1.  Find div F, given that F = grad(f), where f(x,y,z) = x sqrt(z) / e^y

 

2.  Prove that the curl of the gradient of a function f(x, y, z) is always 0.

 

3.  Evaluate Int[ (x / y) ds, C] where C is defined by x = t cos(t/12), y = t sin(t/12),  0 <= t <= 4 pi.

 

4.  Sketch the curve C, sketch the level curves of f(x, y) = 2 x - 3 y for values -4, -2, 0, 2, 4, and use your sketch to estimate the value of the line integral of f(x, y) with respect to arc length along the curve y = x^2 from ( 0, 0 ) to ( 1, 1 ).

 

5.  Find the center of mass of a wire in the shape of the curve x = t, y = t^2, z = t^2 for 0 <= t <= 1 and the density delta(x,y,z) = x - z.

 

6.  Determine whether the vector field F= (2 x cos(x^2 / y) ) i – (x^2 / y^2 * cos(x^2 y) ) j is conservative, and if it is then find a scalar potential function.

 

7.  Find a function g so that g(x) F(x,y) is conservative where F(x,y) = (x^2 + y^2 + x)i + xyj.

 

8.  Use Green's Theorem to evaluate Int[ sqrt(y) dx - x^(3/2) dy, C] where C is the boundary of the rectangle 0 <= x <= 2, 0 <= y <= 1.

 

9. Using a line integral, find the area enclosed in the circle x^2 + y^2 = 9, where a is a constant. You can easily find the area of the region without using Green's Theorem, and you should compare your result with that expression, but you need to find the result using an appropriate line integral. 

 

10.  Let S be the hemisphere x^2 + y^2 + z^2 = a^2 with z >= 0. Evaluate Int[Int[(2 + z) dS,S]].

 

11. Evaluate Int[Int[ F dot N dS, S]], where F = 3 xi - y j + 2z k, S is the surface of the cube bounded by the planes x = 0, x = 1, y = 0, y = 2, z = 0, z = 3. N is the outward directed normal field.

 

12. Use Stokes' theorem to evaluate the line integral Int[2 x y dx + x y^3 dy + 3 z dz,C] where C is the intersection of the sphere x^2 + y^2 + z^2 = 25 in the plane z = 4, with the region traversed counterclockwise when viewed from the top.

 

13. Using Stokes' Theorem, evaluate Int[ F dot ds, C] where F = (x + z) i + (y + x) j + (z + y) k and C is the boundary of the triangular region with vertices (6,0,0), (0,4,0), (0,0,3) traversed counterclockwise as viewed from above.

 

14. Use the divergence theorem to evaluate the surface integral Int[Int[ F dot  N dS, S]] where F = xyz i, S is the sphere x^2 + y^2 + z^2 = 4 and N is the outward unit normal vector field.