Wrong pushforward of vector field definition on wikipedia Wikipedia claims that for $$\mathrm d \varphi_x:T_xM\to T_{\varphi(x)}N\,$$
we have for $X \in T_pM$ and $f \in C^{\infty}(N,\mathbb{R})$
$$\mathrm d\varphi_x(X)(f) = X(f \circ \varphi)$$ whereas I would have assumed that
$$\mathrm d\varphi_x(X)(f) = X(f \circ \varphi) \circ \varphi^{-1}.$$
Presumably, wikipedia is correct, but I currently don't see where I could be wrong here?
EDIT (See this one for explanation)
So yes, probably I would require that $\phi$ is a diffeomorphism, but in another question someone suggested the alternative equation to me and so I got confused which one is correct.
Cause I feel that $\mathrm d\varphi_x(X)(f)$ is a map $C^{\infty}(N,\mathbb{R})$ whereas $X(f \circ \varphi)$ is a map $C^{\infty}(M, \mathbb{R})$, so something does not add up there.

What does your use of $\varphi^{-1}$ mean, when the original smooth map $\varphi:M\to N$ need not have been a bijection (much less a diffeomorphism)?
As to what I suspect was your thought process, I suppose you wanted to use $\varphi^{-1}$ (which, as I've just said, may not mean anything anyway) to "bring things back" to $M$ for $X$ to act on them. However, you should note that $\mathrm{d}\varphi_x(X)(\,\cdot\,)$ as defined on Wikipedia already eats smooth functions on $N$ and outputs real numbers, there's no need to use a $\varphi^{-1}$.
