Question

Evaluate the surface integral s f · ds for the given vector field f and the oriented surface s. in other words, find the flux of f across s. for closed surfaces, use the positive (outward orientation. f(x, y, z = x i ? z j y k s is the part of the sphere x2 y2 z2 = 16 in the first octant, with orientation toward the origin

Answer 1

I'm reading the vector field as


`\mathbf F(x,y,z)=x\,\mathbf i-z\,\mathbf j+y\,\mathbf k`

The surface
`S` can be parameterized by


`\mathbf s(\theta,\varphi)=4\cos\theta\sin\varphi\,\mathbf i+4\sin\theta\sin\varphi\,\mathbf j+4\cos\varphi\,\mathbf k`

with
`0\le\theta\le\frac\pi2` and
`0\le\varphi\le\frac\pi2`.

Then the surface integral is given by


`\displaystyle\iint_S(\mathbf F\cdot\mathbf n)\,\mathrm dS=\iint_S\left(\mathbf F(\mathbf s(\theta,\varphi))\cdot(\mathbf s_\theta*\mathbf s_\varphi)\right)\,\mathrm dS`

`\displaystyle-64\int_(\theta=0)^(\theta=\pi/2)\int_(\varphi=0)^(\varphi=\pi/2)(2\cos\varphi\sin^2\varphi\sin\theta+\cos^2\theta\sin^3\varphi))\,\mathrm d\varphi\,\mathrm d\theta=-\frac{32(\pi+4)}3`