 x + 17 y + 30 z - 3\right)}{9}\\\displaystyle \tfrac{y \left(180 x^{2} + 15 x y + 90 x z - 180 x + 595 y^{2} - 780 y z - 30 y - 60 z^{2} + 60 z + 9\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{5 x^{2} \left(17 x - 15 y + 30 z - 3\right)}{9}\\\displaystyle \tfrac{5 x \left(- 109 x^{2} + 105 x y + 66 x z + 3 x + 6 y^{2} - 36 y z - 6 y\right)}{9}\\\displaystyle \tfrac{595 x^{3}}{9} + \tfrac{5 x^{2} y}{3} - \tfrac{260 x^{2} z}{3} - \tfrac{10 x^{2}}{3} + 20 x y^{2} + 10 x y z - 20 x y - \tfrac{20 x z^{2}}{3} + \tfrac{20 x z}{3} + y\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{665 x^{3}}{9} + 145 x^{2} y + 230 x^{2} z - \tfrac{545 x^{2}}{3} + \tfrac{245 x y^{2}}{3} + 290 x y z - \tfrac{670 x y}{3} + \tfrac{665 x z^{2}}{3} - \tfrac{1090 x z}{3} + \tfrac{425 x}{3} + \tfrac{95 y^{3}}{9} + 80 y^{2} z - 55 y^{2} + 135 y z^{2} - \tfrac{640 y z}{3} + \tfrac{235 y}{3} + \tfrac{590 z^{3}}{9} - 165 z^{2} + \tfrac{400 z}{3} - \tfrac{305}{9}\\\displaystyle \tfrac{95 x^{3}}{9} + \tfrac{245 x^{2} y}{3} + 80 x^{2} z - 55 x^{2} + 145 x y^{2} + 290 x y z - \tfrac{670 x y}{3} + 135 x z^{2} - \tfrac{640 x z}{3} + \tfrac{235 x}{3} + \tfrac{665 y^{3}}{9} + 230 y^{2} z - \tfrac{545 y^{2}}{3} + \tfrac{665 y z^{2}}{3} - \tfrac{1090 y z}{3} + \tfrac{425 y}{3} + \tfrac{590 z^{3}}{9} - 165 z^{2} + \tfrac{400 z}{3} - \tfrac{305}{9}\\\displaystyle - \tfrac{685 x^{3}}{9} - 195 x^{2} y - \tfrac{910 x^{2} z}{3} + 220 x^{2} - 195 x y^{2} - 580 x y z + \tfrac{1220 x y}{3} - 375 x z^{2} + \tfrac{1760 x z}{3} - \tfrac{635 x}{3} - \tfrac{685 y^{3}}{9} - \tfrac{910 y^{2} z}{3} + 220 y^{2} - 375 y z^{2} + \tfrac{1760 y z}{3} - \tfrac{632 y}{3} - \tfrac{1330 z^{3}}{9} + \tfrac{1090 z^{2}}{3} - \tfrac{850 z}{3} + \tfrac{610}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{8} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{5 z \left(6 x^{2} + 18 x y + 51 x z - 6 x - 33 y z - 10 z^{2} + 3 z\right)}{9}\\\displaystyle \tfrac{5 z \left(18 x y - 33 x z + 6 y^{2} + 51 y z - 6 y - 10 z^{2} + 3 z\right)}{9}\\\displaystyle 20 x^{2} y + 20 x y^{2} - 20 x y - \tfrac{25 x z^{2}}{3} - \tfrac{25 y z^{2}}{3} + y - \tfrac{170 z^{3}}{9} + \tfrac{10 z^{2}}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{5 y \left(6 x^{2} + 105 x y - 36 x z - 6 x - 109 y^{2} + 66 y z + 3 y\right)}{9}\\\displaystyle \tfrac{5 y^{2} \left(- 15 x + 17 y + 30 z - 3\right)}{9}\\\displaystyle \tfrac{y \left(180 x^{2} + 15 x y + 90 x z - 180 x + 595 y^{2} - 780 y z - 30 y - 60 z^{2} + 60 z + 9\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{5 x^{2} \left(17 x - 15 y + 30 z - 3\right)}{9}\\\displaystyle \tfrac{5 x \left(- 109 x^{2} + 105 x y + 66 x z + 3 x + 6 y^{2} - 36 y z - 6 y\right)}{9}\\\displaystyle \tfrac{595 x^{3}}{9} + \tfrac{5 x^{2} y}{3} - \tfrac{260 x^{2} z}{3} - \tfrac{10 x^{2}}{3} + 20 x y^{2} + 10 x y z - 20 x y - \tfrac{20 x z^{2}}{3} + \tfrac{20 x z}{3} + y\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{665 x^{3}}{9} + 145 x^{2} y + 230 x^{2} z - \tfrac{545 x^{2}}{3} + \tfrac{245 x y^{2}}{3} + 290 x y z - \tfrac{670 x y}{3} + \tfrac{665 x z^{2}}{3} - \tfrac{1090 x z}{3} + \tfrac{425 x}{3} + \tfrac{95 y^{3}}{9} + 80 y^{2} z - 55 y^{2} + 135 y z^{2} - \tfrac{640 y z}{3} + \tfrac{235 y}{3} + \tfrac{590 z^{3}}{9} - 165 z^{2} + \tfrac{400 z}{3} - \tfrac{305}{9}\\\displaystyle \tfrac{95 x^{3}}{9} + \tfrac{245 x^{2} y}{3} + 80 x^{2} z - 55 x^{2} + 145 x y^{2} + 290 x y z - \tfrac{670 x y}{3} + 135 x z^{2} - \tfrac{640 x z}{3} + \tfrac{235 x}{3} + \tfrac{665 y^{3}}{9} + 230 y^{2} z - \tfrac{545 y^{2}}{3} + \tfrac{665 y z^{2}}{3} - \tfrac{1090 y z}{3} + \tfrac{425 y}{3} + \tfrac{590 z^{3}}{9} - 165 z^{2} + \tfrac{400 z}{3} - \tfrac{305}{9}\\\displaystyle - \tfrac{685 x^{3}}{9} - 195 x^{2} y - \tfrac{910 x^{2} z}{3} + 220 x^{2} - 195 x y^{2} - 580 x y z + \tfrac{1220 x y}{3} - 375 x z^{2} + \tfrac{1760 x z}{3} - \tfrac{635 x}{3} - \tfrac{685 y^{3}}{9} - \tfrac{910 y^{2} z}{3} + 220 y^{2} - 375 y z^{2} + \tfrac{1760 y z}{3} - \tfrac{632 