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Homework Statement
Let S be the part of the plane z=f(x,y)=4x - 8y +5 above the region (x-1)^2 + (y-3)^2 <= 9 oriented with an upward pointing normal. Use Stoke's theorem to evaluate the surface integral for the vector field <2z, x, 1>.
Homework Equations
Stoke's Theorem is surface integral of Fdot product ds = surface integral curl(F) dot product dS
The Attempt at a Solution
I figured out the curl: <0,2,1>. I am having trouble with finding the limits of integration for the normal to the surface. I see that they are asking for the surface that lies on the plane z=4x-8y+5 above the xy plane. Great.
the normal to the surface is easy as well : <-4, 8, 1>. here is where i hit a stop.
reason 1: there are no variables in either of the two vectors and this flags an alarm because I feel I did something wrong.
reason 2: I cannot figure out the limits of integration to the surface. I considered polar/spherical coordinates but since the boundary on the plane is an ellipse, these do not work (unless there is a way to make them work, in which case I do not know what that is).
One thing i considered doing was reasoning that since stoke's theorem depends solely on the boundary, I said "well we basically have a boundary of half an ellipse in the xy plane". So I solve 4x-8y+5=0 for x or y, plug that into the equation for the original circle with radius 3, and solve for the extreme values of x and y. This way I can come up with the equation for an ellipse and set one of the limits equal to the extreme values and the other limit to the upper half of the ellipse... Except that sounds way too complicated for the average problem in my class and I feel that I must have missed something.
Any guidance would be much appreciated. Thanks!