Chain rule while differentiating

Question

I am trying to find the derivative of a function defined in polar coordinates with respect to $x$ and $y$. My function is defined as follows:

$ v_x(r, \theta ) = v_r \cos (\theta ) - v_{\theta }\sin (\theta ) $

To do this, I start by defining the relation between Cartesian and Polar coordinates:

(* Define the mapping between Cartesian and Polar coordinate systems. *)
x[r_, θ_] = r Cos[θ];
y[r_, θ_] = r Sin[θ];

Then I define the function and find its derivative with respect to $x$:

Subscript[v, r][r_, θ_] = Subscript[v, r][r, θ] Cos[θ] - Subscript[v, θ][r, θ] Sin[θ];
D[Subscript[v, r][r, θ], x]

I am getting 0 because Mathematica is not considering the relation between $r$ and $x$. Is there anyway to tell Mathematica to use the chain rule to find the derivative of $v_x$ with respect to $x$?

The other problem is that Mathematica is considering the subscripts to be variables (which is reasonable), is there anyway to tell it that the subscripts are only notational symbols?

EDIT: The function is better defined as:

vx[r_, θ_] = vr[r, θ] Cos[θ] - vtheta[r, θ] Sin[θ];

to avoid evaluating subscripts and possibly having recursion.

Answer 1

In your case it might be more convenient to define the inverse transformation :

rho[x_, y_] = Sqrt[x^2 + y^2]
theta[x_, y_] = ArcTan[x, y]

 vx[r_, \[Theta]_] = vr[r, \[Theta]] Cos[\[Theta]] - vtheta[r, \[Theta]] Sin[\[Theta]];

Then this will use the chain rule :

 D[vx[rho[x, y], theta[x, y]], x]

One can simplify the result in terms of the polar coordinated :

Simplify[D[vx[rho[x, y], theta[x, y]], x] /. {x^2 + y^2 -> rho^2, ArcTan[x, y] -> theta, x -> rho Cos[theta], y -> rho Sin[theta]}, Assumptions -> {rho >= 0}]

Answer 2

You can use the total derivative Dt:

x[r_, \[Theta]_] = r Cos[\[Theta]];
y[r_, \[Theta]_] = r Sin[\[Theta]];

then for instance

Dt[a[r, \[Theta]]*Cos[\[Theta]] - b[r, \[Theta]]*Sin[\[Theta]], x]

does this

I can't test your example because Subscript[v, r][r_, \[Theta]_] = Subscript[v, r][r, \[Theta]] Cos[\[Theta]] - Subscript[v, \[Theta]][r, \[Theta]] Sin[\[Theta]] hits the recursion limit (because the way it's defined and the evaluation sequence works, it'll never finish). However, here's how to indicate that something is a constant. Suppose I try:

Dt[Sin[\[Theta]] + c, x]

but $c$ is a constant; I can indicate this like so:

Dt[Sin[\[Theta]] + c, x, Constants -> {c}]