A differentiable form on an open subset U⊆R2 is a map ω=ωxdx+ωydy for some ωx,ωy∈C1(U) For example, if f∈C1(U), ω=df=∂f∂xdx+∂f∂ydy is a differential form. My question is: what is the difference between ∂x and dx?

Question

A differentiable form on an open subset U⊆R2 is a map   ω=ωxdx+ωydy   for some ωx,ωy∈C1(U)   For example, if f∈C1(U), ω=df=∂f∂xdx+∂f∂ydy is a differential form.   My question is: what is the difference between ∂x and dx?

SlabydouluS62 · Accepted Answer

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 · Accepted Answer

There may be no such things as ∂x. You have the operator ∂∂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(∂∂x)=1 by definition