I have the following (simple) system of ODE's that I wish to express in Cartesian coordinates:
\begin{align*} r' &= r(1-r)\\ \theta' &= 1 \end{align*}
Working it out by hand, by differentiating $x = r\cos\theta$, $y = r\sin\theta$, I find
\begin{align*} x' &= r' \cos\theta - r \sin\theta \theta' \\ &= r(1-r)\cos\theta - r\sin\theta \\ &= x(1 - \sqrt{x^2 + y^2}) - y\\ \\ y' &= r' \sin\theta + r \cos\theta \theta' \\ &= r(1-r)\sin\theta + r\cos\theta \\ &= y(1 - \sqrt{x^2 + y^2}) + x \end{align*}
However, using TransformedField
like so:
TransformedField["Polar" -> "Cartesian", {r (1 - r), 1}, {r, \[Theta]} -> {x, y}]
the result is:
$$\left\{x \left(1-\sqrt{x^2+y^2}\right)-\frac{y}{\sqrt{x^2+y^2}},\frac{x}{\sqrt{x^2+y^2}}+y \left(1-\sqrt{x^2+y^2}\right)\right\}$$
So does the disagreement come from an error in my calculation (which I have been unable to find), or a misuse/misunderstanding of TransformedField
?