Evaluating a simple differential form

Question

Given the vector field $\partial_x$ and the one form $\,dx$, how would I evaluate $\,dx(\partial_x)$ and show that $\,dx(\partial_x)=1$.

Answer 1

This might boil down to a question about defining differentials but in general, if $f$ is any function defined on your manifold, the value of the differential $df$ at a point $p$ and a tangent vector $X_p$ is $$df_p(X_p) = X_p f.$$ To be completely explicit, let's write $\partial_x|_p$ for the value of the vector field $\partial_x$ in a point $p$. In your case, $f(x) = x$ and so $$df_p(\partial_x|_p) = \partial_x|_p f = \frac{\partial f}{\partial x}(p) = \frac{\partial x}{\partial x}(p) = 1,$$ for all points $p$.