- #1
SiggyYo
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Homework Statement
Given a vector field [itex]\textbf{F}[/itex] and a composite (with this I mean cuboids, cylinders, etc. and not spheres for example) surface [itex]S[/itex], how do I calculate the flux through only some of the sides of [itex]S[/itex]? I am interested in a general way to do this, but right now I am struggling with the following:
Let [itex]\textbf{F}=\frac{xz}{x^2+y^2} \textbf{i}+\frac{yz}{x^2+y^2}\textbf{j}+ze^{x^2+y^2}\sqrt{x^2+y^2}\textbf{k},[/itex] defined on [itex]\mathbb{R}^3[/itex]
and [itex]E\subset\mathbb{R}^3[/itex] be the solid body given by
[itex]\sqrt{x^2+y^2}\leq z\leq2,\ x^2+y^2\geq 1.[/itex] (This is a sort of hollow stumped cone)
a) A part of [itex]E[/itex]'s edge is the cylinder part [itex]S_0[/itex] given by
[itex]x^2+y^2=1, 1\leq z\leq 2[/itex]
Show that the flux of [itex]F[/itex] inward through [itex]S_0[/itex] is [itex]-3\pi.[/itex]
Homework Equations
The divergence theorem.
The Attempt at a Solution
[itex]div{\textbf{F}}=e^{x^2+y^2}\sqrt{x^2+y^2}-2z\frac{x^2-y^2}{x^2+y^2},[/itex]
and with [itex]x=\cos{\theta}[/itex] and [itex]y=\sin{\theta}[/itex] we get
[itex]div{\textbf{F}}=e-2z(1-\sin^2{\theta}).[/itex] Our triple integral then becomes
[itex]\int_1^2\int_0^1\int_0^{2\pi}re-2rz(1-\sin^2{\theta})\,d\theta\,dr\,dz=2e\pi^2-\frac{3}{2}\pi[/itex].
And that is how far I've gotten. Assuming all of this is correct, my problem now is that this is the flux through the entire cylinder, along with its top and bottom. How to I calculate the flux through these discs, so that I can subtract them from the total flux?
Thanks in advance.
SiggyYo