Tangent space

In mathematics, the tangent space of a manifold is a generalization of tangent lines to curves in two-dimensional space and tangent planes to surfaces in three-dimensional space in higher dimensions. In the context of physics, the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold.

Informal description

File:Tangent Plane.svg

A pictorial representation of the tangent space of a single point

P(r,\theta ,\phi )

on a sphere. A vector in this tangent space represents a possible velocity (of something moving on the sphere) at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P} . After moving in that direction to a nearby point, the velocity would then be given by a vector in the tangent space of that point—a different tangent space that is not shown.

In differential geometry, one can attach to every point $x$ of a differentiable manifold a tangent space—a real vector space that intuitively contains the possible directions in which one can tangentially pass through $x$ . The elements of the tangent space at $x$ are called the tangent vectors at $x$ . This is a generalization of the notion of a vector, based at a given initial point, in a Euclidean space. The dimension of the tangent space at every point of a connected manifold is the same as that of the manifold itself.

For example, if the given manifold is a $2$ -sphere, then one can picture the tangent space at a point as the plane that touches the sphere at that point and is perpendicular to the sphere's radius through the point. More generally, if a given manifold is thought of as an embedded submanifold of Euclidean space, then one can picture a tangent space in this literal fashion. This was the traditional approach toward defining parallel transport. Many authors in differential geometry and general relativity use it.^[1]^[2] More strictly, this defines an affine tangent space, which is distinct from the space of tangent vectors described by modern terminology.

In algebraic geometry, in contrast, there is an intrinsic definition of the tangent space at a point of an algebraic variety Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V} that gives a vector space with dimension at least that of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V} itself. The points Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p} at which the dimension of the tangent space is exactly that of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V} are called non-singular points; the others are called singular points. For example, a curve that crosses itself does not have a unique tangent line at that point. The singular points of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V} are those where the "test to be a manifold" fails. See Zariski tangent space.

Once the tangent spaces of a manifold have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving in space. A vector field attaches to every point of the manifold a vector from the tangent space at that point, in a smooth manner. Such a vector field serves to define a generalized ordinary differential equation on a manifold: A solution to such a differential equation is a differentiable curve on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field.

All the tangent spaces of a manifold may be "glued together" to form a new differentiable manifold with twice the dimension of the original manifold, called the tangent bundle of the manifold.

Formal definitions

The informal description above relies on a manifold's ability to be embedded into an ambient vector space $\mathbb {R} ^{m}$ so that the tangent vectors can "stick out" of the manifold into the ambient space. However, it is more convenient to define the notion of a tangent space based solely on the manifold itself.^[3]

There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via the velocity of curves is intuitively the simplest, it is also the most cumbersome to work with. More elegant and abstract approaches are described below.

Definition via tangent curves

In the embedded-manifold picture, a tangent vector at a point $x$ is thought of as the velocity of a curve passing through the point $x$ . We can therefore define a tangent vector as an equivalence class of curves passing through $x$ while being tangent to each other at $x$ .

Suppose that $M$ is a $C^{k}$ differentiable manifold (with smoothness $k\geq 1$ ) and that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x\in M} . Pick a coordinate chart $\varphi :U\to \mathbb {R} ^{n}$ , where $U$ is an open subset of $M$ containing $x$ . Suppose further that two curves Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \gamma _{1},\gamma _{2}:(-1,1)\to M} with Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\gamma _{1}}(0)=x={\gamma _{2}}(0)} are given such that both Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \varphi \circ \gamma _{1},\varphi \circ \gamma _{2}:(-1,1)\to \mathbb {R} ^{n}} are differentiable in the ordinary sense (we call these differentiable curves initialized at $x$ ). Then Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \gamma _{1}} and $\gamma _{2}$ are said to be equivalent at $0$ if and only if the derivatives of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \varphi \circ \gamma _{1}} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \varphi \circ \gamma _{2}} at $0$ coincide. This defines an equivalence relation on the set of all differentiable curves initialized at $x$ , and equivalence classes of such curves are known as tangent vectors of $M$ at $x$ . The equivalence class of any such curve $\gamma$ is denoted by $\gamma '(0)$ . The tangent space of $M$ at $x$ , denoted by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T_{x}M} , is then defined as the set of all tangent vectors at $x$ ; it does not depend on the choice of coordinate chart $\varphi :U\to \mathbb {R} ^{n}$ .

File:Tangentialvektor.svg

The tangent space Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T_{x}M} and a tangent vector Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v\in T_{x}M} , along a curve traveling through Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x\in M} .

To define vector-space operations on Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T_{x}M} , we use a chart $\varphi :U\to \mathbb {R} ^{n}$ and define a map $\mathrm {d} {\varphi }_{x}:T_{x}M\to \mathbb {R} ^{n}$ by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\textstyle {\mathrm {d} {\varphi }_{x}}(\gamma '(0)):={\frac {\mathrm {d} (\varphi \circ \gamma )}{\mathrm {d} {t}}}(0),} where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \gamma \in \gamma '(0)} . The map Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathrm {d} {\varphi }_{x}} turns out to be bijective and may be used to transfer the vector-space operations on $\mathbb {R} ^{n}$ over to Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T_{x}M} , thus turning the latter set into an $n$ -dimensional real vector space. Again, one needs to check that this construction does not depend on the particular chart $\varphi :U\to \mathbb {R} ^{n}$ and the curve $\gamma$ being used, and in fact it does not.

