TITLE: What is the difference between $d$ and $\partial$?
QUESTION [17 upvotes]: After seeing the following equation in a lecture about tensor analysis, I became confused.
$$
\frac{d\phi}{ds}=\frac{\partial \phi}{\partial x^m}\frac{dx^m}{ds}
$$
What exactly is the difference between $d$ and $\partial$?

REPLY [12 votes]: As mentioned $d$ means total and $\partial$ partial derivative and are not the same. Total derivative also counts $x$ dependencies in other variables. For instance:
$$
f(x,v) = x^2 + v(x) \\
\frac{\partial f}{\partial x} = 2x \\
\frac{\partial f}{\partial v} = 1 \\
\frac{d f}{d x} = 2x + \frac{\partial v(x)}{\partial x} 
$$
Your formula most probably uses Einstein notation and is only a shorter way to write
$$
\frac{d\phi}{ds}= \sum_m \frac{\partial \phi}{\partial x^m}\frac{dx^m}{ds}
$$