Calculus can be defined on smooth manifolds.
Vector field (向量场) $X: M \mapsto T M$ on a smooth manifold $M$ is a continuous map that takes each point on the manifold to an element of the corresponding fiber of the tangent bundle: $X_p \in T_p M$. In other words, a vector field is a section of (the projection of) the tangent bundle: $\pi \circ X = \text{Id}_M$. We visualize a vector field as such: it attaches an arrow to each point of the manifold, which varies continuously across the manifold. Rough vector field is almost a vector field except that it is not necessarily continuous. Smooth vector field is a vector field that is a smooth map. Space of smooth vector fields $\mathfrak{X}(M)$ on a smooth manifold is the set of all smooth vector fields on the manifold endowed with pointwise addition and scalar multiplication, which is a real vector space and a module over $C^\infty(M)$: $X \in \mathfrak{X}(M)$, $f \in C^\infty(M)$, then $f X \in \mathfrak{X}(M)$. Vector field along a subset of a smooth manifold is a vector field on the subset. Smooth vector field along a subset of a smooth manifold is a vector field along the subset that can be smoothly extended at each point to a neighborhood in the manifold. Any smooth vector field $X$ along a closed subset $A$ of a smooth manifold $M$ can be extended to a smooth vector field $\tilde{X}$ on $M$ that vanishes on any open subset $U$ containing $A$.
Vector fields $(X_i)_{i=1}^k$ on a subset of a smooth manifold are linearly independent if they are linearly independent in the tangent space at each point; they span the tangent bundle if they span the tangent space at each point. Local frame $(e_i)_{i=1}^n$, or $(e_i)$, on an open subset $U$ of a smooth n-manifold $M$ is an n-tuple of vector fields on $U$ that are linearly independent and span the tangent bundle. Smooth frame is a frame consisting of smooth vector fields. Orthonormal vector fields on an open subset of a Euclidean space are vector fields whose values at each point are orthonormal w.r.t. the Euclidean inner product. Orthonormal frame on an open subset of a Euclidean space is a frame consisting of orthonormal vector fields. Gram-Schmidt Algorithm for Frames: A smooth orthonormal frame $(e_j)$ can be contructed from a smooth frame $(X_j)$ on an open subset $U$ of $\mathbb{R}^n$ such that $\text{Span}\{e_i\}_{i=1}^j = \text{Span}\{X_i\}_{i=1}^j$ for all $j$ at each point. Global frame for a smooth manifold is a frame on the entire manifold. Parallelizable manifold is a smooth manifold that admits a smooth global frame. Most smooth manifolds do not admit a smooth global frame, e.g. the sphere $\mathbb{S}^2$, and therefore are not parallelizable.
Coordinate vector field $\partial / \partial x^i$ w.r.t. a smooth chart on a smooth manifold is the vector field consisting of the i-th coordinate vector $\partial / \partial x^i |_p$ at each point of the coordinate domain. Coordinate vector fields are smooth vector fields. Coordinate frame $(\partial / \partial x^i)$ is the smooth local frame consisting of the coordinate vector fields. Component function $X^i: U \mapsto \mathbb{R}$ of a rough vector field in a smooth chart is the real-valued function on the coordinate domain that provides the i-th component of the field w.r.t. the coordinate frame associated with the chart: $X_p = X^i(p) \frac{\partial}{\partial x^i} \bigg{|}_p$. The restriction of a rough vector field to the coordinate domain of a smooth chart is smooth if and only if its component functions w.r.t. this chart are smooth.
Applying a smooth vector field $X \in \mathfrak{X}(M)$ to a smooth real-valued function $f \in C^\infty(U)$ on an open subset $U$ of $M$ is a smooth real-valued function $X f \in C^\infty(U)$, defined by $(X f)(p) = X_p f$. For a rough vector field $X: M \mapsto T M$, the following are equivalent: (1) $X$ is a smooth map; (2) $X$ is a closed operator on smooth functions; (3) $X$ is a closed operator on smooth functions on every open subset. Derivation on smooth real-valued functions is a linear transformation over $\mathbb{R}$ that satisfies the product rule: $D \in \mathcal{L}(C^\infty(M), C^\infty(M))$; $\forall f, g \in C^\infty(M)$, $D (f g) = (D f) g + f (D g)$. A transformation on smooth real-valued functions on a smooth manifold is a derivation if and only if it is a smooth vector field on the manifold.
Given a smooth map $F \in C^\infty(M, N)$, a vector field $X$ on $M$ and a vector field $Y$ on $N$ are $F$-related if $Y$ equals the pushforward of $X$ by $F$: $dF_p(X_p) = Y_{F(p)}$. Given a smooth map $F \in C^\infty(M, N)$ and smooth vector fields $X \in \mathfrak{X}(M)$, $Y \in \mathfrak{X}(N)$, the following are equivalent: (1) $X$ and $Y$ are $F$-related; (2) $X (f \circ F) = (Y f) \circ F$ for all $f \in C^\infty(U)$ on every open subset $U$ of $N$; (3) the composition of $F$ with any integral curve of $X$ is an integral curve of $Y$. Pushforward $F_∗ X$ of a smooth vector field $X \in \mathfrak{X}(M)$ by a diffeomorphism $F: M \mapsto N$ is the smooth vector field on $N$ that is $F$-related to $X$, defined by $(F_∗ X)_q = dF_{F^{-1}(q)}(X_{F^{-1}(q)})$. The pushforward of $X$ by $F$ is the only smooth vector field on $N$ that is $F$-related to $X$. A vector field $X$ on a smooth manifold $M$ is tangent to a smooth submanifold $S$ if it lies in the tangent subspace $T_p S$ at every point $p \in S$. A smooth vector field $X$ on a smooth manifold $M$ is tangent to an embedded submanifold $S$ if and only if applying $X$ to any smooth real-valued function on $M$ that equals zero on $S$ gives a function that also equals zero on $S$: $f \in C^\infty(M)$, $f|_S = 0$, then $(X f)|_S = 0$. Restricting Vector Fields to Submanifolds: If a smooth vector field $Y$ on a smooth manifold $M$ is tangent to a smooth submanifold $S$, then the restricted vector field $Y|_S$ is the only smooth vector field on $S$ that is $\iota$-related to $Y$, where $\iota: S \mapsto M$ is the inclusion map.
