How to calculate the Jacobian matrix?

Question

I want to calculate the Jacobian matrix of $F_a$ for $x = (r\cos\theta, r\sin\theta)$. $F_a$ is the transformation defined below.
$$F_a(r\cos\theta, r\sin\theta) := (a(\theta)r\cos\theta, a(\theta)r\sin\theta) \ \ \ (\theta \in [0, 2\pi))$$
$a(\theta)$ is differentiable.
The book I am now reading suggests that the answer is below.
$${\rm d}F_a(x) = \begin{pmatrix} a(\theta)\cos\theta & a'(\theta)\cos\theta - a(\theta)\sin\theta \\\ a(\theta)\sin\theta & a'(\theta)\sin\theta - a(\theta)\cos\theta \end{pmatrix}$$ However, I have no idea why I can get this answer. Could you show me how to do it?

Answer 1

Let $F = (f_1, f_2, f_3)$. Then the rows of the jacobian are $$(\frac{\partial f_i}{\partial r}, \frac{\partial f_i}{\partial \theta}).$$

Calculate them, and you should get the transpose of what you wrote. Maybe, the book uses a different convention.