Question

Explain or use evidence why each is true or false

If C is any smooth, closed curve then Z C e^{x^2} dx = 0

Let F(x, y) = <P(x, y), Q(x, y)> be a vector field on R 2 and let C be a closed curve formed by the unit circle. If ∂P/∂y = ∂Q/∂x , then R C F · dr = 0.

The coordinate transformation x = u² − v² , y = 2uv maps the quarter-disk S = {(u, v) : u² + v² ≤ 1, v ≥ 0, u ≥ 0} onto the half-disk R = {(x, y) : x² + y² ≤ 1, v ≥ 0}.

If x = g(u, v) and y = h(u, v) is a transformation whose Jacobian is a constant 3, then it must map the square [0, 1] × [0, 1] to a region whose area is 1/3.

The equation in φ = π/12 in spherical coordinates describes a cone in R 3 .

Answer 1