Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PARTIAL DIFFERENTIATION

By consideringd(U−TS−HM) prove that

(

∂M

∂T

)

H

=

(

∂S

∂H

)

T

.

For a particular salt,

M(H, T)=M 0 [1−exp(−αH/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^34 M 0 , then the salt’s entropy

decreasesby an amount

M 0

4 α

(3−ln 4).

5.29 Using the results of section 5.12, evaluate the integral

I(y)=

∫∞

0

e−xysinx

x

dx.

Hence show that

J=

∫∞

0

sinx

x

dx=

π

2

.

5.30 The integral
∫∞

−∞

e−αx

2

dx

has the value (π/α)^1 /^2. Use this result to evaluate

J(n)=

∫∞

−∞

x^2 ne−x

2

dx,

wherenis a positive integer. Express your answer in terms of factorials.
5.31 The functionf(x) is differentiable andf(0) = 0. A second functiong(y) is defined
by

g(y)=

∫y

0

f(x)dx

√

y−x

.

Prove that

dg

dy

=

∫y

0

df

dx

dx

√

y−x

.

For the casef(x)=xn, prove that

dng

dyn

=2(n!)

√

y.

5.32 The functionsf(x, t)andF(x) are defined by

f(x, t)=e−xt,

F(x)=

∫x

0

f(x, t)dt.

Verify, by explicit calculation, that

dF

dx

=f(x, x)+

∫x

0

∂f(x, t)

∂x

dt.