y}{3} - \tfrac{1330 z^{3}}{9} + \tfrac{1090 z^{2}}{3} - \tfrac{850 z}{3} + \tfrac{610}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{9} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(- 60 x^{2} + 90 x y - 780 x z + 60 x + 180 y^{2} + 15 y z - 180 y + 595 z^{2} - 30 z + 9\right)}{9}\\\displaystyle \tfrac{5 z \left(- 36 x y + 66 x z + 6 y^{2} + 105 y z - 6 y - 109 z^{2} + 3 z\right)}{9}\\\displaystyle \tfrac{5 z^{2} \left(30 x - 15 y + 17 z - 3\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{20 x^{2} y}{3} - \tfrac{260 x y^{2}}{3} + 10 x y z + \tfrac{20 x y}{3} + \tfrac{595 y^{3}}{9} + \tfrac{5 y^{2} z}{3} - \tfrac{10 y^{2}}{3} + 20 y z^{2} - 20 y z + z\\\displaystyle \tfrac{5 y^{2} \left(30 x + 17 y - 15 z - 3\right)}{9}\\\displaystyle \tfrac{5 y \left(66 x y - 36 x z - 109 y^{2} + 105 y z + 3 y + 6 z^{2} - 6 z\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{170 x^{3}}{9} - \tfrac{25 x^{2} y}{3} - \tfrac{25 x^{2} z}{3} + \tfrac{10 x^{2}}{3} + 20 y^{2} z + 20 y z^{2} - 20 y z + z\\\displaystyle \tfrac{5 x \left(- 10 x^{2} + 51 x y - 33 x z + 3 x + 6 y^{2} + 18 y z - 6 y\right)}{9}\\\displaystyle \tfrac{5 x \left(- 10 x^{2} - 33 x y + 51 x z + 3 x + 18 y z + 6 z^{2} - 6 z\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{1330 x^{3}}{9} - 375 x^{2} y - 375 x^{2} z + \tfrac{1090 x^{2}}{3} - \tfrac{910 x y^{2}}{3} - 580 x y z + \tfrac{1760 x y}{3} - \tfrac{910 x z^{2}}{3} + \tfrac{1760 x z}{3} - \tfrac{850 x}{3} - \tfrac{685 y^{3}}{9} - 195 y^{2} z + 220 y^{2} - 195 y z^{2} + \tfrac{1220 y z}{3} - \tfrac{635 y}{3} - \tfrac{685 z^{3}}{9} + 220 z^{2} - \tfrac{632 z}{3} + \tfrac{610}{9}\\\displaystyle \tfrac{590 x^{3}}{9} + \tfrac{665 x^{2} y}{3} + 135 x^{2} z - 165 x^{2} + 230 x y^{2} + 290 x y z - \tfrac{1090 x y}{3} + 80 x z^{2} - \tfrac{640 x z}{3} + \tfrac{400 x}{3} + \tfrac{665 y^{3}}{9} + 145 y^{2} z - \tfrac{545 y^{2}}{3} + \tfrac{245 y z^{2}}{3} - \tfrac{670 y z}{3} + \tfrac{425 y}{3} + \tfrac{95 z^{3}}{9} - 55 z^{2} + \tfrac{235 z}{3} - \tfrac{305}{9}\\\displaystyle \tfrac{590 x^{3}}{9} + 135 x^{2} y + \tfrac{665 x^{2} z}{3} - 165 x^{2} + 80 x y^{2} + 290 x y z - \tfrac{640 x y}{3} + 230 x z^{2} - \tfrac{1090 x z}{3} + \tfrac{400 x}{3} + \tfrac{95 y^{3}}{9} + \tfrac{245 y^{2} z}{3} - 55 y^{2} + 145 y z^{2} - \tfrac{670 y z}{3} + \tfrac{235 y}{3} + \tfrac{665 z^{3}}{9} - \tfrac{545 z^{2}}{3} + \tfrac{425 z}{3} - \tfrac{305}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{9} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(- 60 x^{2} + 90 x y - 780 x z + 60 x + 180 y^{2} + 15 y z - 180 y + 595 z^{2} - 30 z + 9\right)}{9}\\\displaystyle \tfrac{5 z \left(- 36 x y + 66 x z + 6 y^{2} + 105 y z - 6 y - 109 z^{2} + 3 z\right)}{9}\\\displaystyle \tfrac{5 z^{2} \left(30 x - 15 y + 17 z - 3\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{20 x^{2} y}{3} - \tfrac{260 x y^{2}}{3} + 10 x y z + \tfrac{20 x y}{3} + \tfrac{595 y^{3}}{9} + \tfrac{5 y^{2} z}{3} - \tfrac{10 y^{2}}{3} + 20 y z^{2} - 20 y z + z\\\displaystyle \tfrac{5 y^{2} \left(30 x + 17 y - 15 z - 3\right)}{9}\\\displaystyle \tfrac{5 y \left(66 x y - 36 x z - 109 y^{2} + 105 y z + 3 y + 6 z^{2} - 6 z\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{170 x^{3}}{9} - \tfrac{25 x^{2} y}{3} - \tfrac{25 x^{2} z}{3} + \tfrac{10 x^{2}}{3} + 20 y^{2} z + 20 y z^{2} - 20 y z + z\\\displaystyle \tfrac{5 x \left(- 10 x^{2} + 51 x y - 33 x z + 3 x + 6 y^{2} + 18 y z - 6 y\right)}{9}\\\displaystyle \tfrac{5 x \left(- 10 x^{2} - 33 x y + 51 x z + 3 x + 18 y z + 6 z^{2} - 6 z\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{1330 x^{3}}{9} - 375 x^{2} y - 375 x^{2} z + \tfrac{1090 x^{2}}{3} - \tfrac{910 x y^{2}}{3} - 580 x y z + \tfrac{1760 x y}{3} - \tfrac{910 x z^{2}}{3} + \tfrac{1760 x z}{3} - \tfrac{850 x}{3} - \tfrac{685 y^{3}}{9} - 195 y^{2} z + 220 y^{2} - 195 y z^{2} + \tfrac{1220 y z}{3} - \tfrac{635 y}{3} - \tfrac{685 z^{3}}{9} + 220 z^{2} - \tfrac{632 z}{3} + \tfrac{610}{9}\\\displaystyle \tfrac{590 