Finding the partial derivatives of a function in terms of the partial derivatives of another function

Question

Let $f : \mathbb R^2 → \mathbb R$ be a differentiable function and let $g : \mathbb R^2 →\mathbb R$ be the function defined by $g(r, θ) := f(r \cos (θ), r \sin (θ))$. Answer the following questions:

1. Find the partial derivatives of $g$ in terms of the partial derivatives of $f$.

2. Find the partial derivatives of $f$ in terms of the partial derivatives of $g$.

My attempts:

1. $\dfrac{\partial g}{\partial r} = \dfrac{\partial f}{\partial x}\cdot \dfrac{\partial x}{\partial r} + \dfrac{\partial f}{\partial y}\cdot \dfrac{\partial y}{\partial r} = \dfrac{\partial f}{\partial x}\cdot\cosθ + \dfrac{\partial f}{\partial y}\cdot\sinθ$

And $\dfrac{\partial g}{\partial θ} = \dfrac{\partial f}{\partial x}\cdot \dfrac{\partial x}{\partial θ} + \dfrac{\partial f}{\partial y}\cdot \dfrac{\partial y}{\partial θ} = -r\dfrac{\partial f}{\partial x}\cdot\sin(θ) +r \dfrac{\partial f}{\partial y}\cdot\cos(θ)$

2. We have $\dfrac{\partial f}{\partial r} = \dfrac{\partial g}{\partial r}$ and $\dfrac{\partial f}{\partial θ} = \dfrac{\partial g}{\partial r}$

Are my answers correct? If not, please can you correct me? Thank you

Answer 1

In 1. your notation indicated that $f$ was a function of $x$ and $y$, so in 2. your writing $\partial f/\partial r$ makes no sense.

I would suggest that in 1. you put in explicitly what $\partial x/\partial r$, etc., are in this case. Then, to do 2. correctly, solve the system of equations you have in 1. for $\partial f/\partial x$ and $\partial f/\partial y$.