Show that the vector \(f_{* x} v\) does not depend on the curve \(\boldsymbol{\varphi}\), but only on the vector \(\boldsymbol{v}\).

Let \(\boldsymbol{\varphi}\) be a curve and \(\boldsymbol{v}\) be a vector in tangent space \(T_x M\) at a point \(x\) on a manifold \(M\). Further, let \(f: M \to N\) be a smooth map between manifolds \(M\) and \(N\). Then, the pushforward \(f_{*x}v\) of the vector \(\boldsymbol{v}\) at point \(x\) is defined by the composition of maps \(f\) and \(d\boldsymbol{\varphi}\). To be more specific, $$ f_{*x}(\boldsymbol{v}) = df_{x}(\boldsymbol{v}) = (\boldsymbol{v} \cdot {\nabla})f $$ Where \(df_{x}\) is the differential of \(f\) at point \(x\) and \({\nabla}\) is the gradient operator.

Now we need to show that the vector \(f_{*x}(\boldsymbol{v})\) depends only on the tangent vector \(\boldsymbol{v}\) and not on the curve \(\boldsymbol{\varphi}\). We'll prove this by contradiction. Suppose that there is a curve \(\boldsymbol{\varphi}_1\) and another curve \(\boldsymbol{\varphi}_2\) such that the pushforward of \(\boldsymbol{v}\) under \(f\) is different for each curve. That is, $$ f_{*x}(\boldsymbol{v}_{\boldsymbol{\varphi}_1}) \neq f_{*x}(\boldsymbol{v}_{\boldsymbol{\varphi}_2}). $$ Now consider a smooth function \(g: N \to \mathbb{R}\) on the manifold \(N\). Using the definition of the pushforward \(f_{*x}(\boldsymbol{v})\), we have $$ f_{*x}(\boldsymbol{v}_{\boldsymbol{\varphi}_1}) = (\boldsymbol{v}_{\boldsymbol{\varphi}_1} \cdot {\nabla})f $$ Similarly, for curve \(\boldsymbol{\varphi}_2\), $$ f_{*x}(\boldsymbol{v}_{\boldsymbol{\varphi}_2}) = (\boldsymbol{v}_{\boldsymbol{\varphi}_2} \cdot {\nabla})f $$ By our assumption, these two vectors are not equal. However, both vectors are only related to \(\boldsymbol{v}\) through the gradient operator \({\nabla}\) and function \(f\), and they do not directly depend on the curve \(\boldsymbol{\varphi}\) itself. This contradicts the assumption that \(f_{*x}(\boldsymbol{v}_{\boldsymbol{\varphi}_1})\) and \(f_{*x}(\boldsymbol{v}_{\boldsymbol{\varphi}_2})\) are different due to the curves \(\boldsymbol{\varphi}_1\) and \(\boldsymbol{\varphi}_2\). Therefore, we conclude that the vector \(f_{*x}v\) is indeed independent of the curve \(\boldsymbol{\varphi}\), and depends only on the vector \(\boldsymbol{v}\).