Mathematical Tools for Physics

Tensors

You can’t walk across a room without using a tensor (the pressure tensor). You can’t balance the wheels on
your car without using a tensor (the inertia tensor). You definitely can’t understand Einstein’s theory of gravity
without using tensors (many of them).
Some of this material overlaps that of chapter 7, but I will extend it in a different direction. The first
examples will then be familiar.

12.1 Examples
A tensor is a particular type of function. Before presenting the definition, some examples will clarify what I mean.
Start with a rotating rigid body, and compute its angular momentum. Pick an origin and assume that the body
is made up ofNpoint massesmiat positions described by the vectors~ri(i= 1, 2 ,...,N). The angular velocity
vector is~ω. For each mass the angular momentum is~ri×~pi=~ri×(mi~vi). The velocity~viis given by~ω×~ri
and so the angular momentum of theithparticle ismi~ri×(~ω×~ri). The total angular momentum is therefore

m 1

m 2

m 3

~ω

L~=

∑N

i=1

mi~ri×(~ω×~ri). (1)

The angular momentum,L~, will depend on the distribution of mass within the body and upon the angular velocity.
Write this as
L~=I(~ω),

where the functionIis called the tensor of inertia.
For a second example, take a system consisting of a mass suspended by six springs. At equilibrium the
springs are perpendicular to each other. If now a (small) force F~ is applied to the mass it will undergo a
displacementd~. Clearly, ifF~ is along the direction of any of the springs (call these thex,y, andzaxes), then

the displacementd~will be in the same direction asF~. Suppose however thatF~is halfway between thek 1 andk 2
springs, and further that the springk 2 was taken from a railroad locomotive whilek 1 is a watch spring. Obviously

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