Definition via derivations

Suppose now that $M$ is a $C^{\infty }$ manifold. A real-valued function Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f:M\to \mathbb {R} } is said to belong to Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {C^{\infty }}(M)} if and only if for every coordinate chart $\varphi :U\to \mathbb {R} ^{n}$ , the map Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f\circ \varphi ^{-1}:\varphi [U]\subseteq \mathbb {R} ^{n}\to \mathbb {R} } is infinitely differentiable. Note that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {C^{\infty }}(M)} is a real associative algebra with respect to the pointwise product and sum of functions and scalar multiplication.

A derivation at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x\in M} is defined as a linear map Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D:{C^{\infty }}(M)\to \mathbb {R} } that satisfies the Leibniz identity $\forall f,g\in {C^{\infty }}(M):\qquad D(fg)=D(f)\cdot g(x)+f(x)\cdot D(g),$ which is modeled on the product rule of calculus.

(For every identically constant function Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f={\text{const}},} it follows that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D(f)=0} ).

Denote Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T_{x}M} the set of all derivations at $x.$ Setting

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (D_{1}+D_{2})(f):={D}_{1}(f)+{D}_{2}(f)} and
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (\lambda \cdot D)(f):=\lambda \cdot D(f)}

turns $T_{x}M$ into a vector space.

Generalizations

Generalizations of this definition are possible, for instance, to complex manifolds and algebraic varieties. However, instead of examining derivations $D$ from the full algebra of functions, one must instead work at the level of germs of functions. The reason for this is that the structure sheaf may not be fine for such structures. For example, let $X$ be an algebraic variety with structure sheaf Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\mathcal {O}}_{X}} . Then the Zariski tangent space at a point Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p\in X} is the collection of all Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {k} } -derivations Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D:{\mathcal {O}}_{X,p}\to \mathbb {k} } , where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {k} } is the ground field and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\mathcal {O}}_{X,p}} is the stalk of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\mathcal {O}}_{X}} at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p} .

Equivalence of the definitions

For $x\in M$ and a differentiable curve Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \gamma :(-1,1)\to M} such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \gamma (0)=x,} define Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {D_{\gamma }}(f):=(f\circ \gamma )'(0)} (where the derivative is taken in the ordinary sense because Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f\circ \gamma } is a function from Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-1,1)} to $\mathbb {R}$ ). One can ascertain that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D_{\gamma }(f)} is a derivation at the point $x,$ and that equivalent curves yield the same derivation. Thus, for an equivalence class Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma'(0), } we can define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {D_{\gamma'(0)}}(f) := (f \circ \gamma)'(0), } where the curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma \in \gamma'(0) } has been chosen arbitrarily. The map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma'(0) \mapsto D_{\gamma'(0)} } is a vector space isomorphism between the space of the equivalence classes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma'(0) } and the space of derivations at the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x.}

Definition via cotangent spaces

Again, we start with a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^\infty } manifold Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M } and a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in M } . Consider the ideal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I } of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^\infty(M) } that consists of all smooth functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f } vanishing at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } , i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = 0 } . Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I^2 } are both real vector spaces, and the quotient space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I / I^2 } can be shown to be isomorphic to the cotangent space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^{*}_x M } through the use of Taylor's theorem. The tangent space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_x M } may then be defined as the dual space of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I / I^2 } .

While this definition is the most abstract, it is also the one that is most easily transferable to other settings, for instance, to the varieties considered in algebraic geometry.

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D } is a derivation at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D(f) = 0 } for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \in I^2 } , which means that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D } gives rise to a linear map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I / I^2 \to \mathbb{R} } . Conversely, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r: I / I^2 \to \mathbb{R} } is a linear map, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D(f) := r\left((f - f(x)) + I^2\right) } defines a derivation at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } . This yields an equivalence between tangent spaces defined via derivations and tangent spaces defined via cotangent spaces.

Properties

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M } is an open subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^{n} } , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M } is a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^{\infty} } manifold in a natural manner (take coordinate charts to be identity maps on open subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^{n} } ), and the tangent spaces are all naturally identified with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^{n} } .

Tangent vectors as directional derivatives

Another way to think about tangent vectors is as directional derivatives. Given a vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v } in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^{n} } , one defines the corresponding directional derivative at a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \mathbb{R}^{n} } by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall f \in {C^{\infty}}(\mathbb{R}^{n}): \qquad (D_{v} f)(x) := \left. \frac{\mathrm{d}}{\mathrm{d}{t}} [f(x + t v)] \right|_{t = 0} = \sum_{i = 1}^{n} v^{i} {\frac{\partial f}{\partial x^{i}}}(x). }

This map is naturally a derivation at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } . Furthermore, every derivation at a point in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^{n} } is of this form. Hence, there is a one-to-one correspondence between vectors (thought of as tangent vectors at a point) and derivations at a point.