Parametrized curve $\gamma: J \mapsto M$ in a manifold $M$ is a continuous map from an interval $J \subset \mathbb{R}$ to the manifold. The range of a smooth parametrized curve in a smooth manifold need not be a 1-submanifold, e.g. if a curve crosses itself, the subspace topology on the curve fails to be a manifold topology. Starting point $\gamma(0)$ of a curve $\gamma$ is its value at $t = 0$ if $0 \in J$. Curve segment is a curve whose domain is a compact interval: $J = [a, b]$. Starting point $\gamma(a)$ and ending point $\gamma(b)$ of a curve segment are the values of the ends of its domain. Closed curve segment is a curve segment with identical end points: $\gamma(a) = \gamma(b)$. Smooth curve in a smooth manifold is a curve that is a smooth map. Velocity $\gamma'(t)$ or $\dot{\gamma}(t)$ of a smooth curve at a time instance is the pushforward of the coordinate vector $d/dt|_t$ by the curve: $\gamma'(t) = d \gamma_t \left(\frac{d}{d t} \bigg{|}_t \right)$. Velocity $\gamma'(t)$ is a tangent vector of $M$ at $\gamma(t)$. Regular curve in a smooth manifold is a smooth curve with nonzero velocities. Piecewise regular curve segment, or "admissible curve" for short, in a smooth manifold is a curve segment that can be partitioned into regular curve segments. Any two points of a connected smooth manifold can be connected by a piecewise regular curve segement.
Integral curve $\gamma: J \mapsto M$ of a vector field $V$ on a smooth manifold $M$ is a differentiable curve whose velocity equals the vector field everywhere on the curve: $\forall t \in J$, $\gamma'(t) = V_{\gamma(t)}$. The local coordinate representation of integral curves in a smooth chart is equivalent to the solutions of the system of ordinary differential equations (ODEs) $\dot{\gamma}^i(t) = V^i (\gamma^i(t))_{i=1}^n$, which is why such curves are called "integral curves". Maximal integral curve is an integral curve that cannot be extended to an integral curve on a larger open interval.
Global flow $\theta: \mathbb{R} \times M \mapsto M$ on a smooth manifold $M$ is a continuous map such that $\forall s, t \in \mathbb{R}$, $\forall p \in M$, $\theta(0, p) = p$, $\theta(t, \theta(s, p)) = \theta(s+t, p)$. Equivalently, a global flow is a continuous left $\mathbb{R}$-action on $M$, aka a "one-parameter group action". Every global flow induces a family $(\theta_t)_{t \in \mathbb{R}}$ of transformations on $M$ by $\forall p \in M$, $\theta_t(p) = \theta(t, p)$, and a family $(\theta^{(p)})_{p \in M}$ of curves in $M$ by $\forall t \in \mathbb{R}$, $\theta^{(p)}(t) = \theta(t, p)$. Every transformation induced by a global flow is a homeomorphism; it is a diffeomorphism if the global flow is a smooth map. Flow domain $\mathscr{D}$ for a smooth manifold $M$ is a subset of $\mathbb{R} \times M$ such that for each $p \in M$ the subset $\mathscr{D}^{(p)} = \{t: (t, p) \in \mathscr{D}\}$ is an open interval containing zero. Flow domains looks like open tubes around $\{0\} \times M$. Local flow (流) $\theta: \mathscr{D} \mapsto M$ on a smooth manifold $M$ is a continuous map from a flow domain to the manifold such that $\forall p \in M$, $\forall s \in \mathscr{D}^{(p)}$, $\forall t \in \mathscr{D}^{(\theta(s, p))} \cap (\mathscr{D}^{(p)} -s)$, $\theta(0, p) = p$, $\theta(t, \theta(s, p)) = \theta(s+t, p)$.
Maximal flow is a flow that that cannot be extended to a flow on a larger flow domain. Infinitesimal generator $V$ of a smooth flow $\theta$ on $M$, is the rough vector field on $M$ defined by $V_p = \theta^{(p)'}(0)$. The infinitesimal generator $V$ of $\theta$ is a smooth vector field on $M$, and each curve $\theta^{(p)}$ is an integral curve of $V$. Flow generated by a smooth vector field is a smooth maximal flow, if exists, whose infinitesimal generator is the field. Fundamental theorem on flows: Every smooth vector field $V$ on a smooth manifold $M$ (tangent to the boundary) generates a unique smooth maximal flow $\theta$. The curve $\theta^{(p)}: \mathscr{D}^{(p)} \mapsto M$ is the unique maximal integral curve of $V$ starting at each $p \in M$. If $s \in \mathscr{D}^{(p)}$, then $\mathscr{D}^{(\theta(s, p))} = \mathscr{D}^{(p)} - s$. For all $t \in \mathbb{R}$, $M_t = \{p: (t, p) \in \mathscr{D}\}$ is an open subset of $M$, and $\theta_t: M_t \mapsto M_{-t}$ is a diffeomorphism with inverse $\theta_{-t}$. Complete vector field is a smooth vector field that generates a global flow. Every compactly-supported smooth vector field on a smooth manifold is complete. Every smooth vector field on a compact smooth manifold is complete.
