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Vector Calculus Independent Study

Unit 8 Sample Test

1. [25 points] Let R be a region in the plane with area A, and let ∂R be R’s boundary. Use Green’s theorem to show that the center of mass (x̄, ȳ) of R has coordinates

x̄ = \frac{1}{2A} ∫_{C} x^{2} dy

and

ȳ = \frac{1}{2A} ∫_{C} y^{2} dy

2. [25 points] Suppose

(a) ∇ · F⃗ = 0 everywhere except at (1, 0, 0) and (3, 0, 0),

(b) ∫∫ F⃗ ·dS⃗ = 5 over the sphere x^{2}+y^{2}+z^{2} = 4 oriented with outward pointing normal, and

(c) ∫∫ F⃗ · dS⃗ = 7 over the sphere x^{2} + y^{2} + z^{2} = 16 oriented with outward pointing normal.

Use Gauss’ Theorem to determine all the other possible values of

∫ ∫ F⃗ · dS⃗

evaluated over spheres (not necessarily centered at the origin) with outward pointing normals.

3. [25 points] Calculate the surface integral

∫_{S}∫ (∇ × F⃗) · dS⃗

where S is the hemisphere x^{2} + y^{2} + z^{2} = 1, x ≤ 0 and

F⃗ (x, y, z) = (x^{3}, −y^{3}, 0).

4. [25 points] Prove that the work done by a particle moving along a closed path against a constant force field F⃗ (x, y, z) = v⃗ is 0.