Formulas#
Derivatives#
Cartesian Coordinates#
(482)#\[
dl = dx\hat{\textbf{x}} + dy\hat{\textbf{y}} + dz\hat{\textbf{z}}
\]
(483)#\[d\tau = dxdydz\]
Gradient#
(484)#\[
\nabla t = \frac{\partial t}{\partial x}\hat{\textbf{x}} +\frac{\partial t}{\partial y}\hat{\textbf{y}} + \frac{\partial t}{\partial z}\hat{\textbf{z}}
\]
Divergence#
(485)#\[\nabla \cdot \textbf{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}\]
Curl#
(486)#\[\nabla \times \textbf{v} = (\frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z})\hat{\textbf{x}} + (\frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x})\hat{\textbf{y}} + (\frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y})\hat{\textbf{z}}
\]
(487)#\[\begin{split}
\nabla \times \textbf{v} = \begin{vmatrix}
\hat{\textbf{x}} & \hat{\textbf{y}} & \hat{\textbf{z}} \\
\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} \\
v_x & v_y & v_z \\
\end{vmatrix}
\end{split}\]
Laplacian#
(488)#\[
\nabla ^2 T = \nabla \cdot (\nabla T) = \frac{\partial^2T}{\partial x} + \frac{\partial^2T}{\partial y} + \frac{\partial^2T}{\partial z}
\]
Spherical Coordinates#
(489)#\[
dl = dr\hat{\textbf{r}} + rd\theta \hat{\bm{\theta}} + r \sin{\bm{\theta}} d\phi \hat{\bm{\phi}}
\]
Gradient#
Divergence#
Curl#
Laplacian#
\[\]
$$