Two methods of integration for Divergence Thm

[mooeypoo]

Hey again,

 

Okay, I have a hw problem that asks for a solution to an integral using 2 methods. I've solved them both but got two slightly different answers, and I can't figure out where I went wrong.... help!

 

Estimate the integral

[math]I=\int e^{-ar} \left(\bigtriangledown \cdot \frac{\hat{r}}{r^2} \right)d^3r [/math]

(over a sphere with radius R, centered on the origin)

 

by two different methods:

(a) carrying out the divergence operation

(b) integrating by parts.

So here are my solutions, please help me out, it's all VERY new and I had to go through the book about 3 times before I could understand what it is they're doing. I thought I did well, but.. well, apparently I didn't, since the answers don't match.

 

(a)

[math]I=\int e^{-ar} \left(\bigtriangledown \cdot \frac{\hat{r}}{r^2} \right)d^3r = \int e^{-ar} 4\pi\delta^3® d^3r

[/math]

Since we integrate over all space, r is in the domain of the integration, the delta function is equal to 1, and we solve for f(a) which is f(0), because the function is delta (r-0).

 

So:

 

[math]I=\int e^{-ar} \left(\bigtriangledown \cdot \frac{\hat{r}}{r^2} \right)d^3r = e^{-a*0}*4\pi = 4\pi[/math]

 

 

(b)

 

[math]I=\int e^{-ar} \left(\bigtriangledown \cdot \frac{\hat{r}}{r^2} \right)d^3r [/math]

 

We know that:

 

[math]

\int f(\bigtriangledown \cdot A) d\tau = -\int_{V} A \cdot (\bigtriangledown f) d\tau + \oint_{S} fA \cdot da

[/math]

 

Where

 

[math]

f(\bigtriangledown \cdot A) = e^{-ar} \left(\bigtriangledown \cdot \frac{\hat{r}}{r^2} \right)

[/math]

 

So:

 

[math]

\int e^{-ar} \left( \bigtriangledown \cdot \frac{\hat{r}}{r^2} \right) d^3r =[/math]

 

[math] - \int_{V} \left( \frac{\hat{r}}{r^2} \cdot \left( \bigtriangledown e^{-ar} \right) \right) d \tau + \oint_{S} \left( \frac{e^{-ar}}{r^2}\hat{r} \right) \cdot da =

[/math]

 

[math]

- \int_{V} \left( \frac{1}{r^2}\hat{r} \right) \cdot \left( e^{-ar}\hat{r} \right) d\tau + \oint_{S} \left( \frac{e^{-ar}}{r^2} \right) \cdot da=

[/math]

 

[math]

- \int_{V} \left( \frac{e^{-ar}}{r^2} \right) r^2 \sin{\theta} dr d\theta d\phi + \oint_{S} \left( \frac{e^{-ar}}{r^2} \right) r^2 \sin{\theta} dr d\theta d\phi =

[/math]

 

[math]

\int^{R}_{0} e^{-ar}dr \int^{\pi}_{0} \sin{\theta} d\theta \int^{2\pi}_{0} d\phi + \oint_{R} e^{-ar} \int^{pi}_{0} \sin{\theta} d\theta\int^{2\pi}_{0} \phi d\phi =

[/math]

 

[math]

\left( \frac{e^{-ar}}{-a} \right) |^{R}_{0} (-\cos{\theta})|^{\pi}_{0}\phi |^{2\pi}_{0} + \left( \frac{e^{-aR}}{-a} \right) (- \cos{\theta}) |^{\pi}_{0}\phi |^{2\pi}_{0}=

[/math]

 

[math]

\left( \dfrac{1-e^{-aR}}{-a} \right) 2*2\pi + \left( \dfrac{e^{-aR}}{-a} \right)2*2\pi

[/math]

 

[math]

= - \dfrac{4\pi}{a}

[/math]

 

Meh!!!

 

Help? What did I do wrong? Is my method even right????

 

help help I'm so confused...

 

~moo