Jacobian of an ellipse
An ellipse is given by
$$ \frac {x^2}{a^2} + \frac {y^2}{b^2} = 1$$
You want to find the area by using a change of coordinates: $x = r\cos θ$, $y = \frac{br}{a}\sin θ$.
Find the range of values of $r$ and $θ$ that correspond to the interior of the ellipse.
Find the Jacobian of the transformation and the area of the ellipse.
To find the Jacobian, do I need to find $\frac{\delta x}{\delta a},\frac{\delta x}{\delta a},\frac{\delta y}{\delta a},\frac{\delta y}{\delta b} $ and work out the determinant? I get 0 for determinant :/ When the answer is $\frac{br}{a}$
$\endgroup$ 21 Answer
$\begingroup$Continue from the comment...Then the area of the circle in the $uv$-plane becomes the Jacobian times the area of the cirle, namely $J=ab$ times the are of the cirle which is $\pi$. The result will be $ab\pi$.
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