x^{3}}{9} + \tfrac{665 x^{2} y}{3} + 135 x^{2} z - 165 x^{2} + 230 x y^{2} + 290 x y z - \tfrac{1090 x y}{3} + 80 x z^{2} - \tfrac{640 x z}{3} + \tfrac{400 x}{3} + \tfrac{665 y^{3}}{9} + 145 y^{2} z - \tfrac{545 y^{2}}{3} + \tfrac{245 y z^{2}}{3} - \tfrac{670 y z}{3} + \tfrac{425 y}{3} + \tfrac{95 z^{3}}{9} - 55 z^{2} + \tfrac{235 z}{3} - \tfrac{305}{9}\\\displaystyle \tfrac{590 x^{3}}{9} + 135 x^{2} y + \tfrac{665 x^{2} z}{3} - 165 x^{2} + 80 x y^{2} + 290 x y z - \tfrac{640 x y}{3} + 230 x z^{2} - \tfrac{1090 x z}{3} + \tfrac{400 x}{3} + \tfrac{95 y^{3}}{9} + \tfrac{245 y^{2} z}{3} - 55 y^{2} + 145 y z^{2} - \tfrac{670 y z}{3} + \tfrac{235 y}{3} + \tfrac{665 z^{3}}{9} - \tfrac{545 z^{2}}{3} + \tfrac{425 z}{3} - \tfrac{305}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{10} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{5 z \left(6 x^{2} - 36 x y + 105 x z - 6 x + 66 y z - 109 z^{2} + 3 z\right)}{9}\\\displaystyle \tfrac{z \left(180 x^{2} + 90 x y + 15 x z - 180 x - 60 y^{2} - 780 y z + 60 y + 595 z^{2} - 30 z + 9\right)}{9}\\\displaystyle \tfrac{5 z^{2} \left(- 15 x + 30 y + 17 z - 3\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{5 y \left(6 x^{2} + 51 x y + 18 x z - 6 x - 10 y^{2} - 33 y z + 3 y\right)}{9}\\\displaystyle 20 x^{2} z - \tfrac{25 x y^{2}}{3} + 20 x z^{2} - 20 x z - \tfrac{170 y^{3}}{9} - \tfrac{25 y^{2} z}{3} + \tfrac{10 y^{2}}{3} + z\\\displaystyle \tfrac{5 y \left(- 33 x y + 18 x z - 10 y^{2} + 51 y z + 3 y + 6 z^{2} - 6 z\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{5 x^{2} \left(17 x + 30 y - 15 z - 3\right)}{9}\\\displaystyle \tfrac{595 x^{3}}{9} - \tfrac{260 x^{2} y}{3} + \tfrac{5 x^{2} z}{3} - \tfrac{10 x^{2}}{3} - \tfrac{20 x y^{2}}{3} + 10 x y z + \tfrac{20 x y}{3} + 20 x z^{2} - 20 x z + z\\\displaystyle \tfrac{5 x \left(- 109 x^{2} + 66 x y + 105 x z + 3 x - 36 y z + 6 z^{2} - 6 z\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{665 x^{3}}{9} + 230 x^{2} y + 145 x^{2} z - \tfrac{545 x^{2}}{3} + \tfrac{665 x y^{2}}{3} + 290 x y z - \tfrac{1090 x y}{3} + \tfrac{245 x z^{2}}{3} - \tfrac{670 x z}{3} + \tfrac{425 x}{3} + \tfrac{590 y^{3}}{9} + 135 y^{2} z - 165 y^{2} + 80 y z^{2} - \tfrac{640 y z}{3} + \tfrac{400 y}{3} + \tfrac{95 z^{3}}{9} - 55 z^{2} + \tfrac{235 z}{3} - \tfrac{305}{9}\\\displaystyle - \tfrac{685 x^{3}}{9} - \tfrac{910 x^{2} y}{3} - 195 x^{2} z + 220 x^{2} - 375 x y^{2} - 580 x y z + \tfrac{1760 x y}{3} - 195 x z^{2} + \tfrac{1220 x z}{3} - \tfrac{635 x}{3} - \tfrac{1330 y^{3}}{9} - 375 y^{2} z + \tfrac{1090 y^{2}}{3} - \tfrac{910 y z^{2}}{3} + \tfrac{1760 y z}{3} - \tfrac{850 y}{3} - \tfrac{685 z^{3}}{9} + 220 z^{2} - \tfrac{632 z}{3} + \tfrac{610}{9}\\\displaystyle \tfrac{95 x^{3}}{9} + 80 x^{2} y + \tfrac{245 x^{2} z}{3} - 55 x^{2} + 135 x y^{2} + 290 x y z - \tfrac{640 x y}{3} + 145 x z^{2} - \tfrac{670 x z}{3} + \tfrac{235 x}{3} + \tfrac{590 y^{3}}{9} + \tfrac{665 y^{2} z}{3} - 165 y^{2} + 230 y z^{2} - \tfrac{1090 y z}{3} + \tfrac{400 y}{3} + \tfrac{665 z^{3}}{9} - \tfrac{545 z^{2}}{3} + \tfrac{425 z}{3} - \tfrac{305}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{10} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{5 z \left(6 x^{2} - 36 x y + 105 x z - 6 x + 66 y z - 109 z^{2} + 3 z\right)}{9}\\\displaystyle \tfrac{z \left(180 x^{2} + 90 x y + 15 x z - 180 x - 60 y^{2} - 780 y z + 60 y + 595 z^{2} - 30 z + 9\right)}{9}\\\displaystyle \tfrac{5 z^{2} \left(- 15 x + 30 y + 17 z - 3\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{5 y \left(6 x^{2} + 51 x y + 18 x z - 6 x - 10 y^{2} - 33 y z + 3 y\right)}{9}\\\displaystyle 20 x^{2} z - \tfrac{25 x y^{2}}{3} + 20 x z^{2} - 20 x z - \tfrac{170 y^{3}}{9} - \tfrac{25 y^{2} z}{3} + \tfrac{10 y^{2}}{3} + z\\\displaystyle \tfrac{5 y \left(- 33 x y + 18 x z - 10 y^{2} + 51 y z + 3 y + 6 z^{2} - 6 z\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{5 x^{2} \left(17 x + 30 y - 15 z - 3\right)}{9}\\\displaystyle \tfrac{595 x^{3}}{9} - \tfrac{260 x^{2} y}{3} + \tfrac{5 