1. (10) Let F(x, y, z) = < 7x + arctan(yz), 3ln (x² + 1) + 4y − √√z, cos(x) + 3ey² – 2z>, and let S be the tetrahedron w

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1 10 Let F X Y Z 7x Arctan Yz 3ln X 1 4y Z Cos X 3ey 2z And Let S Be The Tetrahedron W 1 (126.76 KiB) Viewed 44 times

1. (10) Let F(x, y, z) = < 7x + arctan(yz), 3ln (x² + 1) + 4y − √√z, cos(x) + 3ey² – 2z>, and let S be the tetrahedron with its corners at (0, 0, 0), (2, 0, 0), (0, 2, 0), and (0, 0, 2). We want to calculate the flux of F out of S. The trouble is that this would require doing 4 (impossible) surface integrals, which would not be fun. So, you need to apply the divergence theorem (and be happy this isn't "verify"). a) Find the equation of the plane that is the "top" of the tetrahedron, in the form z=f(x, y). (Hints: You're finding the equation of a plane! Not parametric equations. Plain old equation. "Corners" makes this easy.) b) c) d) Sketch and label the "base" of the tetrahedron (in the xy-plane). (Hint: It's a triangle!) Use (a) and (b) to write down the inequalities in x, y, and z which determine S. Compute div(F). (This should be a nice function, leading to a nice integral.) e) Compute the correct volume integral, so that you know (using the divergence theorem) the desired flux. (Your answer should be a single, simplified, integer. This has been rigged to be nice.)