12—Tensors 388As in Eq. ( 36 ), the transformation matrices for the direct and the reciprocal basis are inverses of each other,
In the present context, this becomes
e′k.~e′j=δjk=~e`∂yk
∂x`.~ei∂xi
∂yj=δi`∂yk
∂x`∂xi
∂yj=∂yk
∂xi∂xi
∂yj=
∂yk
∂yjThe matrices∂xi/
∂yjand its inverse matrix,∂yk/
∂xiare called Jacobian matrices. When you do multiple
integrals and have to change coordinates, the determinant of one or the other of these matrices will appear as a
factor in the integral.
As an example, compute the change from rectangular to polar coordinates
x^1 =x y^1 =r x^2 =y y^2 =θ
x=rcosθ r=√
x^2 +y^2 y=rsinθ θ= tan−^1 y/x~e′j=~ei∂xi
∂yj~e′ 1 =~e 1∂x^1
∂y^1+~e 2∂x^2
∂y^1=ˆx∂x
∂r+ˆy∂y
∂r
=xˆcosθ+ˆysinθ=ˆr~e′ 2 =~e 1∂x^1
∂y^2+~e 2∂x^2
∂y^2=ˆx∂x
∂θ+ˆy∂y
∂θ
=xˆ(−rsinθ) +ˆy(rcosθ) =rθˆ
Knowing the change in the basis vectors, the change of the components of any tensor follows as before as
it did after Eq. ( 34 ).
A realistic example using non-orthonormal bases appears in special relativity. Here the manifold is four
dimensional instead of three and the coordinate changes of interest represent Lorentz transformations. Points in