When $x$ is given, what is $\mathrm{d}x$?

Question

In the spherical coordinate, we know that $x = r\sin\theta\cos\phi$. Why does it imply that $$\mathrm{d}x = r\cos\theta\cos\phi \mathrm{d}\theta - r\sin \theta\sin \phi \mathrm{d} \phi?$$ It is first time I have faced this one. I tried to find some informations on Google, but I do not know what to search. Which rules do you use here?

Answer 1

This is called the total derivative. If $f=f(x_1,\dots,x_n)$ then $$df=\sum_{i=1}^n\frac{\partial f}{\partial x_i}dx_i.$$ You can take a look at the Wikipedia entry: Total dervative for more information.

Answer 2

hint

If $A=f(u,v,w)$

then under some conditions $$dA=\frac{\partial f}{\partial u}du+...$$

Answer 3

To understand this, you must know that $x,r,\theta,\phi$ are (almost) smooth functions defined on the usual Euclidean space, and $dx,dr,d\theta,d\phi$ are the derivatives of these functions.

Derivatives abide by a few rules: Linearity: if $u,v$ are smooth and if $t$ is a scalar, then $d(u+tv)=du+t\cdot dv$. Leibniz: if $u,v$ are smooth, then $d(uv)=u dv+vdu$. Chain rule: if $f$ is smooth and $u$ is a smooth function from $\mathbb{R}$ to itself, then $d(u \circ f)=u’\circ f df

As a consequence, constant functions’ derivatives vanish. So to get your formula, you just need to apply Leibniz’s rule and assume that in your context $r$ is a constant function.