# Derivatives and the cotangent space

In Differentiable Manifolds, the derivative of a function $$f: M \rightarrow \mathbb{R}$$ at $$a$$ denoted by $$(df)_a$$ is defined as its image in the cotangent space: $$T_a^* = C^\infty(M)/Z_a$$, where $$Z_a$$ is the subspace of functions whose derivatives vanishes a $$a$$.

I'm having trouble understanding the following proposition:

Proposition 3.1 Let $$M$$ be an $$n$$-dimensional manifold, then

• the cotangent space $$T_a^*$$ at $$a \in M$$ is an n-dimensional vector space

• if $$(U, \varphi)$$ is a coordinate chart around $$x$$ with coordinates $$x_1, \dots, x_n$$, then the elements $$(dx_1)_a, \dots, (dx_n)_a$$ form a basis for $$T_a^*$$

• if $$f \in C^\infty(M)$$ and in the coordinate chart, $$f \varphi^{-1} = \phi(x_1, \dots, x_n)$$ then $$(df)_a = \sum_i \frac{\partial \phi}{\partial x_i} (\varphi(a))(dx_i)_a$$

Proof: If $$f \in C^\infty(M)$$, with $$f \varphi^{-1} = \phi(x_1, \dots, x_n)$$ then $$f - \sum \frac{\partial \phi}{\partial x_i} (\varphi(a))x_i$$ is a (locally defined) smooth function whose derivative vanishes at $$a$$, so $$(df)_a = \sum \frac{\partial f}{\partial x_i} (\varphi(a))(dx_i)_a$$ and $$(dx_1)_a, \dots, (dx_n)_a$$ span $$T_a^*$$.

More specifically, I wonder what is meant by $$f - \sum \frac{\partial \phi}{\partial x_i} (\varphi(a))x_i$$ since $$f$$ is a function on $$M$$ and the sum is a function on $$\mathbb{R}^n$$. I also wonder what is meant by $$(dx_i)_a$$ in this context since $$x_i$$ is not a function on $$M$$.

Let $\phi:U\to\mathbb{R}^n$. Then $x_i:U\to\mathbb{R}^n$ is defined to be the $i$th component function of $\phi$. In other words, $$\phi(p)=(x_1(p),\ldots,x_n(p))$$ for each $p\in U$.
So $x_i$ is in fact a function whose domain is the open set $U\subset M$.