Chapter 19: Chapter 19

Express the integral: $$ \psi_{0}\left[\left(\sin ^{2} \psi d \psi\right) / \sqrt{\left.\left(1-k^{2} \sin ^{2} \psi\right)\right](0

The electrostatic field produced by a unit positive charge at 0 is $$ \mathrm{E}=\left(1 / \mathrm{r}^{3}\right) \mathrm{A}^{-} $$ where \(\mid \mathrm{A} \|=\mathrm{r}\). Find the divergence of this field (wherever the field is defined, i.e. at all points except 0).

Suppose that the electrical potential at the point $(\mathrm{x}, \mathrm{y}, \mathrm{z})$ is $$ E(x, y, z)=x^{2}+y^{2}-2 z^{2} $$ What is the direction of the acceleration at the point \((1,3,2) ?\)

Let \(\mathrm{G}\) be the gravitational constant, \(\mathrm{r}=\|\mathrm{p}\|\), and \(\mathrm{F}=-\left(\mathrm{GM} / \mathrm{r}^{3}\right) \mathrm{p}\) where \(\mathrm{M}\) is the mass at \(0 .\) That is, (1) describes the gravitational field of a mass concentrated at \(0 .\) Show that \(F\) is irrotational.

A particle moves in the plane according to $$ x=64 \sqrt{3} t, \quad y=64 t-16 t^{2} $$ and is acted on by a force \(\mathrm{F}\) which is directly proportional to the velocity but opposite in direction. Find the work done by \(F\) from \(t=0\) to \(t=4\)

Find the center of mass of a hemisphere of radius \(a=0\) assuming the surface is a homogeneous lamina.

A plane lamina is bounded by $\mathrm{x}=\mathrm{a}, \mathrm{x}=\mathrm{b}, \mathrm{y}=0, \mathrm{f}(\mathrm{x})>0$ where \(\mathrm{f} \in \mathrm{C}[\mathrm{a}, \mathrm{b}]\). The density is constant along all vertical lines $\mathrm{x}=\mathrm{c}, \mathrm{c} \in[\mathrm{a}, \mathrm{b}]$, so it can be described as a function of one variable \(\mathrm{M}(\mathrm{x})\). Find the moment of inertia of the lamina about the \(\mathrm{y}\) -axis.

Derive the Equation of Continuity for fluid flows: $$ (\partial p / \partial t)=-\operatorname{div} \rho \mathrm{V}^{-} $$ where \(\rho(\mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{t})\) and \(\mathrm{V}^{-}(\mathrm{x}, \mathrm{y}, z, \mathrm{t})\) are, respectively, the fluid density and velocity at the point \((x, y, z)\) at time t. Conclude \(\operatorname{div} \mathrm{V}^{-}=0\) if the fluid is incompressible.

What force field would account for a particle of unit mass moving around in a unit circle in the plane according to the function $$ f(t)=(\cos t, \sin t) ? $$

Suppose there is a force field defined by $$ \mathrm{F}(\mathrm{x}, \mathrm{y}, z, \mathrm{t})=(-\mathrm{x},-\mathrm{y}, 0) $$ If a particle of unit mass is at \((1,0,0)\) with an initial velocity of $(0,1, a)$, what is its path of motion?