Flowout Theorem: Given an embedded submanifold $S$ of a smooth manifold $M$ and a smooth vector field $V$ on $M$ that is nowhere tangent to $S$, let $V$ generates flow $\theta$ with flow domain $\mathscr{D}$, denote restricted flow domains $\mathscr{O} = \{(t, p) \in \mathscr{D}: p \in S\}$ and $\mathscr{O}_\delta = \{(t, p) \in \mathscr{O} : |t| < \delta(p)\}$, where $\delta$ is a smooth positive function on $S$, then: (1) the restricted flow $\theta|_\mathscr{O}$ is a smooth immersion; (2) the coordinate vector field $\partial / \partial t$ on $\mathscr{O}$ is $\theta|_\mathscr{O}$-related to $V$; (3) the restricted flow $\theta|_{\mathscr{O}_\delta}$ can be injective for some $\delta$, and thus its range $\theta(\mathscr{O}_\delta)$---called a flowout (流出) from $S$ along $V$--- is an immersed submanifold of $M$ containing $S$, and $V$ is tangent to this submanifold; (4) if $S$ has codimension one, then the restricted flow $\theta|_{\mathscr{O}_\delta}$ is a diffeomorphism onto the flowout, which is an open submanifold of $M$.
Equilibrium point of a flow $\theta: \mathscr{D} \mapsto M$ on a smooth manifold $M$ is a point $p$ in $M$ such that $\forall t \in \mathscr{D}^{(p)}$, $\theta(t, p) = p$. Singular point (奇点) or zero of a vector field on a smooth manifold is a point where the vector field is zero: $V_p = 0$. Regular point (常点) of a vector field on a smooth manifold is a point where the field is nonzero. The singular points of a smooth vector field are precisely the equilibrium points of the flow it generates. Canonical Form Near a Regular Point: A smooth vector field $V$ matches the first coordinate vector field w.r.t. a smooth chart on a neighborhood $U$ of any regular point $p$, and the first coordinate can be a local defining function for any embedded hypersurface $S$ containing $p$ given that $V$ is not tangent to $S$ at $p$: $V|_U = \partial/\partial x^1$, $S \cap U = (x^1)^{-1}(0)$.
Real-valued first-order partial differential equations (PDEs) can be reduced to ODEs by the theory of flows. Linear first-order Cauchy problem is a problem of finding a smooth real-valued function $u$ in a neighborhood of an embedded hypersurface $S$ in a smooth manifold $M$ that satisfies a linear first-order PDE $A u + b u = f$ and an initial condition $u|_S = \phi$, where $A$ is a smooth vector field on $M$, $b$ and $f$ are smooth real-valued functions on $M$, and $\phi$ is a smooth real-valued function on $S$. Characteristic line (特征线)... A linear first-order Cauchy problem is noncharacteristic if $A$ is nowhere tangent to $S$. If a linear first-order Cauchy problem is noncharacteristic, then it has a unique solution in a flowout from the initial hypersurface along the vector field. Given a restricted flow domain $\mathscr{O}_\delta$ that satisfies the Flowout Theorem, composition with the restricted flow $\theta_\delta := \theta|_{\mathscr{O}_\delta}$ of $A$ transforms the flowout to the restricted flow domain where $A$ is in its canonical form $\partial/\partial t$, so the PDE becomes a linear first-order ODE $\frac{\partial \hat{u}}{\partial t} + \hat{b} \hat{u} = \hat{f}$ with initial condition $\hat{u}(0) = \phi$, where $\hat{u} = u \circ \theta_\delta$, and $\hat{b}, \hat{f}$ are similarly defined. Thus the solution in the flowout is $u = \hat{u} \circ \theta_\delta^{-1}$, where $\hat{u}(t) = e^{-B(t)} \left(\phi + \int_0^t \hat{f}(\tau) e^{B(\tau)}~d \tau\right)$ and $B(t) = \int_0^t \hat{b}(\tau)~d\tau$.
1-jet bundle $J^1 M$ of a smooth manifold $M$ is the smooth vector bundle $J^1 M = \mathbb{R} \times T^∗ M \mapsto M$, with fibers $\mathbb{R} \times T_x^∗ M$. 1-jet $j^1 u$ of a smooth function $u \in C^\infty(M)$ is the section of the 1-jet bundle $J^1 M$ defined by $j^1 u = (u, d u)$. First-order Cauchy problem is a problem of finding a real-valued function $u$ in a neighborhood of an embedded hypersurface $S$ in a smooth manifold $M$ that satisfies a first-order PDE $F(x, u, d u) = 0$ and an initial condition $u|_S = \phi$, where $\phi$ is a smooth real-valued function on $S$ and $F$ is a smooth real-valued function on an open subset of the 1-jet bundle $J^1 M$. A first-order Cauchy problem is noncharacteristic if there is a smooth section $\sigma$ of $T^∗ M|_S$ that takes values in $W$ and satisfies $\sigma(x)|_{T_x S} = d \phi(x)$ and $F(x, \phi(x), \sigma(x)) = 0$ on all points $x$ in $S$, and the vector field $A^{\phi, \sigma}$ along $S$ is nowhere tangent to $S$, defined as $A^{\phi, \sigma}|_x = \sum_{i=1}^n \frac{\partial F}{\partial \xi_i}(x, \phi(x), \sigma(x)) \frac{\partial}{\partial x^i}$. If a first-order Cauchy problem is noncharacteristic, then it has a smooth solution on a neighborhood of each point on the hypersurface.
