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Multivariable and Vector Calculus · Surface integrals, divergence theorem, and Stokes' theorem

Orient a triangular Stokes calculation

Problem

Let $S$ be the part of the plane $x+y+z=1$ in the first octant, oriented with upward-pointing normal. Let $C=\partial S$ carry the induced orientation, and let \[ \mathbf F(x,y,z)=(z^2,x^2,y^2). \] Compute $\oint_C\mathbf F\cdot d\mathbf r$ by Stokes' theorem. State the induced order of the vertices $(0,0,1)$, $(1,0,0)$, and $(0,1,0)$, derive the oriented surface element rather than using only a unit normal, and then verify the answer by integrating directly along all three edges.

Hint

The upward induced order is $(0,0,1)\to(1,0,0)\to(0,1,0)\to(0,0,1)$.

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