Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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.
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