Answer to Question #297329 in Calculus for Goatie

Question

Let F(x,y)=2x^2+3sin y and let G(x,y)=e^x-2y.Compute the Jacobian determinant d(F ,G)/d(x,y)

Answer 1

Solution:

$F(x,y)=2x^2+3\sin y \\G(x,y)=e^x-2y$

$\dfrac{\partial (F,G)}{\partial (x,y)}=\begin{vmatrix} \dfrac{\partial F}{\partial x}&\dfrac{\partial F}{\partial y} \\ \dfrac{\partial G}{\partial x}&\dfrac{\partial G}{\partial y}\end{vmatrix}$

$=\begin{vmatrix} \dfrac{\partial (2x^2+3\sin y)}{\partial x}&\dfrac{\partial (2x^2+3\sin y)}{\partial y} \\ \dfrac{\partial (e^x-2y)}{\partial x}&\dfrac{\partial (e^x-2y)}{\partial y}\end{vmatrix}$

$=\begin{vmatrix} 4x&3\cos y \\ e^x&-2\end{vmatrix} \\=-8x-3e^x\cos y$