Question

For each vector field F~ (x, y) = hP(x, y), Q(x, y)i, find a function f(x,...

For each vector field F~ (x, y) = hP(x, y), Q(x, y)i, find a function f(x, y) such that F~ (x, y) = ∇f(x, y) = h ∂f ∂x , ∂f ∂y i by integrating P and Q with respect to the appropriate variables and combining answers. Then use that potential function to directly calculate the given line integral (via the Fundamental Theorem of Line Integrals):

a) F~ 1(x, y) = hx 2 , y2 i Z C F~ 1 · d~r C is the line segment from (1, 0) to (3, −2).

b) F~ 2(x, y) = h2xy, x2 + 2yi Z C F~ 2 · d~r C is the parabola ~r(t) = ht 2 + 2, −3ti, 0 ≤ t ≤ 1.

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