Exterior differentiation allows for a generalization of differential operators such as gradient, divergence, curl, and Laplacian.
Covector field $\omega$ is a local or global section of the cotangent bundle. The value $\omega_p$ of a covector field at a point is denoted by subscript, while parentheses are reserved for the action $\omega(v)$ of a covector on a vector. We visualize a covector field as such: in each tangent space, it defines a linear hyperplane as the zero set and a parallel affine hyperplane as the level set of one, both of which vary continuously across the manifold. As with vector fields, a rough field needs not be continuous, and a smooth field is smooth. Action $\omega(X)$ of a rough covector field on a vector field on a smooth manifold is the real-valued function on the manifold that equals the action of the covector on the vector at each point: $\forall p \in M$, $\omega(X)(p) = \omega_p(X_p)$. Space of smooth covector fields $\mathfrak{X}^∗ (M)$ on a smooth manifold, endowed with pointwise vector addition and scalar multiplication, is a real vector space and a module over $C^\infty(M)$.
Local coframe $(\varepsilon^i)_{i=1}^n$ is a local frame for the cotangent bundle. Smooth coframe is a coframe consisting of smooth covector fields. Global coframe is a coframe on the entire manifold. Component functions $\omega_i: U \mapsto \mathbb{R}$ of a rough covector field w.r.t. a coframe $(\varepsilon^i)$ are the maps whose values form the coordinate representation of the covector at each point: $\omega_i(p) = \omega_p(e^i |_p)$, where $(e_i)$ is the dual frame. Given a coframe, a covector field can be written uniquely as $\omega = \omega_i \varepsilon^i$. Coordinate coframe $(\lambda^i)$ is a smooth local coframe consisting of the coordinate covector fields associated with a smooth chart. A coframe $(\varepsilon^i)$ and a frame $(e_i)$ are dual to each other if their values at each point are dual basis: $\varepsilon^i(e_j) = \delta^i_j$. Component functions of a rough covector field w.r.t. a smooth chart are the component functions of the field w.r.t. the coordinate coframe: $\omega_i(p) = \omega_p(\partial/\partial x^i |_p)$. The action of a rough covector field on a vector field equals the sum of products of their component functions in any smooth frame and its dual coframe: $\omega(X)(p) = \omega_i X^i$.
Pullback $F^∗ \omega$ of a covector field $\omega$ on $N$ by a smooth map $F \in C^\infty(M, N)$ is the rough covector field on $M$ whose value at each point equals the pullback of the covector field at that point: $(F^∗ \omega)_p = d F_p^∗ (\omega_{F(p)})$, i.e. $\forall v \in T_p M$, $(F^∗ \omega)_p (v) = \omega_{F(p)}(d F_p (v))$. The pullback of any covector field by a smooth map is a covector field; if the covector field is smooth, its pullback is also smooth. Restriction $\iota^∗ \omega$ of a smooth covector field $\omega \in \mathfrak{X}^∗ (M)$ to a smooth submanifold is the pullback of the field by the inclusion map $\iota: S \mapsto M$; equivalently, it is the restriction of the covector field to vectors tangent to the submanifold.
Differential form of degree $k$ or k-form $\omega$ is a section of the alternating k-tensor bundle, i.e. an alternating k-tensor field, aka a k-covector field. Space of smooth k-forms $\Omega^k(M)$ on a smooth manifold is the space of smooth alternating k-tensor fields: $\Omega^k(M) = \Gamma(\Lambda^k T^∗ M)$. The space of smooth 1-forms is just the space of smooth covector fields: $\Omega^1(M) = \mathfrak{X}^∗ (M)$. Sum space of smooth differential forms $\Omega^∗ (M)$ on a smooth n-manifold is the direct sum of all the smooth k-form spaces on the manifold: $\Omega^∗ (M) = \oplus_{k=0}^n \Omega^k(M)$. Exterior algebra $(\Omega^∗ (M), \wedge)$ of a smooth n-manifold $M$ is the associative, anticommutative graded algebra consisting of its space of smooth differential forms and the pointwise wedge product.
Component function $\omega_I$ of a rough k-form w.r.t. a smooth chart is the action of the k-form on the k-tuple of coordinate vector fields indexed by an increasing multi-index: $\omega_I = \omega(\partial/\partial x^i)_{i \in I}$. Given a smooth chart, every k-form can be written uniquely as a linear combintion of elementary k-forms based on the coordinate coframe and increasing multi-indices of length $k$: $\omega = \sum_I' \omega_I d x^I$, where $d x^I = \wedge_{i \in I} d x^i$.
Pullback $F^∗ \omega$ of a k-form on $N$ by a smooth map $F \in C^\infty(M, N)$ is the rough k-form on $M$ whose value at each point equals the pullback of the k-covector at that point: $(F^∗ \omega)_p = d F_p^∗ (\omega_{F(p)})$, i.e. $\forall v_i \in T_p M$, $(F^∗ \omega)_p (v_i)_{i=1}^k = \omega_{F(p)}(d F_p (v_i))_{i=1}^k$. The pullback of any k-form by a smooth map is a k-form; if the k-form is smooth, its pullback is also smooth. Given a smooth chart $(y^i)$ on the codomain, the pullback of a k-form by a smooth map can be written as: $F^∗ (\sum_I' \omega_I d y^I) = \sum_I' (\omega_I \circ F) \bigwedge_{i \in I} d (y^i \circ F)$. Given a smooth chart $(x^i)$ on the domain and a smooth chart $(y^i)$ on the codomain, the pullback of an n-form by a smooth map can be written as: $F^∗ (u (\wedge_{i=1}^n d y^i)) = (u \circ F) (\det DF) (\wedge_{i=1}^n d y^i)$, where $\det DF$ is the determinant of the Jacobian matrix of the map in these coordinates.
