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Explain or use evidence why each is true or false If C is any smooth, closed curve...

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Basic Math

Explain or use evidence why each is true orfalse

If C is any smooth, closed curve then Z C ex^2 dx =0

Let F(x, y) = be a vector field on R 2and 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 whoseJacobian 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 acone in R 3 .

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