M2313-07 calculus three homework let vector field fx y z

Question

Given line integral ∫_C F·dr = _a∫^bF (r(t))· r'(t) dt and

Surface integral ∫∫_SF· n dS = ∫∫_DF (r(t)) · (r_u x r_v) 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 r_Φ x r_θ)

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. Findby using Divergence Theorem.