The most important application of covector field is to allow for an invariant definition of the differential of a smooth real-valued function on a smooth manifold. Differential (微分) $d f$ of a smooth real-valued function on a smooth manifold is the covector field defined by $\forall p \in M$, $\forall v \in T_p M$, $d f_p(v) = v f$. Due to the canonical identification $T_p \mathbb{R} \leftrightarrow \mathbb{R}$, the definitions of the differential of a smooth real-valued function as a tangent map $d f: T M \mapsto T \mathbb{R}$ and as a covector field where $d f_p: T_p M \mapsto \mathbb{R}$ are the same. The action of the differential of a smooth real-valued function on a vector field is thus $d f(X) = X f$. The differential of a smooth function is a smooth covector field: $d: C^\infty(M) \mapsto \mathfrak{X}^∗ (M)$. The component functions of a differential in a smooth chart are the partial derivatives w.r.t. those coordinates: $d f = \frac{\partial f}{\partial x^i} \lambda^i$. The differential of a coordinate function is the corresponding coordinate covector field: $d x^i = \lambda^i$; we therefore use $d x^i$ to denote a coordinate covector field.
Map of degree m on a graded algebra $A = \oplus_k A^k$ is a linear transformation that maps each subspace to the subspace $m$ indices higher. Antiderivation on a graded algebra is a linear transformation such that $T (x \times y) = (T x) \times y + (-1)^k x \times (T y)$ where $x \in A^k$. Exterior differentiation (外微分) $d: \Omega^* (M) \mapsto \Omega^* (M)$ of smooth forms is the unique extension of the differential $d: C^\infty(M) \mapsto \mathfrak{X}^* (M)$ to an antiderivation of degree +1 on the exterior algebra whose square is zero. Exterior differentiation has the following properties: (1) map of degree +1: $\forall k \in \{i\}_{i=0}^n$, $d \in \mathcal{L}(\Omega^k(M), \Omega^{k+1}(M))$; (2) antiderivation: $d(\omega \wedge \eta) = d \omega \wedge \eta + (-1)^k \omega \wedge d \eta$, where $\omega \in \Omega^k(M)$; (3) repeated action vanishes: $d \circ d = 0$. (4) commutes with pullbacks: $\forall F \in C^\infty(M, N)$, $\forall \omega \in \Omega^* (N)$, $F^* (d \omega) = d(F^* \omega)$. Exterior derivative (外导数) $d \omega$ of a smooth k-form on an open submanifold or a regular domain of a Euclidean space is the (k+1)-form defined by $d(\sum_I' \omega_I d x^I) = \sum_I' d \omega_I \wedge d x^I$, where $(d x^i)$ is the standard coordinate coframe. In particular, the exterior derivative of a smooth 1-form can be written as: $d(\omega_j d x^j) = \sum_{i<j} \left( \frac{\partial \omega_j}{\partial x^i} - \frac{\partial \omega_i}{\partial x^j} \right) d x^i \wedge d x^j$. Given a smooth chart, the exterior differentiation can be written in the form of the exterior derivative, where $(d x^i)$ is the coordinate coframe.
Exact covector field or exact differential is a smooth covector field that equals the differential of a smooth real-valued function: $\exists f \in C^\infty(M)$, $\omega = d f$. We call this function a potential for the exact covector field. The potentials for an exact covector field differ only by a constant on each component of the manifold. Conservative covector field is a smooth covector field whose line integral over every piecewise smooth closed curve segment is zero; equivalently, its line integrals over piecewise smooth curve segments are path-independent, i.e. only depend on the starting and ending points. A smooth covector field is conservative if and only if it is exact. Closed covector field is a smooth covector field whose Jacobian in every smooth chart is symmetric, or equivalently, whose Jacobian in every chart in a smooth atlas is symmetric: $\frac{\partial \omega_j}{\partial x^i} = \frac{\partial \omega_i}{\partial x^j}$. Every exact covector field is closed. The pullback of a covector field by a local diffeomorphism preserves closedness and exactness of the covector field. Star-shaped subset of a vector space is a subset that includes the line segment between one point and any point in the subset: $\exists c \in A$, $A = \cup_{p \in A} \overline{cp}$. Every convex subset is star-shaped. Poincaré Lemma for Covector Fields: Every closed covector field on a star-shaped open subset of a Euclidean space $\mathbb{R}^n$ or a closed upper half-space $\mathbb{H}^n$ is exact. Every closed covector field is exact on a collection of open sets that cover the manifold. Every closed covector field is exact on any simply connected manifold. Exact k-form is a k-form that equals the exterior differentiation of a smooth (k-1)-form: $\exists \eta \in \Omega^{k-1}(M)$, $\omega = d \eta$. Closed k-form is a smooth k-form whose exterior differentiation is zero: $d \omega = 0$. Every exact differential form is closed. Every closed differential form is locally exact.