x^{2} z}{3} - \tfrac{10 x^{2}}{3} - \tfrac{20 x y^{2}}{3} + 10 x y z + \tfrac{20 x y}{3} + 20 x z^{2} - 20 x z + z\\\displaystyle \tfrac{5 x \left(- 109 x^{2} + 66 x y + 105 x z + 3 x - 36 y z + 6 z^{2} - 6 z\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{665 x^{3}}{9} + 230 x^{2} y + 145 x^{2} z - \tfrac{545 x^{2}}{3} + \tfrac{665 x y^{2}}{3} + 290 x y z - \tfrac{1090 x y}{3} + \tfrac{245 x z^{2}}{3} - \tfrac{670 x z}{3} + \tfrac{425 x}{3} + \tfrac{590 y^{3}}{9} + 135 y^{2} z - 165 y^{2} + 80 y z^{2} - \tfrac{640 y z}{3} + \tfrac{400 y}{3} + \tfrac{95 z^{3}}{9} - 55 z^{2} + \tfrac{235 z}{3} - \tfrac{305}{9}\\\displaystyle - \tfrac{685 x^{3}}{9} - \tfrac{910 x^{2} y}{3} - 195 x^{2} z + 220 x^{2} - 375 x y^{2} - 580 x y z + \tfrac{1760 x y}{3} - 195 x z^{2} + \tfrac{1220 x z}{3} - \tfrac{635 x}{3} - \tfrac{1330 y^{3}}{9} - 375 y^{2} z + \tfrac{1090 y^{2}}{3} - \tfrac{910 y z^{2}}{3} + \tfrac{1760 y z}{3} - \tfrac{850 y}{3} - \tfrac{685 z^{3}}{9} + 220 z^{2} - \tfrac{632 z}{3} + \tfrac{610}{9}\\\displaystyle \tfrac{95 x^{3}}{9} + 80 x^{2} y + \tfrac{245 x^{2} z}{3} - 55 x^{2} + 135 x y^{2} + 290 x y z - \tfrac{640 x y}{3} + 145 x z^{2} - \tfrac{670 x z}{3} + \tfrac{235 x}{3} + \tfrac{590 y^{3}}{9} + \tfrac{665 y^{2} z}{3} - 165 y^{2} + 230 y z^{2} - \tfrac{1090 y z}{3} + \tfrac{400 y}{3} + \tfrac{665 z^{3}}{9} - \tfrac{545 z^{2}}{3} + \tfrac{425 z}{3} - \tfrac{305}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{11} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(- 60 x^{2} - 180 x y - 132 x z + 96 x + 330 y z + 430 z^{2} - 192 z - 9\right)}{9}\\\displaystyle \tfrac{z \left(- 180 x y + 330 x z - 60 y^{2} - 132 y z + 96 y + 430 z^{2} - 192 z - 9\right)}{9}\\\displaystyle \tfrac{2 z^{2} \left(75 x + 75 y + 44 z - 48\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(- 60 x^{2} - 132 x y - 180 x z + 96 x + 430 y^{2} + 330 y z - 192 y - 9\right)}{9}\\\displaystyle \tfrac{y \left(150 x y + 88 y^{2} + 150 y z - 96 y - 9\right)}{9}\\\displaystyle \tfrac{110 x y^{2}}{3} - 20 x y z + \tfrac{430 y^{3}}{9} - \tfrac{44 y^{2} z}{3} - \tfrac{64 y^{2}}{3} - \tfrac{20 y z^{2}}{3} + \tfrac{32 y z}{3} - y + z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(88 x^{2} + 150 x y + 150 x z - 96 x - 9\right)}{9}\\\displaystyle \tfrac{x \left(430 x^{2} - 132 x y + 330 x z - 192 x - 60 y^{2} - 180 y z + 96 y - 9\right)}{9}\\\displaystyle \tfrac{430 x^{3}}{9} + \tfrac{110 x^{2} y}{3} - \tfrac{44 x^{2} z}{3} - \tfrac{64 x^{2}}{3} - 20 x y z - \tfrac{20 x z^{2}}{3} + \tfrac{32 x z}{3} - x + z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{112 x^{3}}{9} - 6 x^{2} y - 6 x^{2} z + \tfrac{52 x^{2}}{3} + \tfrac{56 x y^{2}}{3} + 24 x y z - \tfrac{52 x y}{3} + \tfrac{56 x z^{2}}{3} - \tfrac{52 x z}{3} - \tfrac{x}{3} + \tfrac{110 y^{3}}{9} + 30 y^{2} z - 28 y^{2} + 30 y z^{2} - \tfrac{148 y z}{3} + \tfrac{61 y}{3} + \tfrac{110 z^{3}}{9} - 28 z^{2} + \tfrac{61 z}{3} - \tfrac{41}{9}\\\displaystyle \tfrac{110 x^{3}}{9} + \tfrac{56 x^{2} y}{3} + 30 x^{2} z - 28 x^{2} - 6 x y^{2} + 24 x y z - \tfrac{52 x y}{3} + 30 x z^{2} - \tfrac{148 x z}{3} + \tfrac{61 x}{3} - \tfrac{112 y^{3}}{9} - 6 y^{2} z + \tfrac{52 y^{2}}{3} + \tfrac{56 y z^{2}}{3} - \tfrac{52 y z}{3} - \tfrac{y}{3} + \tfrac{110 z^{3}}{9} - 28 z^{2} + \tfrac{61 z}{3} - \tfrac{41}{9}\\\displaystyle \tfrac{110 x^{3}}{9} + 30 x^{2} y + \tfrac{56 x^{2} z}{3} - 28 x^{2} + 30 x y^{2} + 24 x y z - \tfrac{148 x y}{3} - 6 x z^{2} - \tfrac{52 x z}{3} + \tfrac{61 x}{3} + \tfrac{110 y^{3}}{9} + \tfrac{56 y^{2} z}{3} - 28 y^{2} - 6 y z^{2} - \tfrac{52 y z}{3} + \tfrac{61 y}{3} - \tfrac{112 z^{3}}{9} + \tfrac{52 z^{2}}{3} + \tfrac{2 z}{3} - \tfrac{41}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{11} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(- 60 x^{2} - 180 x y - 132 x z + 96 x + 330 y z + 430 z^{2} - 192 z - 9\right)}{9}\\\displaystyle \tfrac{z \left(- 180 x y + 330 x z - 60 y^{2} - 132 y z + 96 y + 430 z^{2} - 192 z - 9\right)}{9}\\\displaystyle \tfrac{2 