| Notation | Meaning/Definition | | --------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | $(U_i, \varphi_i)_{i \in I}$ | [[Atlas]] | | $(U, \varphi)$ | Chart | | $\widehat{U_i}$ | $\varphi_i(U_i)$ | | $(U, \varphi) \in X$ | Atlas in manifold structure | | $p \in X$ | Point in $X$ | | $\hat p$ | $\varphi(p)$, when context is clear | | $(U, \varphi, p, v)$ | Concrete tangent vector | | $T_{(U, \varphi, p)}X$ | [[Concrete Tangent Space\|Concrete tangent space]] at $p$ with respect to $(U, \varphi)$ | | $T_{p}X$ | [[Tangent Space\|Tangent space]] at $p$ | | $\hat{v}: v \in T_pX$ | Short for $(U, \varphi, p, v)$, when context is clear | | $\ol{v}: v \in T_pX$ | Short for $(U, \text{Id}, p, v)$, when the manifold is a subset of a Banach space | | $\ol{v}: v \in T_{(U, \varphi, p)}X$ | Short for the equivalence class of $(U, \varphi, p, v)$ | | $\widetilde{U}: U \subset X$ | $\pi^{-1}(U)$ in a [[Vector Bundle\|vector bundle]] | | $p = (x^1, \cdots, x^n)$ | Coordinates of a point in context of a chart $(U, \varphi)$. | | $E(x) = x^iE_i$ | Short for $E(x) = \sum_{i}x^iE_i$. Identifiable by the index showing up exactly twice. The *basis vectors* will have indices subscripted, and *coordinates* will have their indices superscripted. | | $D_v\vert_p f = v^i\frac{\partial}{\partial x^i}f$ | Derivative at $v$ in the direction of $p$. | | $\frac{\partial}{\partial x^i}\vert_a$ | Standard basis vector for the tangent space. The $i$-th partial derivative at $a$. | | $dF_p(v)$ | Differential of $F$ at $p$, evaluated in the direction of $v$. Defined by $dF_p(v) = v(f \circ F)$ where $f$ is a differentiable function. | | $\frac{\partial}{\partial x^i}\vert_{\varphi(p)}$ or $\frac{\partial}{\partial x^i}\vert_{p}$ | Standard basis vector for the abstract tangent space, in the context of a chart. | | $v = v^i\frac{\partial}{\partial x^i}\vert_p$ | Coordinate representation for an abstract tangent vector, in the context of a chart. | | $\frac{\partial F^i}{\partial x^j}(p)$ | Entry in the Jacobian matrix of $F$ at $p$, in the context of a pair of charts. | | $\psi \circ \varphi^{-1}(x) = (\tilde x^1(x), \cdots, \tilde x^n(x))$ | Coordinate representation of the transition map. | | $(\widetilde U, \widetilde \psi)$ | Adapted chart for the tangent bundle. | | $\gamma'(t_0) = d\gamma_{t_0}(1) = d\gamma \cdot \frac{d}{dt}\vert_{t_0}$ | Velocity of a curve at $t_0$. | | $\gamma'(t_0) = \frac{d\gamma^i}{dt}(t_0)\frac{\partial}{\partial x^i}\vert_{\gamma(t_0)}$ | Coordinate representation of the velocity of a curve, in the context of a chart. |