Every pseudo-Riemannian metric $g$ is equivalent to a smooth bundle isomorphism $\hat{g}: T M \mapsto T^∗ M$ defined by $\hat{g}(v)(w) = g_p(v, w)$. Musical isomorphisms between smooth vector fields and smooth covector fields on a pseudo-Riemannian manifold $(M, g)$ are the two vector space (and module) isomorphisms flat (降X) or lower an index (降指标) $\flat: \mathfrak{X}(M) \mapsto \mathfrak{X}^∗ (M)$ and sharp (升ω) or raise an index (升指标) $\sharp: \mathfrak{X}^∗ (M) \mapsto \mathfrak{X}(M)$ defined by $X^\flat (Y) = \hat{g}(X)(Y) = g(X, Y)$ and $\omega^\sharp = \hat{g}^{-1}(\omega)$. Given a smooth local frame $(e_i)$ and its dual coframe $(\varepsilon^i)$, the musical isomorphisms have coordinate representations $X^\flat = g_{ij} X^i \varepsilon^j$ and $\omega^\sharp = g^{ij} \omega_i e_j$, where $(g^{ij})$ is the inverse of the matrix representation of the pseudo-Riemannian metric so that $g^{ij} g_{jk} = \delta^i_k$.
Gradient $\text{grad}~f$ of a smooth real-valued function on a pseudo-Riemannian manifold is the vector field obtained from the differential of the function by raising an index: $\text{grad}~f = (d f)^\sharp$. Gradient and differential on a Riemannian manifold are related by $d f_p = \langle \text{grad}~f|_p, \cdot \rangle_g$. Given a smooth local frame, gradient can be written as $\text{grad}~f = g^{ij} (e_i f) e_j$. Gradient and differential have the same coordinate representation in any orthonormal frame.
Divergence $\text{div}~X$ of a smooth vector field on a Riemannian n-manifold $(M, g)$ is the smooth real-valued function that locally satisfies the equation $d(X \lrcorner d V_g) = (\text{div}~X) d V_g$, where $d V_g$ is the Riemannian density. Given a smooth coordinate frame, divergence can be written as $\text{div}~X = (\sqrt{\det g})^{-1} \frac{\partial}{\partial x^i} (X^i \sqrt{\det g})$, where $\det g$ the determinant of the component matrix of the Riemannian metric in these coordinates.
Curl $\text{curl}~X$ of a smooth vector field on an oriented Riemannian 3-manifold is the smooth 2-form defined by $\text{curl}~X = \beta^{-1} d(X^\flat)$, where $\beta: TM \mapsto \Lambda^2 T^* M$ is the smooth bundle isomorphism defined by $\beta(X) = X \lrcorner d V_g$.
Geometric Laplacian or Laplace–Beltrami operator $\Delta f$ of a smooth real-valued function on a Riemannian manifold is the smooth real-valued function defined by the divergence of the gradient of the function: $\Delta f = \text{div}(\text{grad}~f)$. Many authors define the Laplacian with a negative sign so that its eigenvalues are nonnegative, but the given definition is much more common in Riemannian geometry. Given a smooth coordinate frame, Laplacian can be written as $\Delta f = (\sqrt{\det g})^{-1} \frac{\partial}{\partial x^i} \left( g^{ij} \frac{\partial f}{\partial x^j} \sqrt{\det g} \right)$.
Hodge star operator $∗$ is the smooth bundle homomorphism between alternating tensor bundles $\Lambda^k T^∗ M$ and $\Lambda^{n-k} T^∗ M$ on an oriented Riemannian n-manifold $(M, g)$ for each $k \in \{i\}_{i=0}^n$, determined by $\forall \omega, \eta \in \Omega^k(M)$, $\omega \wedge ∗ \eta = \langle \omega, \eta \rangle_g d V_g$. In particular, for smooth real-valued functions, $∗ f = f d V_g$. Laplace–Beltrami operator $\Delta \omega$ of a smooth k-form on an oriented compact Riemannian n-manifold is the smooth k-form defined by $\Delta \omega = d d^∗ \omega + d^∗ d \omega$, where $d^∗$ is the map of degree -1 defined by $d^∗ \omega = (-1)^{n(k+1)+1} ∗ d ∗ \omega$, where $∗$ is the Hodge star operator. Harmonic k-form is a smooth k-form in the kernel of the Laplace–Beltrami operator: $\Delta \omega = 0$. Harmonic function is a Harmonic 0-form. Harmonic analysis of real-valued functions on smooth manifolds, e.g. spherical harmonics.
Line integral $\int_J \omega$ of a smooth covector field over a compact interval is the ordinary integral of the standard coordinate representation of the field over the interval: $J = [a, b]$, $\omega \in \mathfrak{X}^∗ (J)$, $\omega_t = \hat \omega(t) d t$, then $\int_J \omega = \int_a^b \hat \omega(t) dt$. Line integral $\int_\gamma \omega$ of a smooth covector field over a smooth curve segment is the integral of the pullback of the field by the curve: $\forall \omega \in \mathfrak{X}^∗ (M)$, $\forall \gamma \in C^\infty(J, M)$, $\int_\gamma \omega = \int_J \gamma^∗ \omega$. Rewinding the definitions, the line integral equals the ordinary integral of the action of the covector field on curve velocity over the parameter interval: $\int_\gamma \omega = \int_a^b \omega_{\gamma(t)}(\gamma'(t))~dt$.
The line integral of the pullback of a smooth covector field over a piecewise smooth curve segment equals the line integral of the field over the composite curve: $\forall F \in C^\infty(M, N)$, $\forall \eta \in \mathfrak{X}^∗ (N)$, $\forall \gamma \in C^\infty(J, M)$, $\int_\gamma F^∗ \eta = \int_{F \circ \gamma} \eta$.