z^{2} \left(75 x + 75 y + 44 z - 48\right)}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(- 60 x^{2} - 132 x y - 180 x z + 96 x + 430 y^{2} + 330 y z - 192 y - 9\right)}{9}\\\displaystyle \tfrac{y \left(150 x y + 88 y^{2} + 150 y z - 96 y - 9\right)}{9}\\\displaystyle \tfrac{110 x y^{2}}{3} - 20 x y z + \tfrac{430 y^{3}}{9} - \tfrac{44 y^{2} z}{3} - \tfrac{64 y^{2}}{3} - \tfrac{20 y z^{2}}{3} + \tfrac{32 y z}{3} - y + z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(88 x^{2} + 150 x y + 150 x z - 96 x - 9\right)}{9}\\\displaystyle \tfrac{x \left(430 x^{2} - 132 x y + 330 x z - 192 x - 60 y^{2} - 180 y z + 96 y - 9\right)}{9}\\\displaystyle \tfrac{430 x^{3}}{9} + \tfrac{110 x^{2} y}{3} - \tfrac{44 x^{2} z}{3} - \tfrac{64 x^{2}}{3} - 20 x y z - \tfrac{20 x z^{2}}{3} + \tfrac{32 x z}{3} - x + z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{112 x^{3}}{9} - 6 x^{2} y - 6 x^{2} z + \tfrac{52 x^{2}}{3} + \tfrac{56 x y^{2}}{3} + 24 x y z - \tfrac{52 x y}{3} + \tfrac{56 x z^{2}}{3} - \tfrac{52 x z}{3} - \tfrac{x}{3} + \tfrac{110 y^{3}}{9} + 30 y^{2} z - 28 y^{2} + 30 y z^{2} - \tfrac{148 y z}{3} + \tfrac{61 y}{3} + \tfrac{110 z^{3}}{9} - 28 z^{2} + \tfrac{61 z}{3} - \tfrac{41}{9}\\\displaystyle \tfrac{110 x^{3}}{9} + \tfrac{56 x^{2} y}{3} + 30 x^{2} z - 28 x^{2} - 6 x y^{2} + 24 x y z - \tfrac{52 x y}{3} + 30 x z^{2} - \tfrac{148 x z}{3} + \tfrac{61 x}{3} - \tfrac{112 y^{3}}{9} - 6 y^{2} z + \tfrac{52 y^{2}}{3} + \tfrac{56 y z^{2}}{3} - \tfrac{52 y z}{3} - \tfrac{y}{3} + \tfrac{110 z^{3}}{9} - 28 z^{2} + \tfrac{61 z}{3} - \tfrac{41}{9}\\\displaystyle \tfrac{110 x^{3}}{9} + 30 x^{2} y + \tfrac{56 x^{2} z}{3} - 28 x^{2} + 30 x y^{2} + 24 x y z - \tfrac{148 x y}{3} - 6 x z^{2} - \tfrac{52 x z}{3} + \tfrac{61 x}{3} + \tfrac{110 y^{3}}{9} + \tfrac{56 y^{2} z}{3} - 28 y^{2} - 6 y z^{2} - \tfrac{52 y z}{3} + \tfrac{61 y}{3} - \tfrac{112 z^{3}}{9} + \tfrac{52 z^{2}}{3} + \tfrac{2 z}{3} - \tfrac{41}{9}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.
\(\displaystyle \boldsymbol{\phi}_{12} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{2 z \left(60 x^{2} + 180 x y + 132 x z - 96 x - 330 y z - 430 z^{2} + 192 z + 9\right)}{3}\\\displaystyle \tfrac{2 z \left(180 x y - 330 x z + 60 y^{2} + 132 y z - 96 y - 430 z^{2} + 192 z + 9\right)}{3}\\\displaystyle \tfrac{2 z \left(- 150 x z - 150 y z - 88 z^{2} + 96 z + 9\right)}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{2 y \left(60 x^{2} + 132 x y + 180 x z - 96 x - 430 y^{2} - 330 y z + 192 y + 9\right)}{3}\\\displaystyle \tfrac{2 y \left(- 150 x y - 88 y^{2} - 150 y z + 96 y + 9\right)}{3}\\\displaystyle \tfrac{2 y \left(- 330 x y + 180 x z - 430 y^{2} + 132 y z + 192 y + 60 z^{2} - 96 z + 9\right)}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{2 x \left(- 88 x^{2} - 150 x y - 150 x z + 96 x + 9\right)}{3}\\\displaystyle \tfrac{2 x \left(- 430 x^{2} + 132 x y - 330 x z + 192 x + 60 y^{2} + 180 y z - 96 y + 9\right)}{3}\\\displaystyle \tfrac{2 x \left(- 430 x^{2} - 330 x y + 132 x z + 192 x + 180 y z + 60 z^{2} - 96 z + 9\right)}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{224 x^{3}}{3} + 36 x^{2} y + 36 x^{2} z - 104 x^{2} - 112 x y^{2} - 144 x y z + 104 x y - 112 x z^{2} + 104 x z + 2 x - \tfrac{220 y^{3}}{3} - 180 y^{2} z + 168 y^{2} - 180 y z^{2} + 296 y z - 122 y - \tfrac{220 z^{3}}{3} + 168 z^{2} - 122 z + \tfrac{82}{3}\\\displaystyle - \tfrac{220 x^{3}}{3} - 112 x^{2} y - 180 x^{2} z + 168 x^{2} + 36 x y^{2} - 144 x y z + 104 x y - 180 x z^{2} + 296 x z - 122 x + \tfrac{224 y^{3}}{3} + 36 y^{2} z - 104 y^{2} - 112 y z^{2} + 104 y z + 2 y - \tfrac{220 z^{3}}{3} + 168 z^{2} - 122 z + \tfrac{82}{3}\\\displaystyle - \tfrac{220 x^{3}}{3} - 180 x^{2} y - 112 x^{2} z + 168 x^{2} - 180 x y^{2} - 144 x y z + 296 x y + 36 x z^{2} + 104 x z - 122 x - \tfrac{220 y^{3}}{3} - 112 y^{2} z + 168 y^{2} + 36 y z^{2} + 104 y z - 122 y + \tfrac{224 z^{3}}{3} - 104 z^{2} + 2 z + \tfrac{82}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.
