Partial Differentiation: Suppose $f(r,\theta,\phi)$ and $x=r\sin(\theta)\cos(\phi)$. How to find $∂f/∂x$?

Question

I have the following question and no access to solutions.

The variables $x$, $y$, $z$ and $r$, $θ$, $φ$ are connected by the following equations:

\begin{align} x&=r\sin(θ)\cos(φ) \\ y&=r\sin(θ)\sin(φ) \\ z&=r\cos(θ) \end{align}

(a) Find $∂x/∂φ$, $∂y/∂φ$, and $∂z/∂φ$.

(b) Show that for any differentiable function $g(x,y,z)$ we have $g_φ(x,y,z)=x\cdot g_y(x,y,z)−y\cdot g_x(x,y,z)$.

(c) Suppose $f(r,θ,φ)=\cosθ$. Find $∂f/∂x$, giving your answer in terms of $x$, $y$, and $z$.

I am happy with parts a and b. Unfortunately, I am a little stuck with (c).

Any help or pointers would be much appreciated.

Thank-you for your help in advance.

Welcome to MSE! We use something called MathJax here to format mathematics so equations and expressions are easier to read. That's what I did in my edit to your question, and you can find a tutorial to learn how to do it yourself here: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference. It's really easy to pick up, trust me. — Robert Howard, Aug 13 '18 at 22:49
Hi there mate! Thank you for pointing this out. Rather new here as you likely guessed! I will use this going forward. Cheers. — Dan, Aug 13 '18 at 22:50

score 2 · Accepted Answer · answered Aug 13 '18 at 22:43

2

Hint

Use that

$$f(r,\theta,\varphi)=\cos \theta=\dfrac{z}{r}=\dfrac{z}{\sqrt{x^2+y^2+z^2}}.$$

answered Aug 13 '18 at 22:43

mfl

Mate, thank you for this. Still not getting it. Clearly I've reached my saturation point today... – Dan Aug 13 '18 at 22:52
Note that $f=\cos \theta.$ From the definition of $z=r\cos\theta$ you have $\cos \theta =z/r.$ Now, just check that $r^2=x^2+y^2+z^2$ (note that it is a parametrization of the sphere of radius $r$). – mfl Aug 13 '18 at 22:54
Understand that bit mate. It's the next step which has me flummoxed... – Dan Aug 13 '18 at 22:58
Now you have to get the derivative with respect to $x$ of the quotient $\dfrac{z}{\sqrt{x^2+y^2+z^2}}.$ – mfl Aug 13 '18 at 23:00
so $df/dx = -xz(x^2 + y^2 + z^2)^(-3/2)$ – Dan Aug 13 '18 at 23:08
I think that is the correct answer. – mfl Aug 13 '18 at 23:10
Trying to find a way to message you directly and thank you, but I'm yet to master navigating Stack Exchange. Thank you very much indeed for this mate. Much appreciated. – Dan Aug 13 '18 at 23:13
1

You're welcome. It is a pleasure to give hints when the OP does the work. – mfl Aug 13 '18 at 23:15

1 Answers1