Suppose that f(x) is differentiable on [a, b]. Let R = [a, b] [0, 1]...

 

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Suppose that f(x) is differentiable on [a, b]. Let R = [a, b] × [0, 1] = {(x, y)|a ≤ x ≤ b, 0 ≤ y ≤ 1}, and consider F = (0, f(x)) on R. Orient the boundary of R counterclockwise. (a) Find curl F. Y 1 (d) Show that, on the left edge of R, we have [F. T (b) Show that, on the top and bottom edges of R, we have fF.T ds = 0. C (c) Show that, on the right edge of R, we have [F. Tds = f(b). [F.Tds = -f(a). a (e) Apply the circulation form of Green's Theorem to F on R. What theorem do you get? b X Suppose that f(x) is differentiable on [a, b]. Let R = [a, b] × [0, 1] = {(x, y)|a ≤ x ≤ b, 0 ≤ y ≤ 1}, and consider F = (0, f(x)) on R. Orient the boundary of R counterclockwise. (a) Find curl F. Y 1 (d) Show that, on the left edge of R, we have [F. T (b) Show that, on the top and bottom edges of R, we have fF.T ds = 0. C (c) Show that, on the right edge of R, we have [F. Tds = f(b). [F.Tds = -f(a). a (e) Apply the circulation form of Green's Theorem to F on R. What theorem do you get? b X