\(\displaystyle \boldsymbol{\phi}_{12} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{2 z \left(60 x^{2} + 180 x y + 132 x z - 96 x - 330 y z - 430 z^{2} + 192 z + 9\right)}{3}\\\displaystyle \tfrac{2 z \left(180 x y - 330 x z + 60 y^{2} + 132 y z - 96 y - 430 z^{2} + 192 z + 9\right)}{3}\\\displaystyle \tfrac{2 z \left(- 150 x z - 150 y z - 88 z^{2} + 96 z + 9\right)}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{2 y \left(60 x^{2} + 132 x y + 180 x z - 96 x - 430 y^{2} - 330 y z + 192 y + 9\right)}{3}\\\displaystyle \tfrac{2 y \left(- 150 x y - 88 y^{2} - 150 y z + 96 y + 9\right)}{3}\\\displaystyle \tfrac{2 y \left(- 330 x y + 180 x z - 430 y^{2} + 132 y z + 192 y + 60 z^{2} - 96 z + 9\right)}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{2 x \left(- 88 x^{2} - 150 x y - 150 x z + 96 x + 9\right)}{3}\\\displaystyle \tfrac{2 x \left(- 430 x^{2} + 132 x y - 330 x z + 192 x + 60 y^{2} + 180 y z - 96 y + 9\right)}{3}\\\displaystyle \tfrac{2 x \left(- 430 x^{2} - 330 x y + 132 x z + 192 x + 180 y z + 60 z^{2} - 96 z + 9\right)}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{224 x^{3}}{3} + 36 x^{2} y + 36 x^{2} z - 104 x^{2} - 112 x y^{2} - 144 x y z + 104 x y - 112 x z^{2} + 104 x z + 2 x - \tfrac{220 y^{3}}{3} - 180 y^{2} z + 168 y^{2} - 180 y z^{2} + 296 y z - 122 y - \tfrac{220 z^{3}}{3} + 168 z^{2} - 122 z + \tfrac{82}{3}\\\displaystyle - \tfrac{220 x^{3}}{3} - 112 x^{2} y - 180 x^{2} z + 168 x^{2} + 36 x y^{2} - 144 x y z + 104 x y - 180 x z^{2} + 296 x z - 122 x + \tfrac{224 y^{3}}{3} + 36 y^{2} z - 104 y^{2} - 112 y z^{2} + 104 y z + 2 y - \tfrac{220 z^{3}}{3} + 168 z^{2} - 122 z + \tfrac{82}{3}\\\displaystyle - \tfrac{220 x^{3}}{3} - 180 x^{2} y - 112 x^{2} z + 168 x^{2} - 180 x y^{2} - 144 x y z + 296 x y + 36 x z^{2} + 104 x z - 122 x - \tfrac{220 y^{3}}{3} - 112 y^{2} z + 168 y^{2} + 36 y z^{2} + 104 y z - 122 y + \tfrac{224 z^{3}}{3} - 104 z^{2} + 2 z + \tfrac{82}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \boldsymbol{\phi}_{13} = \begin{cases} \left(\begin{array}{c}\displaystyle 2 z \left(- 90 x y + 216 x z + 12 x - 60 y^{2} + 105 y z + 60 y - 55 z^{2} - 54 z - 3\right)\\\displaystyle 2 z \left(- 30 y^{2} - 219 y z + 42 y + 325 z^{2} - 69 z - 3\right)\\\displaystyle 2 z \left(75 y z + z^{2} - 27 z - 3\right)\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 y \left(216 x y - 90 x z + 12 x - 55 y^{2} + 105 y z - 54 y - 60 z^{2} + 60 z - 3\right)\\\displaystyle 2 y \left(y^{2} + 75 y z - 27 y - 3\right)\\\displaystyle 2 y \left(325 y^{2} - 219 y z - 69 y - 30 z^{2} + 42 z - 3\right)\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 172 x^{3} + 150 x^{2} y + 150 x^{2} z - 84 x^{2} - 6 x - 120 y^{2} z - 120 y z^{2} + 120 y z\\\displaystyle 2 x \left(160 x^{2} - 129 x y + 165 x z - 69 x - 30 y^{2} - 90 y z + 42 y - 3\right)\\\displaystyle 2 x \left(160 x^{2} + 165 x y - 129 x z - 69 x - 90 y z - 30 z^{2} + 42 z - 3\right)\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 812 x^{3} + 2214 x^{2} y + 2214 x^{2} z - 2076 x^{2} + 1932 x y^{2} + 3624 x y z - 3624 x y + 1932 x z^{2} - 3624 x z + 1698 x + 530 y^{3} + 1350 y^{2} z - 1488 y^{2} + 1350 y z^{2} - 2736 y z + 1392 y + 530 z^{3} - 1488 z^{2} + 1392 z - 434\\\displaystyle - 320 x^{3} - 1218 x^{2} y - 630 x^{2} z + 822 x^{2} - 1416 x y^{2} - 1596 x y z + 2076 x y - 300 x z^{2} + 984 x z - 678 x - 518 y^{3} - 906 y^{2} z + 1194 y^{2} - 378 y z^{2} + 1236 y z - 852 y + 10 z^{3} + 162 z^{2} - 348 z + 176\\\displaystyle - 320 x^{3} - 630 x^{2} y - 1218 x^{2} z + 822 x^{2} - 300 x y^{2} - 1596 x y z + 984 x y - 1416 x z^{2} + 2076 x z - 678 x + 10 y^{3} - 378 y^{2} z + 162 y^{2} - 906 y z^{2} + 1236 y z - 348 y - 518 z^{3} + 1194 z^{2} - 852 z + 176\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \boldsymbol{\phi}_{13} = \begin{cases} \left(\begin{array}{c}\displaystyle 2 z \left(- 90 x y + 216 x z + 12 x - 60 y^{2} + 105 y z + 60 y - 55 z^{2} - 54 z - 3\right)\\\displaystyle 2 z \left(- 30 y^{2} - 219 y z + 42 y + 325 z^{2} - 69 z - 3\right)\\\displaystyle 2 z \left(75 y z + z^{2} - 27 z - 3\right)\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 y \left(216 x y - 90 x z + 12 x - 55 y^{2} + 105 y z - 54 y - 60 z^{2} + 60 z - 3\right)\\\displaystyle 2 y \left(y^{2} + 75 y z - 27 y - 3\right)\\\displaystyle 2 y \left(325 y^{2} - 219 y z - 69 y - 30 z^{2} + 42 z - 3\right)\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 172 x^{3} + 150 x^{2} y + 150 x^{2} z - 84 x^{2} - 6 x - 120 y^{2} z - 120 y z^{2} + 120 y z\\\displaystyle 2 x \left(160 x^{2} - 129 x y + 165 x z - 69 x - 30 y^{2} - 90 y z + 42 y - 3\right)\\\displaystyle 