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My question is: How can I solve the following problem?

$C$ is the curve resulting from the intersection of $x+y+z=0$ with $x^2+y^2+z^2=4$, parameterized counterclockwise as seen from the positive $x$-direction.

Evaluate $\int { ydx + zdy + xdz }$

Thanks.

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Have you drawn the graph of both equations to determine $C$? That's the first step. If you already have graphed the equations, could you be more specific about where you're struggling? The more specific you can be about what you've done, where you're stuck, etc., the more helpful we can be. It's also a good idea, when posting homework on this site, to include a sketch of what you've already done: e.g., if you've graphed the equations, can you determine the region(s) over which you need to integrate? How have you approached similar problems in the past, etc. –  amWhy May 23 '11 at 1:00
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1 Answer

Hint:

You have the vector field $F(x,y,z)=(y,z,x)$, and you want to evaluate $$\int_C F\cdot ds.$$ What is the Curl of your vector field? Notice that it must be a constant since $F$ is linear. Let $K=\nabla \times F$ denote this constant.

Now, apply Stokes Theorem. What is the area of the enclosed curve? (Hint: it is a circle of radius $2$). If we call that area $A$, then the line integral is equal to $$K A.$$

Hope that helps,

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