Questions from General Calculus

Q: Find (a) the curl and (b) the divergence

Find (a) the curl and (b) the divergence of the vector field. F (x, y, z) = 〈arctan (xy), arctan (yz), arctan (zx)〉

Q: Use Green’s Theorem to evaluate the line integral along the given positively

Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. ∫C y3 dx - x3 dy, C is the circle x2 + y2 = 4

Q: Evaluate the surface integral. ∫∫S x dS, S

Evaluate the surface integral. ∫∫S x dS, S is the surface y = x2 + 4z, 0 < x < 1, 0 < z < 1

Q: Find a parametric representation for the surface. The plane that

Find a parametric representation for the surface. The plane that passes through the point (0, -1, 5) and contains the vectors 〈2, 1, 4〉 and 〈-3, 2, 5〉

Q: Find (a) the curl and (b) the divergence

Find (a) the curl and (b) the divergence of the vector field. F (x, y, z) = xyez i + yzex k

Q: Suppose that f (x, y, z) = g

Suppose that f (x, y, z) = g (√x2 + y2 + z2), where t is a function of one variable such that g (2) = 25. Evaluate ∫∫S f (x, y, z) dS, where S is the sphere x2 + y2 + z2 = 4.

Q: State Stokes’ Theorem.

State Stokes’ Theorem.

Q: Evaluate the surface integral. ∫∫S xyz dS, S

Evaluate the surface integral. ∫∫S xyz dS, S is the cone with parametric equations x = u cos v, y = u sin v, z = u, 0 < u < 1, 0 < v < π/2

Q: Evaluate the surface integral. ∫∫S y dS, S

Evaluate the surface integral. ∫∫S y dS, S is the helicoid with vector equation r (u, v) =〈u cos v, u sin v, v〉, 0 < u < 1, 0 < v < π

Q: Evaluate the surface integral. ∫∫S (x2 + y2

Evaluate the surface integral. ∫∫S (x2 + y2) dS, S is the surface with vector equation r (u, v) =〈2uv, u2 - v2, u2 + v2〉, u2 + v2 < 1