Let $\theta \in \mathbb{R}$ and consider the rotational action $X = x \cos\theta - y \sin\theta$ ; $Y = x \sin\theta + y \cos\theta$. Find the transformed derivatives $Y'$ and $Y''$.
How do I approach this question. What am I supposed to be differentiating with respect to?
The second part is to show the infinitesimals associated with $x, y, y' \text{and } y''$ of which I am able to do using a different method, but i need to work from the transformed derivatives.
If anybody know any reads that will help me out could you please tell me.
It comes in my mind O.Toeplitz The Calculus - A Genetic Approach (2007 edition) pp.144-145.
There it is proved that the concept of acceleration does not depend on the coordinate system, considering $x$ and $y$ as functions of time and differentiating w.r.t. the latter.
Try to read and let me know.