Reparametrization $\tilde{\gamma}$ of a piecewise smooth curve segment $\gamma$ by a strictly monotonic smooth function $\phi: \tilde{J} \mapsto J$ is the composition of the curve with the bijection: $\tilde{\gamma} = \gamma \circ \phi$. Forward reparametrization is a reparametrization by an increasing function. Backward reparametrization is a reparametrization by a decreasing function. The line integral of a smooth covector field over a piecewise smooth curve segment is invariant under forward reparametrization, and flips sign under backward reparametrization. Fundamental Theorem for Line Integrals: The line integral of the differential of a smooth real-valued function over a piecewise smooth curve segment equals the difference of function values at the ends of the curve: $\int_\gamma d f = f(\gamma(b)) - f(\gamma(a))$.
Integral of real-valued functions on oriented smooth manifolds cannot be defined independent of coordinates; however, integral can be defined intrinsically for differential forms. Domain of integration in a Euclidean space is a bounded subset whose boundary has measure zero. Integral $\int_D \omega$ of an n-form $\omega$ on the closure of a domain of integration in the n-dimensional Euclidean space is the integral of the standard coordinate representation of the n-form over the domain: $\omega = f (\wedge_{i=1}^n dx^i)$, then $\int_D \omega = \int_D f d V$, where $d V = \prod_{i=1}^n dx^i$. Any compact subset $K$ of an open subset $U$ of a Euclidean space or a closed upper half-space is included in an open domain of integration $D$ whose closure is also a subset of the open set: $K \subset D \subset \bar{D} \subset U$. Integral $\int_U \omega$ of a compactly supported n-form on an open subset of a Euclidean space or a closed upper half-space is the integral of the n-form on any domain of integration containing its support: $\text{supp}~\omega = K$, $K \subset D \subset \bar{D} \subset U$, then $\int_U \omega = \int_D \omega$. The integral of the pullback of a compactly supported n-form by an orientation-preserving diffeomorphism between open subsets of a Euclidean space or its closed upper half-space equals the integral of the n-form over the codomain: $\int_U F^∗ \omega = \int_{F(U)} \omega$; the integral flips sign if the diffeomorphism is orientation-reversing. Integral $\int_U \omega$ of an n-form on a compact subset of a positively-oriented smooth coordinate domain of an oriented smooth n-manifold is the integral of the pullback of the n-form by the inverse of the chart: $\int_U \omega = \int_{\phi(U)} (\phi^{-1})^∗ \omega$; the integral flips sign if the chart is negatively-oriented. Integral $\int_M \omega$ of a compactly-supported n-form on an oriented smooth n-manifold is the sum of integrals of n-forms $\psi_i \omega$, where $\{\psi_i\}_{i=1}^m$ is any smooth partition of unity subordinate to a finite open cover of the support of the n-form by positively or negatively oriented smooth charts: $\int_M \omega = \sum_{i=1}^m \int_{U_i} \psi_i \omega$. Integration Over Piecewise Parametrizations: Integral of a compactly-supported n-form on an oriented smooth n-manifold equals the sum of integrals of the n-form on a finite partition of its support such that there are positively-oriented smooth charts from their interior onto open domains of integration in the Euclidean n-space: $\text{supp}~\omega = K = \overline{\sqcup_{i=1}^m U_i}$, $\phi_i: U_i \cong D_i$, then $\int_M \omega = \sum_{i=1}^m \int_{D_i} (\phi_i^{-1})^∗ \omega$. Integration over piecewise parametrizations also works for boundary integrals of (n-1)-forms on any compact, oriented smooth n-manifold with corners, and integrals of densities on any compact smooth n-manifold. The integral map on compactly-supported n-forms on oriented smooth n-manifolds is a linear functional that is positive for positively-oriented orientation forms, is invariant under orientation-preserving diffeomorphisms, and flips sign upon orientation reversal.
Riemannian volume form $\omega_g$ or $d V_g$ of an oriented Riemannian n-manifold $(M, g)$ is the unique n-form on the manifold satisfying any of the following equivalent properties: (1) it equals the wedge product of any oriented orthonormal coframe, $\omega_g = \wedge_i ε^i$; (2) it maps any oriented orthonormal frame to one, $\omega_g(e_i) = 1$; (3) it equals the wedge product of any oriented coordinate coframe multiplied by the square root of the determinant of the matrix representation of the Riemannian metric, $\omega_g = \sqrt{\det g_{ij}} (\wedge_i dx^i)$. The notation $d V_g$ for a Riemannian volume form or a Riemannian density is just a convention, which does not mean it is the exterior derivative of an (n-1)-form. The boundary of an oriented Riemannian manifold is orientable if and only if there exists a global unit normal vector field on the boundary. The Riemannian volume form of the boundary of an oriented Riemannian manifold, given a global unit normal vector field, is the interior multiplication of the Riemannian volume form of the manifold by the vector field: $\omega_{\iota^∗ g} = (N \lrcorner \omega_g)|_{\partial M}$. Integral $\int_M f \omega_g$ of a compactly-supported continuous real-valued function over an oriented Riemannian manifold is the integral of the compactly-supported n-form $f \omega_g$ on the manifold.
Integral $\int_S \omega$ of a k-form on an oriented smooth n-manifold over an oriented smooth k-submanifold where the restriction of the form is compactly supported is the integral of the pullback of the k-form by the inclusion map of the k-submanifold: $\int_S \omega = \int_S \iota_S^∗ \omega$. Stokes’s Theorem: $\int_M d \omega = \int_{\partial M} \omega$.
