Show that

$$d f=y\left(1+x-x^2\right) d x+x(x+1) d y$$

is not an exact differential. Find the differential equation that a function $g(x)$ must satisfy if $d \phi=g(x) d f$ is to be an exact differential. Verify that $g(x)=e^{-x}$ is a solution of this equation and deduce the form of $\phi(x, y)$.

The integral

$$\int_{-\infty}^{\infty} e^{-\alpha x^2} d x$$

has the value $(\pi / \alpha)^{1 / 2}$. Use this result to evaluate

$$J(n)=\int_{-\infty}^{\infty} x^{2 n} e^{-x^2} d x,$$

where $n$ is a positive integer. Express your answer in terms of factorials.

Show that the differential

$$d f=x^2 d y-\left(y^2+x y\right) d x$$

is not exact, but that $d g=\left(x y^2\right)^{-1} d f$ is exact.

Using the results of section 5.12, evaluate the integral

$$I(y)=\int_0^{\infty} \frac{e^{-x y} \sin x}{x} d x .$$

Hence show that

$$J=\int_0^{\infty} \frac{\sin x}{x} d x=\frac{\pi}{2} .$$

The entropy S(H, T), the magnetization M(H, T) and the internal energy U(H, T) of a magnetic salt placed in a magnetic field of strength H, at temperature T, are connected by the equation

$$T d S=d U-H d M .$$

By considering $d(U-T S-H M)$ prove that

$$\left(\frac{\partial M}{\partial T}\right)_H=\left(\frac{\partial S}{\partial H}\right)_T .$$

For a particular salt,

$$M(H, T)=M_0[1-\exp (-\alpha H / T)] .$$

Show that if, at a fixed temperature, the applied field is increased from zero to a strength such that the magnetization of the salt is $\frac{3}{4} M_0$, then the salt's entropy decreases by an amount

$$\frac{M_0}{4 \alpha}(3-\ln 4) .$$

As in the previous two exercises on the thermodynamics of a simple gas, the quantity $d S=T^{-1}(d U+P d V)$ is an exact differential. Use this to prove that

$$\left(\frac{\partial U}{\partial V}\right)_T=T\left(\frac{\partial P}{\partial T}\right)_V-P .$$

In the van der Waals model of a gas, $P$ obeys the equation

$$P=\frac{R T}{V-b}-\frac{a}{V^2},$$

where $R, a$ and $b$ are constants. Further, in the limit $V \rightarrow \infty$, the form of $U$ becomes $U=c T$, where $c$ is another constant. Find the complete expression for $U(V, T)$

Functions P(V, T), U(V, T) and S(V, T) are related by

$$T d S=d U+P d V,$$

where the symbols have the same meaning as in the previous question. The pressure $P$ is known from experiment to have the form

$$P=\frac{T^4}{3}+\frac{T}{V},$$

in appropriate units. If

$$U=\alpha V T^4+\beta T,$$

where $\alpha, \beta$, are constants (or, at least, do not depend on $T$ or $V$ ), deduce that $\alpha$ must have a specific value, but that $\beta$ may have any value. Find the corresponding form of $S$.

By considering the differential

$$d G=d(U+P V-S T),$$

where $G$ is the Gibbs free energy, $P$ the pressure, $V$ the volume, $S$ the entropy and $T$ the temperature of a system, and given further that the internal energy $U$ satisfies

$$d U=T d S-P d V,$$

derive a Maxwell relation connecting $(\partial V / \partial T)_P$ and $(\partial S / \partial P)_T$.

A water feature contains a spray head at water level at the centre of a round basin. The head is in the form of a small hemisphere perforated by many evenly distributed small holes, through which water spurts out at the same speed, $v_0$, in all directions. (a) What is the shape of the 'water bell' so formed? (b) What must be the minimum diameter of the bowl if no water is to be lost?

Find the area of the region covered by points on the lines

$$\frac{x}{a}+\frac{y}{b}=1,$$

where the sum of any line's intercepts on the coordinate axes is fixed and equal to $c$.

Show that the envelope of all concentric ellipses that have their axes along the $x$ - and $y$-coordinate axes, and that have the sum of their semi-axes equal to a constant $L$, is the same curve (an astroid) as that found in the worked example in the earlier section.

Determine which of the following are exact differentials: (a) $(3 x+2) y d x+x(x+1) d y$; (b) $y \tan x d x+x \tan y d y$; (c) $y^2(\ln x+1) d x+2 x y \ln x d y$; (d) $y^2(\ln x+1) d y+2 x y \ln x d x$ (e) $\left[x /\left(x^2+y^2\right)\right] d y-\left[y /\left(x^2+y^2\right)\right] d x$.

A barn is to be constructed with a uniform cross-sectional area $A$ throughout its length. The cross-section is to be a rectangle of wall height $h$ (fixed) and width $w$, surmounted by an isosceles triangular roof that makes an angle $\theta$ with the horizontal. The cost of construction is $\alpha$ per unit height of wall and $\beta$ per unit (slope) length of roof. Show that, irrespective of the values of $\alpha$ and $\beta$, to minimise costs $w$ should be chosen to satisfy the equation

$$w^4=16 A(A-w h),$$

and $\theta$ made such that $2 \tan 2 \theta=w / h$.

Two horizontal corridors, $0 \leq x \leq a$ with $y \geq 0$, and $0 \leq y \leq b$ with $x \geq 0$, meet at right angles. Find the length $L$ of the longest ladder (considered as a stick) that may be carried horizontally around the corner.