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Problem.Let D be a disc and let P and Q be functions on D with continuous partial derivatives with respect to x and y. Let C be any closed curve in D.

Given the fact that

[tex]\int_C (P \, dx + Q \, dy) = 0 \leftrightarrow Q_x = P_y[/tex]

show that P dx + Q dy is an exact differential. (Hint: Let (a, b) be any point in D and set

[tex]g(x, y) = \int_a^b P(t, b) \, dt + \int_b^y Q(a, s) \, ds[/tex]

Show that g is the desired function.

Attempt.P dx + Q dy is an exact differential if there is a function g(x, y) with [itex]g_x = P[/itex] and [itex]g_y = Q[/itex]. So if I take the partial derivative with respect to x of the function g defined in the hint, I should get P right?

Let p be the anti-derivative of P with respect to the first variable and let q be the anti-derivative of Q with respect to the second variable. Then by the fundamental theorem of calculus

g(x, y) = p(x, b) - p(a, b) + q(a, y) - q(a, b)

and so [itex]g_x(x, y) = p_x(x, b) = P(x, b)[/tex] right? The problem here is that there's a b instead of a y. I'm mostly likely doing something terrible wrong here because I haven't used the fact stated in the problem. Help!

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# Homework Help: Proving a Fact About Exact Differentials

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