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## Homework Statement

Please help me to check whether I did the right working for this problem. Thanks. The numerical answer is correct but I'm not very sure if the working is correct also.

Find [tex]\int y dx + z dy + x dz[/tex] over the closed curve C which is the intersection of the surfaces whose equations are [tex]x + y = 2[/tex] and [tex]x^2 + y^2 + z^2 = 2(x+y) [/tex]

## The Attempt at a Solution

First, I note that the integral required is the line integral for F . dr where F = (y, z, x). Since the curve is closed, we can apply Stokes' theorem. By Stokes' theorem [tex]\int F . dr = \iint (curl F) \cdot \b{n} dA[/tex].

Curl F = (-1,-1,-1) after applying the cross product.

Then I sketch the surfaces on the x-y axis and pick out the normal vector [tex]n = \frac{1}{\sqrt{2}}(\b{i} + \b{j})[/tex]. Also [tex]\iint dA = \pi r^2[/tex]. Then the answer for the line integral is [tex]-2\sqrt{2}\pi[/tex]