Chapter 19

Express \(\mathrm{x} \int_{0} \sqrt{\left(1-4 \sin ^{2} \mathrm{u}\right) \text { du in terms of incomplete elliptic }}\) integrals where \(0 \leq \mathrm{x} \leq(\pi / 6)\).

Let \(a, b, c\) be the lengths of the sides of a triangle and let \(\theta\) be the angle opposite the side of length \(c\). Find the differential do and approximate c when \(\mathrm{a}=6.20, \mathrm{~b}=5.90\), and \(0=58^{\circ}\).

Suppose a student has \(\$ 90\) with which to buy lecture notes at \(\$ 3\) each and packs of beer, at \(\$ 5\) each. Let the function $$ f(x, y)=x y $$ (1) express how much satisfaction is derived from buying \(\mathrm{x}\) lecture notes and \(\mathrm{y}\) packs of beer. What \(\mathrm{x}\) and \(\mathrm{y}\) will maximize the student's pleasure?

Find the dimensions of the box of largest volume which can be fitted inside the ellipsoid $$ \left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)+\left(z^{2} / c^{2}\right)=1 $$ assuming that each edge of the box is parallel to a coordinate axis.

Which 3 -dimensional rectangular box of a given volume \(\mathrm{V}\) has the least surface area?

Minimize the distance from a point \(p \in R^{n}\) to the hyperplane \(<\mathrm{x}, \mathrm{a}>+\mathrm{b}=0\) where \(\mathrm{a} \in \mathrm{R}^{n}\) and \(\mathrm{b} \in \mathrm{R}\). (Assume \(\left.\mathrm{a} \neq 0 .\right)\)

Show how to find the semiaxes of the ellipse in which the plane $$ \begin{aligned} \mathrm{P}(\mathrm{x}, \mathrm{y}, z) &=\mathrm{pz}+\mathrm{qy}+\mathrm{rz} \\\ &=0(\mathrm{pqr} \neq 0) \end{aligned} $$ cuts the ellipsoid \(E(x, y, z)=\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)+\left(z^{2} / c^{2}\right)\) \(=1 .(0

Construct a moebius strip and show that its normal is not well-defined (ie that it is not orientahle)

Express \(\mathrm{x} \int_{0} \sqrt{\left(1-4 \sin ^{2} \mathrm{u}\right) \text { du in terms of incomplete elliptic }}\) integrals where \(0 \leq \mathrm{x} \leq(\pi / 6)\).

Evaluate a) \(\left.^{2} \int_{0}\left[\mathrm{dx} / \sqrt{\\{}\left(4-\mathrm{x}^{2}\right)\left(9-\mathrm{x}^{2}\right)\right\\}\right]\) \(\quad\) b) \(\left.{ }^{\infty} \mathrm{I}_{1}\left[\mathrm{du} / \sqrt{\\{}\left(\mathrm{u}^{2}-1\right)\left(\mathrm{u}^{2}+3\right)\right\\}\right]\) in terms of elliptic integrals.

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?

Suppose $$ \mathrm{A}(\mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{t})=(-\mathrm{x}, 0, \mathrm{y}) $$ is the acceleration field of a fluid in motion with an initial velocity of \((0,1,0)\). Find the equation of motion and the divergence and curl of the flow.

A body falls in a medium offering resistance proportional to the square of the velocity. If the limiting velocity is numerically equal to \(\mathrm{g} / 2=16.1 \mathrm{ft} / \mathrm{sec}\)., find a) the velocity at the end of \(1 \mathrm{sec}, ; b\) ) the distance fallen at the end of \(1 \mathrm{sec}\). c) the distance fallen when the velocity equals \(1 / 2\) the d) the time required to fall \(100 \mathrm{ft}\). limiting velocity;

A particle slides freely in a tube which rotates in a vertical plane about its midpoint with constant angular velocity w. If \(\mathrm{x}\) is the distance of the particle from the midpoint of the tube at time \(t\) and if the tube is horizontal with \(t=0\), show that the motion of the particle along the tube is given by $$ \left(d^{2} x / d t^{2}\right)-w^{2} x=-g \text { sin wt. } $$ Solve this equation if \(\mathrm{x}=\mathrm{x}_{0}, \mathrm{dx} / \mathrm{dt}=\mathrm{v}_{0}\) when \(\mathrm{t}=0\). For what values of \(\mathrm{x}_{0}\) and \(\mathrm{v}_{0}\) is the motion simple harmonic?