91Ó°ÊÓ

Compute \(\iint_{S} \mathbf{F} \cdot \mathbf{N} d S\), where \(\mathbf{F}(x, y, z)=x y z \mathbf{i}+x y z \mathbf{j}+x y z \mathbf{k}\) and \(\mathrm{N}\) is an outward normal vector \(S\), where \(S\) is the surface of the five faces of the unit cube \(0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\) missing \(z=0\).

Evaluate surface integral \(\iint_{S} y z d S\), where \(S\) is the first-octant part of plane \(x+y+z=\lambda\), where \(\lambda\) is a positive constant.

Evaluate surface integral \(\iint_{S} x^{2} y z d S\), where \(S\) is the part of plane \(z=1+2 x+3 y\) that lies above rectangle \(0 \leq x \leq 3\) and \(0 \leq y \leq 2\).

Evaluate surface integral \(\iint_{S} y z d S\), where \(S\) is the part of plane \(z=y+3\) that lies inside cylinder \(x^{2}+y^{2}=1\).

For the following exercises, Fourier's law of heat transfer states that the heat flow vector \(\mathrm{F}\) at a point is proportional to the negative gradient of the temperature; that is, \(\mathbf{F}=-k \nabla T\), which means that heat energy flows hot regions to cold regions. The constant \(k>0\) is called the conductivity, which has metric units of joules per meter per second-kelvin or watts per meter-kelvin. A temperature function for region \(D\) is given. Use the divergence theorem to find net outward heat flux \(\iint_{S} \mathbf{F} \cdot \mathbf{N} d S=-k \iint_{S} \nabla T \cdot \mathbf{N} d S\) across the boundary \(S\) of \(D\), where \(k=1$$T(x, y, z)=100+e^{-z} ; D=\\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\\}\)

Find the outward flux of vector field \(\mathbf{F}=x y^{2} \mathbf{i}+x^{2} y \mathbf{j}\) across the boundary of annulus \(R=\left\\{(x, y): 1 \leq x^{2}+y^{2} \leq 4\right\\}=\\{(r, \theta): 1 \leq r \leq 2,0 \leq \theta \leq 2 \pi\\}\) using a computer algebra system.