Mathematical Tools for Physics

7—Operators and Matrices 169

f(~v)

α

~v

f(~v 1 +~v 2 )

α ~v^1 +~v^2

What happens if you change the argument of this function, multiplying it by a scalar? You knowf(~v),
what isf(c~v)? Just from the picture, this isctimes the vector that you got by rotating~v. What happens when
you add two vectors and then rotate the result? The whole parallelogram defining the addition will rotate through
the same angleα, so whether you apply the function before or after adding the vectors you get the same result.

This leads to the definition of the wordlinearity:

f(c~v) =cf(~v), and f(~v 1 +~v 2 ) =f(~v 1 ) +f(~v 2 ) (1)

Keep your eye on this pair of equations! They’re central to the whole subject.
Another example of the type of function that I’ll examine is from physics instead of mathematics. A rotating
rigid body has some angular momentum. The greater the rotation rate, the greater the angular momentum will
be. Now how do I compute the angular momentum assuming that I know the shape and the distribution of masses
in the body and that I know the body’s angular velocity? The body is made of a lot of point masses (atoms),
but you don’t need to go down to that level to make sense of the subject. As with any other integral, you start
by dividing the object in to a lot of small pieces.

What is the angular momentum of a single point mass? It starts from basic Newtonian mechanics, and the
equationF~=d~p/dt. (It’s better in this context to work with this form than with the more common expressions

F~=m~a.) Take the cross product with~r, the displacement vector from the origin.

~r×F~=~r×d~p/dt