Problems

In Problems 1-12, find the line integral by Green's Theorem.

In Problems 13-18, find (a) curl F, (b) (c) div F, (d)

19            Use Green's Theorem to find the area inside the curve r = a + cos θ, (α ≥ 1).

20            Use Green's Theorem to find the area inside the ellipse x²/a² + y²/b² = 1 and above the line y = c (0 < c < b).

21             Show that if D has a piecewise smooth boundary, the area of D is

22            Show that for any continuous function f(t) and constants a, b, c, where D is the circle x² + y² ≤ c².

23            Find the value of the line integral

where D is a region with area A.

24            Show that any vector field of the form F(x,y) = xf(x² + y²)i + yf(x² + y²)j is irrotational.

25            Show that any vector field of the form F(x, y) = y f(x² + y²)i - xf(x² + y²)j is incompressible.

26            Show that any vector field of the form F(x,y) = f(x)i + g(y)j is irrotational.