梯度在单位向量偏导的推广

edward076923

关于 $\nabla f \cdot \vec n = {\partial f \over \partial \vec n}$ 的推广。

\[ \frac{\partial^2f}{\partial \vec n^2}=\left(\pmatrix{f_{xx}&f_{xy}\\f_{yx}&f_{yy}} \vec n \right)^T \vec n \]

edward076923

三元素推广。

$
\operatorname{d} \vec r = h_{1} dx_1 \vec {e_1} + h_{2} dx_2 \vec {e_2} + h_{3} dx_3 \vec {e_3}
$
For coordinates in conical (LHS) to arbitrary in (RHS), and $\phi$ is defined as a smooth scalar field. $\vec F$ is defined as a smooth vector field. By previous sheet, we know that Jacobian, $
J = h_1h_2h_3.
$
$
\nabla \phi = {\partial \phi \over \partial x_1} {\vec {e_{x_1}} \over h_1} + {\partial \phi \over \partial x_2} {\vec {e_{x_2}} \over h_2} + {\partial \phi \over \partial x_3} {\vec {e_{x_3}} \over h_3}
$
$
\nabla \cdot \vec F = {1 \over J} \left( {\partial \left( {J F_1 \over h_1} \right) \over \partial x_1 } + {\partial \left( {J F_2 \over h_2} \right) \over \partial x_2 } + {\partial \left( {J F_3 \over h_3} \right) \over \partial x_3}\right)
$
$
\nabla \times \vec F = \det \left(
\begin{matrix}
h_1 e_{x_1} & h_2 e_{x_2} & h_3 e_{x_3}\\
\partial \over \partial x_1 & \partial \over \partial x_2 & \partial \over \partial x_3 \\
h_1F_1 & h_2F_2 & h_3F_3
\end{matrix}
\right )
$