diff-geo-chart.md

Differentiation in local coordinates

Tip

Here, we follow the convention in physics to label vector components by upper indices and the basis by lower indices.

Charts

Given a smooth manifold $M$ of dimension $m$, there is a smooth bijection $x \colon U \to \mathbb{R}^m$ defined on a neighborhood $U \subseteq M$.

The pair $(U,x)$ is called a chart of the manifold and assigns a set of coordinates $x^i(p)$ to points $p \in U$. The component functions $x^i \colon U \to \mathbb{R}$ project to the $i$th component, $p \mapsto {\rm proj}_i(x(p))$.

Tangent vectors in charts

A tangent vector at $p \in M$ is a linear map $v \colon C^{\infty}(M) \to \mathbb{R}$ (see definition in Tangent vectors on manifolds).

The tangent space $T_pM$ at $p$ is an $m$-dimensional vector space. This means that $T_pM$ has a basis of $m$ tangent vectors, usually denoted $\left( \frac{\partial}{\partial x^i} \right)_p$ , which depends on the choice of coordinates $x^i$ such that

$$ \left( \frac{\partial}{\partial x^i} \right)_p(x^j) = \delta _{ij} . $$

This also means that every tangent vector $v \in T_pM$ has a unique expansion

$$ v = \sum_{i=1}^m v^i \left( \frac{\partial}{\partial x^i} \right)_p $$

with coefficients $v^i \in \mathbb{R}$. Since

$$ v(x^j) = \sum _{i=1}^m v^i \left( \frac{\partial}{\partial x^i} \right)_p(x^j) = v^j , $$

the representation of a tangent vector $v$ in the chart $x$ is the column vector

$$ v(x) = \begin{bmatrix} v^1 \ \vdots \ v^m \end{bmatrix}. $$

Differentials in charts

To be continued...