Question

Use the divergence theorem to calculate the surface integral z s ~f d~s ; that is, calculate the ux of ~f across s, where ~f = z~i + y~j + zx~k; and s is the surface of the tetrahedron enclosed by the coordinate planes and the plane x a + y b + z c = 1 with a; b; c > 0.

Answer 1

To calculate the surface integral using the divergence theorem, we need to find the divergence of the vector field. Then, we apply the divergence theorem to find the surface integral over the given surface. The divergence of the vector field is the sum of the partial derivatives of each component of the vector field. In this case, the vector field is F = z*i + y*j + z*x*k.

Step-by-step explanation:

To calculate the surface integral using the divergence theorem, we first need to find the divergence of the vector field. The divergence of a vector field F is given by div(F) = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z.

In this case, the vector field is F = z*i + y*j + z*x*k.

So, the divergence of F is div(F) = ∂(z)/∂x + ∂(y)/∂y + ∂(zx)/∂z = 1 + 1 + x = 2 + x.

The divergence theorem states that the surface integral of F across a closed surface S is equal to the triple integral of the divergence of F over the volume V enclosed by S.

In this case, the surface S is the tetrahedron enclosed by the coordinate planes and the plane x*a + y*b + z*c = 1.

So, the surface integral of F across S is equal to the triple integral of (2 + x) over the volume of the tetrahedron enclosed by the coordinate planes and the plane x*a + y*b + z*c = 1.

Answer 2

By the divergence theorem,


`\displaystyle\iint_S\mathbf f\cdot\mathrm d\mathbf S=\iiint_R(\\abla\cdot\mathbf f)\,\mathrm dV`

where
`R` is the solid whose boundary is
`S`. We have


`\\abla\cdot\mathbf f=(\partial z)/(\partial x)+(\partial y)/(\partial y)+(\partial zx)/(\partial z)=1+x`

so we set up the volume integral as


`\displaystyle\iiint_R(\\abla\cdot\mathbf f)\,\mathrm dV=\int_(x=0)^(x=1/a)\int_(y=0)^(y=(1-ax)/b)\int_(z=0)^(z=(1-ax-by)/c)(1+x)\,\mathrm dz\,\mathrm dy\,\mathrm dx`

`=(4a+1)/(24a^2bc)`