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.

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July 8, 2011 - Posted by John Armstrong | Differential Topology, Topology

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