Difference (from prior minor revision) (no other diffs)

Changed: 103c103
$$\nabla^2 t=\frac{1}{\rho^2} \dd{\rho}\left(r^2 \d{t}{\rho}\right)
$$\nabla^2 t=\frac{1}{\rho^2} \dd{\rho}\left(\rho^2 \d{t}{\rho}\right)

Changed: 123c123
+ \left[\d{v_r}{z}-\d{z}{v_r}\right] \th+\frac{1}{r}\left[\dd{r}(rv_\theta)-\d{v_r}{\theta}\right] \z$$
+ \left[\d{v_r}{z}-\d{v_z}{r}\right] \th+\frac{1}{r}\left[\dd{r}(rv_\theta)-\d{v_r}{\theta}\right] \z$$

This is the MultiVCheatSheetOfDOOM, taken from a lecture handout by Professor Hermann Gluck of U. Penn., who apparently stole it from some electrodynamics book. It has been LaTeXed and math-convention-ified by MicahSmukler; see Micah if you want a copy of the cheat sheet but don't want to deal with LaTeX. Disclaimer: some professors' test policies may not permit the use of this on tests, as you didn't make it. Using it on homework should be okay, though.



 \newcommand{\dd}[1]{\frac{\partial}{\partial #1}}
 \renewcommand{\d}[2]{\frac{\partial #1}{\partial #2}}


 {\bf Vector Identities}

 {\bf Triple Products}
 \item $\A \cdot (\B \times \C)=\B \cdot(\C \times \A)=\C \cdot (\A \times \B)$
 \item $\A \times (\B \times \C)=\B(\A \cdot \C)-\C(\A \cdot \B)$

 {\bf Product Rules}
 \item $\nabla(fg)=f(\nabla g)+g(\nabla f)$
 \item $\nabla(\A \cdot \B)=\A \times (\nabla \times \B)+\B \times(\nabla \times \A)
 +(\A \cdot \nabla)\B+(\B \cdot \nabla)\A$
 \item $\nabla \cdot(f\A)=f(\nabla \cdot \A)+\A \cdot (\nabla f)$
 \item $\nabla \cdot (\A \times \B)=\B \cdot(\nabla \times \A)-\A \cdot (\nabla \times \B)$
 \item $\nabla \times (f \A)=f(\nabla \times \A)-\A \times(\nabla f)$
 \item $\nabla \times (\A \times \B)=(\B \cdot \nabla)\A-(\A \cdot \nabla)\B+\A(\nabla \cdot \B)
 -\B(\nabla \cdot \A)$

 {\bf Second Derivatives}
 \item $\nabla \cdot (\nabla \times \A)=0$
 \item $\nabla \times (\nabla f)=0$
 \item $\nabla \times (\nabla \times \A)=\nabla(\nabla \cdot \A)-\nabla^2 \A$

 {\bf Vector Derivatives}

 {\bf Cartesian} 


 \item $dl=dx \, \i+dy \, \j+dz \, \k$; $d\tau=dx \, dy \, dz$

 \item Gradient:
 $$\nabla t=\d{t}{x} \i + \d{t}{y} \j + \d{t}{z} \k$$

 \item Divergence:
 $$\nabla \cdot \v=\d{v_x}{x} + \d{v_y}{y} + \d{v_z}{z}$$

 \item Curl:
 $$\nabla \times \mathbf{v}=\left(\d{v_z}{y}-\d{v_y}{z}\right) \i
 + \left(\d{v_x}{\z}-\d{v_z}{x}\right) \j
 + \left(\d{v_y}{x}-\d{v_x}{y}\right) \k$$

 \item Laplacian:
 $$\nabla^2 t=\d{^2 t}{x^2}+\d{^2 t}{y^2}+\d{^2 t}{z^2}$$



 {\bf Spherical} 


 \item $dl=d\rho \, \rh + r\, d\phi \, \ph + \rho \sin \phi \, d\theta \, \th$; $d\tau=\rho^2 \sin \phi \, d\rho \, d\theta \, d\phi$

 \item Gradient:
 $$\nabla t=\d{t}{\rho} \rh +\frac{1}{\rho} \d{t}{\phi} \ph
 +\frac{1}{\rho \sin \phi} \d{t}{\theta} \th$$

 \item Divergence:
 $$\nabla \cdot \v=\frac{1}{\rho^2} \dd{\rho} (\rho^2 v_\rho) 
 + \frac{1}{\rho \sin \phi} \dd{\phi} (v_\phi \sin \phi)
 + \frac{1}{\rho \sin \phi} \d{v_\theta}{\theta}$$

 \item Curl:
 $$\nabla \times \v=\frac{1}{\rho^2} \left[\dd{\phi}(\sin \phi v_\theta)-\d{v_\phi}{\theta}\right] \rh
 + \frac{1}{\rho}\left[\frac{1}{\sin \phi} \d{v_\rho}{\theta}-\dd{\rho}(\rho v_\theta) \right] \ph
 + \frac{1}{\rho}\left[\dd{\rho}(\rho v_\phi)-\d{v_\rho}{\phi}\right] \th$$

 \item Laplacian:
 $$\nabla^2 t=\frac{1}{\rho^2} \dd{\rho}\left(\rho^2 \d{t}{\rho}\right) 
 + \frac{1}{\rho^2 \sin \phi} \dd{\phi}\left(\sin \phi \d{t}{\phi}\right)
 + \frac{1}{\rho^2 \sin^2 \phi} \d{^2 t}{\theta^2}$$


 {\bf Cylindrical}


 \item $dl=dr \, \r + r \, d\theta \, \th + dz \z$; $d\tau=r \, dr \, d\theta \, dz$

 \item Gradient:
 $$\nabla t=\d{t}{r} \r + \frac{1}{r} \d{t}{\theta} \th + \d{t}{z} \z$$

 \item Divergence:
 $$\nabla \cdot \v=\frac{1}{r} \dd{r} (rv_r)+\frac{1}{r} \d{v_\theta}{\theta}+\d{v_z}{z}$$

 \item Curl:
 $$\nabla \times \v=\left[\frac{1}{r} \d{v_z}{\theta}-\d{v_\theta}{z}\right]\r
 + \left[\d{v_r}{z}-\d{v_z}{r}\right] \th+\frac{1}{r}\left[\dd{r}(rv_\theta)-\d{v_r}{\theta}\right] \z$$

 \item Laplacian:
 $$\nabla^2 t=\frac{1}{r}\dd{r}\left(r\d{t}{r}\right)+\frac{1}{r^2}\d{^2 t}{\theta^2}+\d{^2 t}{z^2}$$


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Last edited September 30, 2002 19:05 (diff)