- #1
yungman
- 5,723
- 242
Let ##\vec {F}(\vec {r}')## be a vector function of position vector ##\vec {r}'=\hat x x'+\hat y y'+\hat z z'##. I want to find ##\nabla\cdot\frac {\vec {F}(\vec {r}')}{|\vec {r}-\vec{r}'|}##.
My attempt:
Let ##\vec {r}=\hat x x+\hat y y+\hat z z##. Since ##\nabla## work on ##x,y,z##, not ##x',y',z##'
[tex]\Rightarrow\;\nabla\cdot\vec {F}(\vec{r}')=\nabla\times\vec {F}(\vec{r}')=0[/tex]
Then ##\frac {1}{|\vec {r}-\vec{r}'|}## is not even a vector. How can I perform ##\nabla\cdot\frac {\vec {F}(\vec {r}')}{|\vec {r}-\vec{r}'|}##?
BUT
according to http://faculty.uml.edu/cbaird/95.657%282012%29/Helmholtz_Decomposition.pdf. It uses:
[tex]\nabla\cdot (\Phi\vec A)=\vec A\cdot\nabla\Phi+\Phi\nabla\cdot\vec A\;\hbox { Where }\;\Phi=\frac{1}{|\vec {r}-\vec{r}'|}\;\hbox {and }\;\vec A=\vec {F}(\vec {r}')[/tex]
You can see this from the bottom of page 2 to the top of page 3. I cannot agree with this, ##\vec A=\vec {F}(\vec {r}')## is function of ##x',y',z'##, it's a constant respect to ##x,y,z## as indicated above.
Even the Wikipadia use the same method:http://en.wikipedia.org/wiki/Helmholtz_decomposition. You can see the 5th equation under "Proof".
Please tell me what's going on. Thanks
My attempt:
Let ##\vec {r}=\hat x x+\hat y y+\hat z z##. Since ##\nabla## work on ##x,y,z##, not ##x',y',z##'
[tex]\Rightarrow\;\nabla\cdot\vec {F}(\vec{r}')=\nabla\times\vec {F}(\vec{r}')=0[/tex]
Then ##\frac {1}{|\vec {r}-\vec{r}'|}## is not even a vector. How can I perform ##\nabla\cdot\frac {\vec {F}(\vec {r}')}{|\vec {r}-\vec{r}'|}##?
BUT
according to http://faculty.uml.edu/cbaird/95.657%282012%29/Helmholtz_Decomposition.pdf. It uses:
[tex]\nabla\cdot (\Phi\vec A)=\vec A\cdot\nabla\Phi+\Phi\nabla\cdot\vec A\;\hbox { Where }\;\Phi=\frac{1}{|\vec {r}-\vec{r}'|}\;\hbox {and }\;\vec A=\vec {F}(\vec {r}')[/tex]
You can see this from the bottom of page 2 to the top of page 3. I cannot agree with this, ##\vec A=\vec {F}(\vec {r}')## is function of ##x',y',z'##, it's a constant respect to ##x,y,z## as indicated above.
Even the Wikipadia use the same method:http://en.wikipedia.org/wiki/Helmholtz_decomposition. You can see the 5th equation under "Proof".
Please tell me what's going on. Thanks
Last edited: