How do you solve this difficult calculus problem?

Question

You have to let C be the path counterclockwise around the circle given by x^2 + y^2 = 16.

Then if f(x, y) = (sin x)(cosh y) and g(x, y) = (-cos x)(sinh y), how would you evaluate the integral...

∫ f(x, y) dx + g(x, y) dy
C

which of these is the correct response?

a. 4π
b. 6π
c. 8π
d. 0
e. -4π
f. -6π
g. -8π
h. none of these

firat c · Accepted Answer

Notice that in your situation, 
∂f(x,y)/∂y = ∂g(x,y)/∂x, 
and that means the line integral of yours is zero. You can see that by trying to parameterize  the integral which would take somewhat long, or by noticing that
h(x,y) = f(x,y)+ ig(x,y)
defines an analytic function of z=x+iy and using the fact that an integral of a function on a closed path in which the function is analytic, is zero.

Anonymous · Answer

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How do you solve this difficult calculus problem?

2 Answers