Line integral along a plane and sphere intersection, with Stokes and without
In this exercise, we evaluate the line integral y dx + z dy + x dz along the curve of intersection of the two surfaces x^2 + y^2 + z^2 = 1 and x+z=1. I will show you three solutions, each with a different setup.
Method 1: Translate the variables x and z to u and v. This transformation makes the circle easier to parameterize. We compute the vector line integral directly.
Method 2 (best): Use Stokes' theorem in the new coordinate system. This method is "parametrization free" because the curl is constant and the domain has a nice geometric area that we can recognize. Btw, when I parametrize the surface as r(u,v) (the demo to get to the normal vector form), that u is unrelated to my shifted variable u. Sorry!
Method 3: Solve directly in the original coordinates. We parameterize the curve in x, y, z and use Stokes' theorem. You don't need to write down the parametric curve r(t) at the end (you can deduce what the radius is from the line above), but a student asked me about parametrization, so I included it. Enjoy!
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