Understanding Multivariable Calculus: Unpacking a Challenging Step

[name]

It has been a long time since i did multivariable calculus, i can't figure out how they get from the first line to the second (ignore the physics). Any ideas?

http://img517.imageshack.us/img517/5955/40978639pp5.th.jpg

Cheers,

 

[HallsofIvy]

It's a little hard to distinguish between \nabla and V!
What you are asking is how they went from
\int\int\int (\nabla\cdot E)V d\tau
to
\int\int\int E\cdot\nabla V d\tau+ \int\int V (E\cdot dA)
(some people prefer E\cdot n dA rather than E\cdot dA where "n" is the unit normal to the surface.) V here is the scalar potential and E is a vector function.

There are actually two steps in there. First they are using the "product" rule:
\nabla\cdot(VE)= V\nabla\cdot E+ (\nabla V)\cdot E[/itex]<br /> where \nabla\cdot E and \nabla\cdot (VE) are the divergence (div) of the vectors and \nabla V is grad V.<br /> <br /> so<br /> \int\int\int\nabla\cdot(VE)d\tau= \int\int\int V\nabla\cdot Ed\tau +\int\int\int(\nabla V)\cdot E d\tau[/itex]&lt;br /&gt; &lt;br /&gt; Now use the divergence theorem to convert that first integral on the right to&lt;br /&gt; \int\int VE\cdot dA[/itex]&amp;lt;br /&amp;gt; &amp;lt;br /&amp;gt; But I&amp;amp;#039;m not at all clear why the \epsilon_0/2 only multiplies the first integral!