Change of Variables for Tensor Fields
We’ve seen that given a local coordinate patch we can decompose tensor fields in terms of the coordinate bases and on and , respectively. But what happens if we want to pass from the -coordinate system to another coordinate system ?
For vectors and covectors, we know the answers. We pass from the -coordinate basis to the -coordinate basis of by using a Jacobian:
where we calculate the coefficients by writing the coordinate function in terms of the coordinates. That is, we’re calculating the Jacobian of the function .
Similarly, we pass from the -coordinate basis to the -coordinate basis of by using another Jacobian:
Where here we use the Jacobian of the inverse transformation .
Since we build up the coordinates on the tensor bundles as the canonical ones induced on the tensor spaces by the coordinate bases on and , we immediately get coordinate transforms for all these bundles.
As one example in particular, given the basis and the basis on the coordinate patch we can build up two “top forms” in — top, since is the highest possible degree of a differential form. These are and , and it turns out there’s a nice formula relating them. We just work it out from the formula above:
That is, the two forms differ at each point by a factor of the Jacobian determinant at that point. This is the differential topology version of the change of basis formula for top forms in exterior algebras.
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