Surface integral $\int_S \langle X, N \rangle_g dA$ of a smooth vector field $X$ over a compact oriented 2-dimensional smooth submanifold $S$ with boundary in an oriented Riemannian 3-manifold. Stokes’s Theorem for Surface Integrals: $\int_S \langle \text{curl}~X, N \rangle_g dA = \int_{\partial S} \langle X, T \rangle_g ds$.
Density $\mu: V^n \mapsto \mathbb{R}$ on an n-dimensional vector space is an n-variate real-valued function such that its action on linearly transformed vectors equls its action on the original vectors, multiplied by the absolute value of the determinant of the linear transformation: $\forall T \in \mathcal{L}(V, V)$, $\mu(T v_i)_{i=1}^n = |\det T| \mu(v_i)_{i=1}^n$. A density is not a tensor, because it is not linear over the real numbers in any of its arguments. Density space $\mathcal{D}(V)$ on an n-dimensional vector space is the vector space consisting of the set of all densities on the space, and pointwise addition and scalar multiplication. The density space on an n-dimensional vector space is the 1-dimensional vector space consisting of the absolute value map and the negative value map of the n-covectors on the underlying space: $\mathcal{D}(V) = \{|\omega|, -|\omega| : \omega \in \Lambda^n(V^∗)\}$. Positive density on an n-dimensional vector space is one whose values are positive on a basis: $\mu = |\omega|$. Negative density is defined analogously: $\mu = -|\omega|$.
Density bundle $\mathcal{D} M$ of a smooth manifold is the disjoint union of density spaces on all tangent spaces of the manifold, endowed with the natural projection map taking each point-indexed density to its point of tangent. The density bundle of a smooth manifold is a smooth line bundle over the manifold. Density $\mu$ on a smooth manifold is a section of the density bundle of the manifold. Positive density on a smooth manifold is one whose values are positive densities on all the tangent spaces. Any nonvanishing n-form determines a positive density by taking pointwise absolute value: $|\omega|_p = |\omega_p|$. Any density can be written as a positive density multiplied by a real-valued function: $\mu = f |\omega|$. Every smooth manifold admits a smooth positive density. Pullback $F^∗ \mu$ of a density on $N$ by a smooth map $F \in C^\infty(M, N)$ is the density on $M$ whose value at each point equals the pullback of the density at that point: $\forall v_i \in T_p M$, $(F^∗ \mu)_p (v_i)_{i=1}^n = \mu_{F(p)}(d F_p (v_i))_{i=1}^n$. The pullback of a smooth density by a smooth map is a smooth density.
Integral $\int_D \mu$ of a density on (the closure of) a domain of integration in the Euclidean n-space is the integral of the standard coordinate representation of the density over the domain: $\mu = f |\wedge_{i=1}^n dx^i|$, then $\int_D \mu = \int_D f d V$ where $d V = \prod_{i=1}^n dx^i$. Analogous to integral of n-forms, we have: integral of a compacly supported density on an open subset, $\int_U \mu = \int_D \mu$, which is diffeomorphism-invariant, $\int_U F^∗ \mu = \int_{F(U)} \mu$; integral on a compact subset of a smooth coordinate domain of a smooth manifold, $\int_U \mu = \int_{\phi(U)} (\phi^{-1})^∗ \mu$; integral of a compactly-supported density on a smooth manifold via a smooth partition of unity, $\int_M \mu = \sum_i \int_M \psi_i \mu$. The integral map on compactly-supported densities on smooth manifolds is a linear functional that is positive for positive densities and is invariant under diffeomorphisms.
Riemannian density $\mu_g$ or $d V_g$ on a Riemannian manifold $(M, g)$ is the unique smooth positive density that maps any orthonormal frame to one: $\mu_g(e_i) = 1$. Volume $\text{Vol}(M)$ of a compact Riemannian manifold is the integral of the Riemannian density on the manifold: $\text{Vol}(M) = \int_M d V_g$. Integral $\int_M f \mu_g$ of a compactly-supported continuous real-valued function over a Riemannian manifold is the integral of the density $f \mu_g$ on the manifold. For an oriented Riemannian manifold, its Riemannian density equals the absolute value map of its Riemannian volume form: $\mu_g = |\omega_g|$, and thus the integrals of a function as a density and an n-form are the same: $\int_M f \mu_g = \int_M f \omega_g$. Divergence Theorem: The integral of the divergence of a compactly-supported smooth vector field on a Riemannian manifold with boundary equals the integral of the inner product of the vector field and the outward-pointing unit normal vector field along the manifold boundary: $\int_M (\text{div}~X) \mu_g = \int_{\partial M} \langle X, N \rangle_g \mu_{\tilde{g}}$, where $\tilde{g}$ is the induced Riemannian metric on the manifold boundary.
Measure on smooth/Riemannian manifolds. Directional statistics deals with observations on n-spheres [@Brigant2019]. Sampling on manifolds [@Soize2016]. Measures on a smooth manifold are preserved in piecewise parametrizations, despite violating the topology. A unit n-volume $I^n$ should suffice for sampling on a connected compact Riemannian n-manifold, e.g. sampling on a sphere via geographic coordinates.
Distribution on smooth manifolds, a k-dimensional subbundle of the tangent bundle. Integral manifold of a distribution, a smooth k-submanifold. Involutive distribution. Frobenius theorem: Involutivity is (necessary and) sufficient for the existence of an integral manifold through each point.
Foliation (叶状结构). Global Frobenius Theorem.
Overdetermined systems of partial differential equations...