7—Operators and Matrices 181This saves integration.I 11 =I 22 =MR^2 / 4.
For the other term in the sum ( 18 ), you have a point mass at the distanceRalong thex-axis,(x,y,z) =
(R, 0 ,0). Substitute this point mass into Eq. ( 14 ) and you have
M
0 0 0
0 R^20
0 0 R^2
The total about the origin is the sum of these two calculations.
MR^2
1 /4 0 0
0 5/4 0
0 0 3/ 2
Components of the Derivative
The set of all polynomials inxhaving degree≤ 2 forms a vector space. There are three independent vectors
that I can choose to be 1 ,x, andx^2. Differentiation is a linear operator on this space because the derivative of
a sum is the sum of the derivatives and the derivative of a constant times a function is the constant times the
derivative of the function. With this basis I’ll compute the components ofd/dx. Start the indexing for the basis
from zero instead of one because it will cause less confusion between powers and subscripts.
~e 0 = 1, ~e 1 =x, ~e 2 =x^2By the definition of the components of an operator — I’ll call this oneD,
D(~e 0 ) =d
dx1 = 0, D(~e 1 ) =d
dxx= 1 =~e 0 , D(~e 2 ) =d
dxx^2 = 2x= 2~e 1These define the three columns of the matrix.
(D) =
0 1 0
0 0 2
0 0 0
check:dx2
dx= 2xis
0 1 0
0 0 2
0 0 0
0
0
1
=
0
2
0
There’s nothing here about the basis being orthonormal. It isn’t.