Mathematical Tools for Physics

8—Multivariable Calculus 233

The quantityλ 2 is usually denotedβin this type of problem, and it is related to temperature byβ= 1/kT
where as usual the Lagrange multiplier is important on its own. It is usual to manipulate this by defining the
“partition function”

Z(β) =

∑^3

`=1

e−βE` (25)

In terms of this functionZyou have

C=N/Z, and E=−

N

Z

dZ

dβ

(26)

For a lot more on this subject, you can refer to one of many books on thermodynamics or statistical physics.
There for example you can find the reason thatβis related to the temperature.

8.13 Solid Angle
The extension of the concept of angle to three dimensions is called “solid angle.” To explain what this is, I’ll
first show a definition of ordinary angle that’s different from what you’re accustomed to. When you see that, the
extension to one more dimension is easy.
Place an object in the plane somewhere not at the origin. You are at the origin and look at it. I want a
definition that describes what fraction of the region around you is spanned by this object. For this, draw a circle
of radiusRcentered at the origin and draw all the lines from everywhere on the object to the origin. These lines
will intersect the circle on an arc of lengths.Definethe angle subtended by the object to beθ=s/R.

s

R

A

R