Given $$x=\frac{u+v}{1-uv}$$ and $$y=\arctan\left(u\right)+\arctan\left(v\right)$$
Then find $$\frac{\partial (x,y)}{\partial (u,v)}$$
I don't understand the notation $(x,y)$ or $(u,v)$,since I've always seen a function instead of $(x,y)$ and a variable instead of $(u,v)$,can someone help?
I don't understand the notation $(x,y)$ or $(u,v)$,since I've always seen a function instead of $(x,y)$ and a variable instead of $(u,v)$,can someone help?
It could be confusing if you're thinking about the standard notation for a partial derivative of a function of several variables, but it's not that different: $x$ and $y$ are in fact functions of $u$ and $v$.
The notation $$\frac{\partial (x,y)}{\partial (u,v)} \tag{1}$$ refers to the Jacobian (determinant) which is, in this case: $$ \det\begin{bmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} \\[4pt] \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} \end{bmatrix}\tag{2} $$ So you should remember $(1)$ as a (compact) notation for $(2)$.
I would expect your text(book) to introduce this notation, if you're giving this kind of exercise.
As far as I understand, $$\frac{\partial (x,y)}{\partial (u,v)}$$ means, that you have to find the determinant of a Jacobian matrix, given $x(u,v)$ and $y(u,v)$, i.e.:
$$\frac{\partial (x,y)}{\partial (u,v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}.$$ Taking partial derivatives of your given $x(u,v)$ and $y(u,v)$.
