Correction of Vector Operator computations

[Bibhatsu]

Homework Statement

A. For the vector $$\vec{A}=a\vec{e}_{x}+b\vec{e}_{y}+c\vec{e}_{z}$$, compute $$div(\vec{A})$$ and $$rot(\vec{A})$$

B. Show that $$rot(grad(\phi (x,y,z)))=0$$ for a potential function $$\phi(x,y,z)$$

C. Compute $$div \ grad \frac{1}{r}\equiv \vec{\nabla} \cdot \vec{\nabla} \frac{1}{r}\equiv \Delta \frac{1}{r}$$

The Attempt at a Solution

A. $$div(\vec{A})= \vec{\nabla}}\vec{A}=\vec{\nabla}\begin{pmatrix}{a\\b \\c} \end{pmatrix} =\begin{pmatrix} \frac{\delta}{\delta x}\\ \frac{\delta}{\delta y} \\ \frac{\delta}{\delta z} \end{pmatrix} \begin{pmatrix}a\\b\\c \end{pmatrix}=\frac{\delta}{\delta x}a+ \frac{\delta}{\delta y}b + \frac{\delta}{\delta z}c= 0+0+0= 0$$

$$rot(\vec{A})=\vec{\nabla}\times \vec{A}=\vec{\nabla}\times \begin{pmatrix}a\\b\\c \end{pmatrix}= \begin{pmatrix}\frac{\delta}{\delta x}\\ \frac{\delta}{\delta y} \\ \frac{\delta}{\delta z} \end{pmatrix} \times \begin{pmatrix}a\\b\\c \end{pmatrix}=\begin{pmatrix}\frac{\delta}{\delta y}c-\frac{\delta}{\delta z}b \\ \frac{\delta}{\delta z}a-\frac{\delta}{\delta x}c \\ \frac{\delta}{\delta x}b-\frac{\delta}{\delta y}a \end{pmatrix}= \begin{pmatrix} 0\\ 0\\ 0 \end{pmatrix}$$

B. $$rot(grad(\phi)))=\vec{\nabla}\times \begin{pmatrix}\frac{\delta \phi}{\delta x} \\ \frac{\delta \phi}{\delta y} \\ \frac{\delta \phi}{\delta z} \end{pmatrix}= \begin{pmatrix}\frac{\delta}{\delta x}\\ \frac{\delta}{\delta y} \\ \frac{\delta}{\delta z} \end{pmatrix} \times \begin{pmatrix}\frac{\delta \phi}{\delta x} \\ \frac{\delta \phi}{\delta y} \\ \frac{\delta \phi}{\delta z}} \end{pmatrix}=\begin{pmatrix}\frac{\delta}{\delta y} \frac{\delta \phi}{\delta z}- \frac{\delta}{\delta z} \frac{\delta \phi}{\delta y} \\ \frac{\delta}{\delta z}\frac{\delta \phi}{\delta x} - \frac{\delta}{\delta x} \frac{\delta \phi}{\delta z} \\ \frac{\delta}{\delta x} \frac{\delta \phi}{\delta \delta y}- \frac{\delta}{\delta y} \frac{\delta \phi}{\delta x} \end{pmatrix} =\begin{pmatrix}0\\ 0 \\ 0 \end{pmatrix}$$

C. $$\operatorname{div} \operatorname{grad}\frac{1}{r}= \nabla \nabla \frac{1}{r}= \begin{pmatrix} \frac{\delta}{\delta x} \\ \frac{\delta}{\delta y} \\ \frac{\delta}{\delta z} \end{pmatrix} \begin{pmatrix}\frac{-x}{r^{3}}\\ \frac{-y}{r^{3}}\\ \frac{-z}{r^{3}} \end{pmatrix}= \frac{\delta}{\delta x}\frac{-x}{r^{3}}+ \frac{\delta}{\delta y}\frac{-y}{r^{3}} + \frac{\delta}{\delta z}\frac{-z}{r^{3}} = \frac{2x^{2}-y^{2}-z^{2}}{r^{5}}+\frac{2y^{2}-x^{2}-z^{2}}{r^{5}}+\frac{2z^{2}-x^{2}-y^{2}}{r^{5}}= 0$$I would be very grateful for corrections and comments. Thank you.Bibhatsu

 

[_coyote_]

:A. div(\vec{A})= \vec{\nabla}}\vec{A}=\vec{\nabla}\begin{pmatrix}{a\\b \\c} \end{pmatrix} =\begin{pmatrix} \frac{\partial}{\partial x}\\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} \begin{pmatrix}a\\b\\c \end{pmatrix}=\frac{\partial a}{\partial x}+ \frac{\partial b}{\partial y} + \frac{\partial c}{\partial z}= 0+0+0= 0 rot(\vec{A})=\vec{\nabla}\times \vec{A}=\vec{\nabla}\times \begin{pmatrix}a\\b\\c \end{pmatrix}= \begin{pmatrix}\frac{\partial}{\partial x}\\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} \times \begin{pmatrix}a\\b\\c \end{pmatrix}=\begin{pmatrix}\frac{\partial c}{\partial y}-\frac{\partial b}{\partial z} \\ \frac{\partial a}{\partial z}-\frac{\partial c}{\partial x} \\ \frac{\partial b}{\partial x}-\frac{\partial a}{\partial y} \end{pmatrix}= \begin{pmatrix} 0\\ 0\\ 0 \end{pmatrix} B. rot(grad(\phi)))=\vec{\nabla}\times \begin{pmatrix}\frac{\partial \phi}{\partial x} \\ \frac{\partial \phi}{\partial y} \\ \frac{\partial \phi}{\partial z} \end{pmatrix}= \begin{pmatrix}\frac{\partial}{\partial x}\\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} \times \begin{pmatrix}\frac{\partial \phi}{\partial