$Evaluate the surface integral $\iint_{S}F \cdot dn$ over the given surface $S$$

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$Evaluate the surface integral $\iint_{S}F \cdot dn$ over the given surface $S$$

Evaluate $\iint_{S}F \cdot dn$ where $F=(x^2,0,3y^2)$ and $S$ is the potion of the plane $x+y+z=1$ in the first octant.

Multivariable Calculus Integrals Stokes' Theorem

Elvin

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The surface $S$ projected to $xy$ plane is $T=\{(x,y): 0\leq x \leq 1, 0\leq y \leq 1-x\}.$ Also the normal to the surface $S$ with $z=1-x-y$ is $n=(1,1,1)/\sqrt{3}$. note that the surface $S$ is given by $g(x,y,z)=x+y+z=1$, and $|\nabla g|=|(1,1,1,)|=\sqrt{3}$.

So
\[\iint_S F\cdot n=\iint F \cdot n \sqrt{3}dx dy=\iint_T x^2+3y^2 dx dy=\int_0^1\int_0^{1-x}x^2+3y^2dydx\]
\[=\int_0^1 (x^2y+y^3)|_0^{1-x}dx=\int_0^1 x^2(1-x)+(1-x)^3=\int_0^1 (1-3x+4x^2-2x^3)=\frac{1}{3}.\]

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$Evaluate the surface integral $\iint_{S}F \cdot dn$ over the given surface $S$$

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