- #1
silverwhale
- 84
- 2
I am wroking through an electrodynamics textbook and there is this Taylor expansion to do later a multipole expansion. But I can't figure out how the author does it. Please any help?
the expansion:
[tex] \frac{1}{|\vec{r}-\vec{r'}|} = \frac{1}{r} - \sum^3_{i=1} x'_i \frac{\partial}{\partial x_i} \frac{1}{r} + \frac{1}{2} \sum^3_{i,j=1} x'_i x'_j \frac{\partial}{\partial x_i} \frac{\partial}{\partial x_j}\frac{1}{r} + \mathellipsis [/tex]
And he writes "the occurring differenciations were changed using:"
[tex] (\frac{\partial}{\partial x'_i} \frac{1}{|\vec{r}-\vec{r'}|})_{r'=0} = - (\frac{\partial}{\partial x_i} \frac{1}{|\vec{r}-\vec{r'}|})_{r'=0} = - \frac{\partial}{\partial x_i} \frac{1}{r} [/tex]
I just can't follow his argument..
the expansion:
[tex] \frac{1}{|\vec{r}-\vec{r'}|} = \frac{1}{r} - \sum^3_{i=1} x'_i \frac{\partial}{\partial x_i} \frac{1}{r} + \frac{1}{2} \sum^3_{i,j=1} x'_i x'_j \frac{\partial}{\partial x_i} \frac{\partial}{\partial x_j}\frac{1}{r} + \mathellipsis [/tex]
And he writes "the occurring differenciations were changed using:"
[tex] (\frac{\partial}{\partial x'_i} \frac{1}{|\vec{r}-\vec{r'}|})_{r'=0} = - (\frac{\partial}{\partial x_i} \frac{1}{|\vec{r}-\vec{r'}|})_{r'=0} = - \frac{\partial}{\partial x_i} \frac{1}{r} [/tex]
I just can't follow his argument..