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In isoparameteric finite element of second order tetrahedron element, the original coordinate $(x,y,z)$ would be transformed to standard coordinate $(\xi, \eta, \zeta)$ by a shape function. As a result, for any $(\xi_j, \eta_j, \zeta_j)$ in the element, the original coordinate $(x_j,y_j,z_j)$ can be calculated: $$ x_j=\sum_{i=1}^{10} N(i)\cdot x_i, $$ $$ y_j=\sum_{i=1}^{10}N(i)\cdot y_i, $$ $$ z_j=\sum_{i=1}^{10}N(i)\cdot z_i. $$ Now my question is: if I give a rotation to the original element, the $(x_j, y_j, z_j)$ is transformed to $(x_{jj}, y_{jj}, z_{jj})$, which can be calculated from $(\xi_{jj}, \eta_{jj}, \zeta_{jj})$, what's the relationship between $(\xi_j, \eta_j, \zeta_j)$ and $(\xi_{jj}, \eta_{jj}, \zeta_{jj})$? are they just the same? how to prove this relationship?

Thanks, Tang Laoya

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