As tangent vectors to a general manifold at a point can be defined as derivations at that point, it is natural to think of them as directional derivatives. Specifically, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v } is a tangent vector to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M } at a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } (thought of as a derivation), then define the directional derivative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_{v} } in the direction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v } by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall f \in {C^{\infty}}(M): \qquad {D_{v}}(f) := v(f). }

If we think of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v } as the initial velocity of a differentiable curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma } initialized at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } , i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = \gamma'(0) } , then instead, define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_{v} } by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall f \in {C^{\infty}}(M): \qquad {D_{v}}(f) := (f \circ \gamma)'(0). }

Basis of the tangent space at a point

For a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^{\infty} } manifold $M$ , if a chart $\varphi =(x^{1},\ldots ,x^{n}):U\to \mathbb {R} ^{n}$ is given with Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p\in U} , then one can define an ordered basis Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\textstyle \left\{\left.{\frac {\partial }{\partial x^{1}}}\right|_{p},\dots ,\left.{\frac {\partial }{\partial x^{n}}}\right|_{p}\right\}} of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T_{p}M} by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall i \in \{ 1,\ldots,n \}, ~ \forall f \in {C^{\infty}}(M): \qquad { \left. \frac{\partial}{\partial x^{i}} \right|_{p}}(f) := \left( \frac{\partial}{\partial x^{i}} \Big( f \circ \varphi^{- 1} \Big) \right) \Big( \varphi(p) \Big) . }

Then for every tangent vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v \in T_{p} M } , one has

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = \sum_{i = 1}^{n} v^{i} \left. \frac{\partial}{\partial x^{i}} \right|_{p}. }

This formula therefore expresses Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v } as a linear combination of the basis tangent vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \left. \frac{\partial}{\partial x^{i}} \right|_{p} \in T_{p} M } defined by the coordinate chart Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi: U \to \mathbb{R}^{n} } .^[4]

The derivative of a map

Every smooth (or differentiable) map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi: M \to N } between smooth (or differentiable) manifolds induces natural linear maps between their corresponding tangent spaces:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{d}{\varphi}_{x}: T_{x} M \to T_{\varphi(x)} N. }

If the tangent space is defined via differentiable curves, then this map is defined by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathrm{d}{\varphi}_{x}}(\gamma'(0)) := (\varphi \circ \gamma)'(0). }

If, instead, the tangent space is defined via derivations, then this map is defined by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\mathrm{d}{\varphi}_{x}(D)](f) := D(f \circ \varphi). }

The linear map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{d}{\varphi}_{x} } is called variously the derivative, total derivative, differential, or pushforward of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi } at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } . It is frequently expressed using a variety of other notations:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D \varphi_{x}, \qquad (\varphi_{*})_{x}, \qquad \varphi'(x). }

In a sense, the derivative is the best linear approximation to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi } near Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } . Note that when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = \mathbb{R} } , then the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{d}{\varphi}_{x}: T_{x} M \to \mathbb{R} } coincides with the usual notion of the differential of the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi } . In local coordinates the derivative of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi } is given by the Jacobian.

An important result regarding the derivative map is the following: Template:Math theorem This is a generalization of the inverse function theorem to maps between manifolds.

Notes

↑ do Carmo, Manfredo P. (1976). Differential Geometry of Curves and Surfaces. Prentice-Hall.:
↑ Dirac, Paul A. M. (1996) [1975]. General Theory of Relativity. Princeton University Press. ISBN 0-691-01146-X.
↑ Chris J. Isham (1 January 2002). Modern Differential Geometry for Physicists. Allied Publishers. pp. 70–72. ISBN 978-81-7764-316-9.
↑ Lerman, Eugene. "An Introduction to Differential Geometry" (PDF). p. 12. Archived from the original (PDF) on 2023-06-08. Retrieved 2021-04-09.

References

Lee, Jeffrey M. (2009), Manifolds and Differential Geometry, Graduate Studies in Mathematics, 107, Providence: American Mathematical Society.
Michor, Peter W. (2008), Topics in Differential Geometry, Graduate Studies in Mathematics, 93, Providence: American Mathematical Society.
Spivak, Michael (1965), Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus, W. A. Benjamin, Inc., ISBN 978-0-8053-9021-6.

External links

Tangent Planes at MathWorld

Template:Manifolds

[1] Carmo, Manfredo P. (1976). Differential Geometry of Curves and Surfaces. Prentice-Hall.:

[2] Dirac, Paul A. M. (1996) [1975]. General Theory of Relativity. Princeton University Press. ISBN 0-691-01146-X.

[Isham2002-3] Chris J. Isham (1 January 2002). Modern Differential Geometry for Physicists. Allied Publishers. pp. 70–72. ISBN 978-81-7764-316-9.

[4] Lerman, Eugene. "An Introduction to Differential Geometry" (PDF). p. 12. Archived from the original (PDF) on 2023-06-08. Retrieved 2021-04-09.

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Tangent space

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Informal description