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frank0520
 one year ago
Use the Divergence Theorem to calculate the flux of
F(x,y,z)=(x^5 +y^5 +z^5 2x3y4z)i+sin(2y)j+4z(sin(y))^2 k
across the surface of the tetrahedron bounded by the coordinate planes and the plane
x+y+z=1.
frank0520
 one year ago
Use the Divergence Theorem to calculate the flux of F(x,y,z)=(x^5 +y^5 +z^5 2x3y4z)i+sin(2y)j+4z(sin(y))^2 k across the surface of the tetrahedron bounded by the coordinate planes and the plane x+y+z=1.

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ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0dw:1436838723223:dw

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0\[\iint\limits_{S}\vec{F}\bullet d\vec{S}~=~\iiint \limits_{E}\text{div}(\vec{F}) dV \]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0work the divergence and plug it in

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0we can write this as a volume integral over \(E=\{(x,y,z)\in\mathbb{R}^3:x+y+z\le 1, x\ge 0, y\ge0, z\ge0\}\) using the divergence theorem as ganeshie8 pointed out, which is something like: $$\iint\limits_S F\cdot dS=\iiint\limits_E\nabla\cdot F\ dV$$

frank0520
 one year ago
Best ResponseYou've already chosen the best response.0\[\int\limits_{0}^{1}\int\limits_{0}^{1x}\int\limits_{1}^{1xy}dzdydx\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0now, using Cartesian coordinates so \(dV=dz\ dy\ dx\) $$\int_0^1\int_0^{1x}\int_0^{1xy}\nabla\cdot F\ dz\ dy\ dx$$ clearly \(\nabla\cdot F=(5x^42) +2\cos(2y)+4\sin^2(y)\) so: $$\int_0^1\int_0^{1x}\int_0^{1xy}\left(5x^42+2\cos(2y)+4\sin^2(y)\right)\,dz\,dy\,dx$$

frank0520
 one year ago
Best ResponseYou've already chosen the best response.0can you tell me if I have the correct limits

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0no, \(0\le z\le 1xy\)

frank0520
 one year ago
Best ResponseYou've already chosen the best response.0Thanks for the help, I had a typo with the dz limit

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0At this point, you may use wolfram to evaluate the triple integral for you http://www.wolframalpha.com/input/?i=%5Cint_0%5E1%5Cint_0%5E%7B1x%7D%5Cint_0%5E%7B1xy%7D%5Cleft%285x%5E42%2B2%5Ccos%282y%29%2B4%5Csin%5E2%28y%29%5Cright%29%5C%2Cdz%5C%2Cdy%5C%2Cdx
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