2 x \left(160 x^{2} + 165 x y - 129 x z - 69 x - 90 y z - 30 z^{2} + 42 z - 3\right)\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 812 x^{3} + 2214 x^{2} y + 2214 x^{2} z - 2076 x^{2} + 1932 x y^{2} + 3624 x y z - 3624 x y + 1932 x z^{2} - 3624 x z + 1698 x + 530 y^{3} + 1350 y^{2} z - 1488 y^{2} + 1350 y z^{2} - 2736 y z + 1392 y + 530 z^{3} - 1488 z^{2} + 1392 z - 434\\\displaystyle - 320 x^{3} - 1218 x^{2} y - 630 x^{2} z + 822 x^{2} - 1416 x y^{2} - 1596 x y z + 2076 x y - 300 x z^{2} + 984 x z - 678 x - 518 y^{3} - 906 y^{2} z + 1194 y^{2} - 378 y z^{2} + 1236 y z - 852 y + 10 z^{3} + 162 z^{2} - 348 z + 176\\\displaystyle - 320 x^{3} - 630 x^{2} y - 1218 x^{2} z + 822 x^{2} - 300 x y^{2} - 1596 x y z + 984 x y - 1416 x z^{2} + 2076 x z - 678 x + 10 y^{3} - 378 y^{2} z + 162 y^{2} - 906 y z^{2} + 1236 y z - 348 y - 518 z^{3} + 1194 z^{2} - 852 z + 176\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.
\(\displaystyle \boldsymbol{\phi}_{14} = \begin{cases} \left(\begin{array}{c}\displaystyle 2 z \left(30 x^{2} + 219 x z - 42 x - 325 z^{2} + 69 z + 3\right)\\\displaystyle 2 z \left(60 x^{2} + 90 x y - 105 x z - 60 x - 216 y z - 12 y + 55 z^{2} + 54 z + 3\right)\\\displaystyle 2 z \left(- 75 x z - z^{2} + 27 z + 3\right)\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 y \left(30 x^{2} + 129 x y + 90 x z - 42 x - 160 y^{2} - 165 y z + 69 y + 3\right)\\\displaystyle 120 x^{2} z - 150 x y^{2} + 120 x z^{2} - 120 x z - 172 y^{3} - 150 y^{2} z + 84 y^{2} + 6 y\\\displaystyle 2 y \left(- 165 x y + 90 x z - 160 y^{2} + 129 y z + 69 y + 30 z^{2} - 42 z + 3\right)\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 x \left(- x^{2} - 75 x z + 27 x + 3\right)\\\displaystyle 2 x \left(55 x^{2} - 216 x y - 105 x z + 54 x + 90 y z - 12 y + 60 z^{2} - 60 z + 3\right)\\\displaystyle 2 x \left(- 325 x^{2} + 219 x z + 69 x + 30 z^{2} - 42 z + 3\right)\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 518 x^{3} + 1416 x^{2} y + 906 x^{2} z - 1194 x^{2} + 1218 x y^{2} + 1596 x y z - 2076 x y + 378 x z^{2} - 1236 x z + 852 x + 320 y^{3} + 630 y^{2} z - 822 y^{2} + 300 y z^{2} - 984 y z + 678 y - 10 z^{3} - 162 z^{2} + 348 z - 176\\\displaystyle - 530 x^{3} - 1932 x^{2} y - 1350 x^{2} z + 1488 x^{2} - 2214 x y^{2} - 3624 x y z + 3624 x y - 1350 x z^{2} + 2736 x z - 1392 x - 812 y^{3} - 2214 y^{2} z + 2076 y^{2} - 1932 y z^{2} + 3624 y z - 1698 y - 530 z^{3} + 1488 z^{2} - 1392 z + 434\\\displaystyle - 10 x^{3} + 300 x^{2} y + 378 x^{2} z - 162 x^{2} + 630 x y^{2} + 1596 x y z - 984 x y + 906 x z^{2} - 1236 x z + 348 x + 320 y^{3} + 1218 y^{2} z - 822 y^{2} + 1416 y z^{2} - 2076 y z + 678 y + 518 z^{3} - 1194 z^{2} + 852 z - 176\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.
\(\displaystyle \boldsymbol{\phi}_{14} = \begin{cases} \left(\begin{array}{c}\displaystyle 2 z \left(30 x^{2} + 219 x z - 42 x - 325 z^{2} + 69 z + 3\right)\\\displaystyle 2 z \left(60 x^{2} + 90 x y - 105 x z - 60 x - 216 y z - 12 y + 55 z^{2} + 54 z + 3\right)\\\displaystyle 2 z \left(- 75 x z - z^{2} + 27 z + 3\right)\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 y \left(30 x^{2} + 129 x y + 90 x z - 42 x - 160 y^{2} - 165 y z + 69 y + 3\right)\\\displaystyle 120 x^{2} z - 150 x y^{2} + 120 x z^{2} - 120 x z - 172 y^{3} - 150 y^{2} z + 84 y^{2} + 6 y\\\displaystyle 2 y \left(- 165 x y + 90 x z - 160 y^{2} + 129 y z + 69 y + 30 z^{2} - 42 z + 3\right)\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 x \left(- x^{2} - 75 x z + 27 x + 3\right)\\\displaystyle 2 x \left(55 x^{2} - 216 x y - 105 x z + 54 x + 90 y z - 12 y + 60 z^{2} - 60 z + 3\right)\\\displaystyle 2 x \left(- 325 x^{2} + 219 x z + 69 x + 30 z^{2} - 42 z + 3\right)\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 518 x^{3} + 1416 x^{2} y + 906 x^{2} z - 1194 x^{2} + 1218 x y^{2} + 1596 x y z - 2076 x y + 378 x z^{2} - 1236 x z + 852 x + 320 y^{3} + 630 y^{2} z - 822 y^{2} + 300 y z^{2} - 984 y z + 678 y - 10 z^{3} - 162 z^{2} + 348 z - 176\\\displaystyle - 530 x^{3} - 1932 x^{2} y - 1350 x^{2} z + 1488 x^{2} - 2214 x y^{2} - 3624 x y z + 3624 x y - 1350 x z^{2} + 2736 x z - 1392 x - 812 y^{3} - 2214 y^{2} z + 2076 y^{2} - 1932 y z^{2} + 3624 y z - 1698 y - 530 z^{3} + 1488 z^{2} - 1392 z + 434\\\displaystyle - 10 x^{3} + 300 x^{2} y + 378 x^{2} z - 162 x^{2} + 630 x y^{2} + 1596 x y z - 984 x y + 906 x z^{2} - 1236 x z + 348 x + 320 y^{3} + 1218 y^{2} z - 822 y^{2} + 1416 y z^{2} - 2076 y z + 678 y + 518 z^{3} - 1194 z^{2} + 852 z - 176\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3rd face;
and \(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3.
