Multi-Variable Calculus (Green's Thm, Stoke's Thm)
Hey everyone, I just had a few questions I ran into studying and need help!
Here they are:
1: Using spherical coordinates, find parametric equations for the part of the surface x^2 + y^2 + z^2 = 4 between the planes z = 0 and z = 1
2: Compute the surface area of the surface T described by r[vector] (u,v) = <u^2, u*sin(v), u*cos(v)> and 1<=u<=2 and 0<=v<=pi
3: Let F(x,y,z) = <y, xz, sin(xyz)> and let T be the cone z = sqrt(x^2 + y^2) between the planes z = 0 and z = 3. Evaluate ∫∫_T curl F *(dot) dS
4: Evaluate ∫F*(dot) dr where F(x,y,z) = <xy, yz, zx> and T is the surface z = xe^y, 0<=x<=1, 0<=y<=1 with downward orientation.
5: Let F(x,y,z) = <z, 5x, e^z>, and let C be the curve given by the parametric equation r(t) = <cos(t), sin(t), 3 - cos(t) + 2sin(t)>, 0 <=t<= 2pi. Use Stokes' Thm to evaluate ∫_c F*(dot) dr. (Hint. The curve C lies in the plane z = 3 - x + 2y)
Thanks and by no means do all of these need answering...as much would be nice...the most important thing for me is that I understand them so a step by step explanation would be so much more helpful as opposed to a "it is (answer) by using (blah formula)" answer.