Joan Thompson

2022-01-14

A differentiable form on an open subset $U\subseteq {\mathbb{R}}^{2}$ is a map
$\omega ={\omega }_{x}dx+{\omega }_{y}dy$
for some ${\omega }_{x},{\omega }_{y}\in {C}^{1}\left(U\right)$
For example, if is a differential form.
My question is: what is the difference between $\partial x$ and $dx$?

SlabydouluS62

Expert

Step 1
dx is a basis vector in the space of 1-forms for functions in C1. On the other hand, ∂x is not a symbol with any meaning: it does denote any particular mathematical object. The symbol ∂x is merely an artifact of notation: ∂f∂x is merely how we choose to denote the partial derivative of f with respect to x. We could choose to denote it differently, avoiding the symbol ∂x altogether, since it has no mathematical significance. Why was this symbol chosen? I am sure there is some history behind the symbol, though I am not knowledgeable in it. But one could very well choose to denote it as fx instead, and now you communicate the same idea, without appealing to strange notation.

Lindsey Gamble

Expert

There may be no such things as $\partial x$. You have the operator $\frac{\partial }{\partial x}$. It is a member of the tangent space. As such, it is a differential operator on the space of germs of fucntions around a point.
dx is in the co-tangent space. It is a function on the tangent space.
You have $dx\left(\frac{\partial }{\partial x}\right)$=1 by definition

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