The Unapologetic Mathematician

Mathematics for the interested outsider

Change of Variables for Tensor Fields

We’ve seen that given a local coordinate patch (U,x) we can decompose tensor fields in terms of the coordinate bases \frac{\partial}{\partial x^i} and dx^i on \mathcal{T}_pM and \mathcal{T}^*_pM, respectively. But what happens if we want to pass from the x-coordinate system to another coordinate system y?

For vectors and covectors, we know the answers. We pass from the x-coordinate basis to the y-coordinate basis of \mathcal{T}_pM by using a Jacobian:

\displaystyle\frac{\partial}{\partial x^i}=\sum\limits_{j=1}^n\frac{\partial y^j}{\partial x^i}\frac{\partial}{\partial y^j}

where we calculate the coefficients by writing the coordinate function y^j in terms of the x coordinates. That is, we’re calculating the Jacobian of the function y\circ x^{-1}:\mathbb{R}^n\to\mathbb{R}^n.

Similarly, we pass from the x-coordinate basis to the y-coordinate basis of \mathcal{T}^*_pM by using another Jacobian:

\displaystyle dx^i=\sum\limits_{j=1}^n\frac{\partial x^i}{\partial y^j}dy^j

Where here we use the Jacobian of the inverse transformation x\circ y^{-1}:\mathbb{R}^n\to\mathbb{R}^n.

Since we build up the coordinates on the tensor bundles as the canonical ones induced on the tensor spaces by the coordinate bases on \mathcal{T}_pM and \mathcal{T}^*_pM, we immediately get coordinate transforms for all these bundles.

As one example in particular, given the basis \{dx^i\} and the basis \{dy^j\} on the coordinate patch (U,x) we can build up two “top forms” in \Lambda^*_n(U) — top, since n is the highest possible degree of a differential form. These are dx^1\wedge\dots\wedge dx^n and dy^1\wedge\dots\wedge dy^n, and it turns out there’s a nice formula relating them. We just work it out from the formula above:

\displaystyle\begin{aligned}dx^1\wedge\dots\wedge dx^n&=\left(\sum\limits_{j_1=1}^n\frac{\partial x^1}{\partial y^{j_1}}dy^{j_1}\right)\wedge\dots\wedge\left(\sum\limits_{j_n=1}^n\frac{\partial x^n}{\partial y^{j_n}}dy^{j_n}\right)\\&=\sum\limits_{j_1,\dots,j_n=1}^n\frac{\partial x^1}{\partial y^{j_1}}\dots\frac{\partial x^n}{\partial y^{j_n}}dy^{j_1}\wedge\dots\wedge dy^{j_n}\\&=\sum\limits_{\pi\in S_n}\prod\limits_{i=1}^n\frac{\partial x^i}{\partial y^{\pi(i)}}dy^{\pi(1)}\wedge\dots\wedge dy^{\pi(n)}\\&=\left(\sum\limits_{\pi\in S_n}\prod\limits_{i=1}^n\frac{\partial x^i}{\partial y^{\pi(i)}}\mathrm{sgn}(\pi)\right)dy^1\wedge\dots\wedge dy^n\\&=\det\left(\frac{\partial x^i}{\partial y^j}\right)dy^1\wedge\dots\wedge dy^n\end{aligned}

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.

July 8, 2011 Posted by | Differential Topology, Topology | Leave a comment