91Ó°ÊÓ

Use a computer algebra system to find the rate of mass flow of a fluid of density \(\rho\) through the surface \(S\) oriented upward if the velocity field is given by \(\mathbf{F}(x, y, z)=\mathbf{0 . 5 z} \mathbf{k}\). \(S: z=16-x^{2}-y^{2}, \quad z \geq 0\)

A function \(f\) is called harmonic if \(\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}=0\). Prove that if \(f\) is harmonic, then \(\int_{C}\left(\frac{\partial f}{\partial y} d x-\frac{\partial f}{\partial x} d y\right)=0\) where \(C\) is a smooth closed curve in the plane.

What is a conservative vector field and how do you test for it in the plane and in space?

A tractor engine has a steel component with a circular base modeled by the vector-valued function \(\mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j} .\) Its height is given by \(z=1+y^{2}\) (All measurements of the component are given in centimeters.) (a) Find the lateral surface area of the component. (b) The component is in the form of a shell of thickness \(0.2\) centimeter. Use the result of part (a) to approximate the amount of steel used in its manufacture. (c) Draw a sketch of the component.