I'm reading the following in a calculus text book:
$$\frac{\partial(x,y)}{\partial(r,\theta)} = \left|\begin{matrix}\cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \end{matrix}\right| = r$$ $$\frac{\partial(y,z)}{\partial(r,\theta)} = \left|\begin{matrix}\sin\theta & r\cos\theta \\ 1 & 0 \end{matrix}\right| = -r\cos\theta$$ $$\frac{\partial(x,z)}{\partial(r,\theta)} = \left|\begin{matrix}\cos\theta & -r\sin\theta \\ 1 & 0 \end{matrix}\right| = r\sin\theta$$
These operations appear to be computing the Jacobian (the matrix of first partial derivatives), which I expect.
But then they appear to take the determinant of the Jacobian. I didn't expect that the notation $\frac{\partial(x,y)}{\partial(r,\theta)}$ implied the "determinant of the Jacobian". I thought that notation only represented the Jacobian itself.
I'm further confused by the fact that the book I'm reading doesn't actually use the term "determinant" in the text anywhere in this chapter (on surface area integrals).
Am I simply wrong in my understanding? Or perhaps I'm missing some fundamental detail here?
In my head this is true:
$$\frac{\partial(x,y)}{\partial(r,\theta)} = \begin{bmatrix}\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \end{bmatrix} \ne \det \left( \begin{bmatrix}\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \end{bmatrix}\right)$$
Also, can you confirm that the single bars $| \cdot |$ represent the determinant of the matrix contained within them?