\(\displaystyle \boldsymbol{\phi}_{15} = \begin{cases} \left(\begin{array}{c}\displaystyle 2 z \left(- 30 x^{2} - 90 x y - 129 x z + 42 x + 165 y z + 160 z^{2} - 69 z - 3\right)\\\displaystyle 2 z \left(- 90 x y + 165 x z - 30 y^{2} - 129 y z + 42 y + 160 z^{2} - 69 z - 3\right)\\\displaystyle - 120 x^{2} y - 120 x y^{2} + 120 x y + 150 x z^{2} + 150 y z^{2} + 172 z^{3} - 84 z^{2} - 6 z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 y \left(- 30 x^{2} - 219 x y + 42 x + 325 y^{2} - 69 y - 3\right)\\\displaystyle 2 y \left(75 x y + y^{2} - 27 y - 3\right)\\\displaystyle 2 y \left(- 60 x^{2} + 105 x y - 90 x z + 60 x - 55 y^{2} + 216 y z - 54 y + 12 z - 3\right)\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 x \left(x^{2} + 75 x y - 27 x - 3\right)\\\displaystyle 2 x \left(325 x^{2} - 219 x y - 69 x - 30 y^{2} + 42 y - 3\right)\\\displaystyle 2 x \left(- 55 x^{2} + 105 x y + 216 x z - 54 x - 60 y^{2} - 90 y z + 60 y + 12 z - 3\right)\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 518 x^{3} - 906 x^{2} y - 1416 x^{2} z + 1194 x^{2} - 378 x y^{2} - 1596 x y z + 1236 x y - 1218 x z^{2} + 2076 x z - 852 x + 10 y^{3} - 300 y^{2} z + 162 y^{2} - 630 y z^{2} + 984 y z - 348 y - 320 z^{3} + 822 z^{2} - 678 z + 176\\\displaystyle 10 x^{3} - 378 x^{2} y - 300 x^{2} z + 162 x^{2} - 906 x y^{2} - 1596 x y z + 1236 x y - 630 x z^{2} + 984 x z - 348 x - 518 y^{3} - 1416 y^{2} z + 1194 y^{2} - 1218 y z^{2} + 2076 y z - 852 y - 320 z^{3} + 822 z^{2} - 678 z + 176\\\displaystyle 530 x^{3} + 1350 x^{2} y + 1932 x^{2} z - 1488 x^{2} + 1350 x y^{2} + 3624 x y z - 2736 x y + 2214 x z^{2} - 3624 x z + 1392 x + 530 y^{3} + 1932 y^{2} z - 1488 y^{2} + 2214 y z^{2} - 3624 y z + 1392 y + 812 z^{3} - 2076 z^{2} + 1698 z - 434\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face;
and \(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3.
\(\displaystyle \boldsymbol{\phi}_{15} = \begin{cases} \left(\begin{array}{c}\displaystyle 2 z \left(- 30 x^{2} - 90 x y - 129 x z + 42 x + 165 y z + 160 z^{2} - 69 z - 3\right)\\\displaystyle 2 z \left(- 90 x y + 165 x z - 30 y^{2} - 129 y z + 42 y + 160 z^{2} - 69 z - 3\right)\\\displaystyle - 120 x^{2} y - 120 x y^{2} + 120 x y + 150 x z^{2} + 150 y z^{2} + 172 z^{3} - 84 z^{2} - 6 z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 y \left(- 30 x^{2} - 219 x y + 42 x + 325 y^{2} - 69 y - 3\right)\\\displaystyle 2 y \left(75 x y + y^{2} - 27 y - 3\right)\\\displaystyle 2 y \left(- 60 x^{2} + 105 x y - 90 x z + 60 x - 55 y^{2} + 216 y z - 54 y + 12 z - 3\right)\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 x \left(x^{2} + 75 x y - 27 x - 3\right)\\\displaystyle 2 x \left(325 x^{2} - 219 x y - 69 x - 30 y^{2} + 42 y - 3\right)\\\displaystyle 2 x \left(- 55 x^{2} + 105 x y + 216 x z - 54 x - 60 y^{2} - 90 y z + 60 y + 12 z - 3\right)\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 518 x^{3} - 906 x^{2} y - 1416 x^{2} z + 1194 x^{2} - 378 x y^{2} - 1596 x y z + 1236 x y - 1218 x z^{2} + 2076 x z - 852 x + 10 y^{3} - 300 y^{2} z + 162 y^{2} - 630 y z^{2} + 984 y z - 348 y - 320 z^{3} + 822 z^{2} - 678 z + 176\\\displaystyle 10 x^{3} - 378 x^{2} y - 300 x^{2} z + 162 x^{2} - 906 x y^{2} - 1596 x y z + 1236 x y - 630 x z^{2} + 984 x z - 348 x - 518 y^{3} - 1416 y^{2} z + 1194 y^{2} - 1218 y z^{2} + 2076 y z - 852 y - 320 z^{3} + 822 z^{2} - 678 z + 176\\\displaystyle 530 x^{3} + 1350 x^{2} y + 1932 x^{2} z - 1488 x^{2} + 1350 x y^{2} + 3624 x y z - 2736 x y + 2214 x z^{2} - 3624 x z + 1392 x + 530 y^{3} + 1932 y^{2} z - 1488 y^{2} + 2214 y z^{2} - 3624 y z + 1392 y + 812 z^{3} - 2076 z^{2} + 1698 z - 434\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
This DOF is associated with face 3 of the reference element.