i have to prove that:
Given $x:\mathbb{R}^2\to\mathbb{R},x(r,\theta)=r\cos(\theta)$ and $y:\mathbb{R}^2\to\mathbb{R},y(r,\theta)=r\sin(\theta)$. Prove that $$ \frac{\partial(x,y)}{\partial(r,\theta)}(r_0,\theta_0)=\mbox{Determinant of }Df(r_0,\theta_0)=r_0 $$ I demonstrated that the determinant evaluated in $(r_0,\theta_0)$ is equal to $r_0$. But im confused with the partials, because i have only worked with partials of the style $\frac{\partial f}{\partial x}$ so i don't know how even start, any help would be appriciated.


1 Answer 1


The partial derivative notation $$ \frac{\partial(x,y)}{\partial(r,\theta)} $$ is used to represent the Jacobian matrix (in particular see example 2):

$$ \displaystyle \frac{\partial(x,y)}{\partial(r,\theta)}= \left[ \begin{array}{cc} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta}\\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta}\\ \end{array} \right] $$

The notation is also sometimes given as (as can be seen in the Wikipedia article): $\frac{d \mathbf{v}}{d\mathbf{x}}=\frac{\partial \langle v_1,v_2 \rangle}{\partial \langle x_1,x_2 \rangle}$.


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