2.II.13F

For a smooth mapping $F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$, the Jacobian $J(F)$ at a point $(x, y)$ is defined as the determinant of the derivative $D F$, viewed as a linear map $\mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$. Suppose that $F$ maps into a curve in the plane, in the sense that $F$ is a composition of two smooth mappings $\mathbf{R}^{2} \rightarrow \mathbf{R} \rightarrow \mathbf{R}^{2}$. Show that the Jacobian of $F$ is identically zero.

Conversely, let $F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ be a smooth mapping whose Jacobian is identically zero. Write $F(x, y)=(f(x, y), g(x, y))$. Suppose that $\partial f /\left.\partial y\right|_{(0,0)} \neq 0$. Show that $\partial f / \partial y \neq 0$ on some open neighbourhood $U$ of $(0,0)$ and that on $U$

$(\partial g / \partial x, \partial g / \partial y)=e(x, y)(\partial f / \partial x, \partial f / \partial y)$

for some smooth function $e$ defined on $U$. Now suppose that $c: \mathbf{R} \rightarrow U$ is a smooth curve of the form $t \mapsto(t, \alpha(t))$ such that $F \circ c$ is constant. Write down a differential equation satisfied by $\alpha$. Apply an existence theorem for differential equations to show that there is a neighbourhood $V$ of $(0,0)$ such that every point in $V$ lies on a curve $t \mapsto(t, \alpha(t))$ on which $F$ is constant.

[A function is said to be smooth when it is infinitely differentiable. Detailed justification of the smoothness of the functions in question is not expected.]

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