Reference no: EM131043581
Given line integral ∫C F·dr = a∫bF (r(t))· r'(t) dt and
Surface integral ∫∫SF· n dS = ∫∫DF (r(t)) · (ru x rv) dA.
Exercise 1- Assume the surface to be z = 1 - x2 - y2. Find the tangent plane at the point (x, y) = (1/2, 1/2) by using
1. gradient property approach. (Hint: that is using ∇F, where F is a function of three variables.)
2. the normal vector came from the surface parametrization S = r(x, y).
Exercise 2- Let vector field to be F(x, y, z) = (0, 0, -1 + z) and the surface to be lower-semi-unit sphere. Find the surface integral ∫∫SF·n dS with positive orientation.
(Hint: be very careful about the angle Φ and the surface orientation rΦ x rθ)
Exercise 3- Let vector field F~(x, y, z) = (0, 0, -2).
1. Find the surface integral of F~ when the surface is z = 4 - x2 - y2 above the xy-plane.
2. Find the surface integral of F~ when the surface is z = 8 - 2x2 - 2y2 above the xy-plane.
3. From part 1 and part 2, do you see surface independence?
4. Find the line integral of F(x, y, z) = (y, -x, 0) along the curve which is the closed circle x2 ± y2 = 4 in R3.
5. From part 1, 2 and 4, do you see Stokes' theorem hold? Explain.
Exercise 4- Let vector field F(x, y, z) = (0, 0, -1 + z) . Consider the surface S = S1 + S2 where S1 is the lower-semi-unit sphere and S2 is the unit circle on the xy-plane.
1. Find 
directly. You can cite results from Exercise 2.
2. Find
by using Divergence Theorem.
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