Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have a map $f(t,g,h)$ where $f:[0,1]\times C^1 \times C^1 \to \mathbb{R}.$

I want to define $$F(t,g,h) = \frac{d}{dt}f(t,g,h)$$ where $g$ and $h$ have no $t$-dependence in them. So $g(x) = t^2x$ would not be admissible if you want to calculate what $F$ is. How do I write this properly? Is it correct to write instead

Define $F$ by $F(t,\cdot,\cdot) = \frac{d}{dt}f(t,\cdot,\cdot)$.

But there is some ambiguity in the arguments. What is the best way to write it?

share|cite|improve this question
up vote 1 down vote accepted

This is called a partial derivative:

$$ F(t,g,h) = \frac{\partial}{\partial t} f(t,g,h) $$

The definition of a partial derivative is that you differentiate with respect to one variable as usual while keeping all others fixed. The different kind of $d$ is there to remind you of that, because in this case writing $\frac{d}{dt}f(t,g,h)$ would imply that $g$ and $h$ have some kind of $t$-dependence.

share|cite|improve this answer

Since $g$ and $h$ do not depend on $t$ one has $$F(t,g,h) := \frac{d}{dt}f(t,g,h)=f_{.1}(t,g,h)\ ,$$ where $f_{.1}$ denotes differentiation with respect to the first variable, whatever its name.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.