Exercise 1798Calculus › Multivariable calculus › Partial derivativesM1Added 3 Nov 2026

Inverse function theorem (illustration)

#inverse function theorem#Jacobian#local invertibility

Statement

Let $F(x,y)=(e^x\cos y,e^x\sin y)$ . Compute the Jacobian matrix at $(0,0)$ and check the inverse function theorem.

Hint

Jacobian columns: derivatives with respect to $x$ and $y$ .

Show solutionHide solution

Partial derivatives. $\partial_x F=(e^x\cos y,e^x\sin y)$ , $\partial_y F=(-e^x\sin y,e^x\cos y)$ .

Jacobian at $(0,0)$ . $J_F(0,0)=\begin{pmatrix}1&0\\0&1\end{pmatrix}$ .

Conclusion. $\det J=1\neq0$ , so the inverse function theorem applies: $F$ is a local diffeomorphism near $(0,0)$ .