Suppose that φ : M → N is a smooth map between smooth manifolds; then the differential of φ at a point x is, in some sense, the best linear approximation of φ near x. It can be viewed as generalization of the total derivative of ordinary calculus. Explicitly, it is a linear map from the tangent space of M at x to the tangent space of N at φ(x). Hence it can be used to push forward tangent vectors on M to tangent vectors on N. In mathematics, a total derivative may be either. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ...

The differential of a map φ is also called, by various authors, the derivative or total derivative of φ, and is sometimes itself called the pushforward.

Motivation

Let φ:U→V be a smooth map from an open subset U of R^m to an open subset V of Rⁿ. For any point x in U, the Jacobian of φ at x (with respect to the standard coordinates) is the matrix representation of the total derivative of φ at x, which is a linear map In mathematics, a smooth function is one that is infinitely (indefinitely) differentiable, i. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

from R^m to Rⁿ.

We wish to generalize this to the case that φ is a smooth function between any smooth manifolds M and N. On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...

The differential of a smooth map

Let φ : M → N be a smooth map of smooth manifolds. Given some x ∈ M, the differential (or (total) derivative) of φ at x is a linear map

from the tangent space of M at x to the tangent space of N at φ(x). The application of dφ_x to a tangent vector X is sometimes called the pushforward of X by φ. The exact definition of this pushforward depends on the definition one uses for tangent vectors (for the various definitions see tangent space). The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ...

If one defines tangent vectors as equivalence classes of curves through x then the differential is given by

Here γ is a curve in M with γ(0) = x. In other words, the pushforward of the tangent vector to the curve γ at 0 is just the tangent vector to the curve φγ at 0.

Alternatively, if tangent vectors are defined as derivations acting on smooth real-valued functions, then the differential is given by In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ...

Here X ∈ T_xM, therefore X is a derivation defined on M and f is a smooth real-valued function on N. By definition, the pushforward of X at a given x in M is in T_φ(x)N and therefore itself is a derivation.

After choosing charts around x and φ(x), F is locally determined by a smooth map On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...

between open sets of R^m and Rⁿ, and dφ_x has representation (at x)

in the Einstein summation notation, where the partial derivatives are evaluated at the point in U corresponding to x in the given chart. For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ...

Extending by linearity gives the following matrix

Thus the differential is a linear transformation, between tangent spaces, associated to the smooth map φ at each point. Therefore, in some chosen local coordinates, it is represented by Jacobian of the corresponding smooth map from R^m to Rⁿ. In general the differential need not be invertible. If φ is a local diffeomorphism, then the pushforward at x is invertible and its inverse gives the pullback of T_φ(x)N. In mathematics, a local diffeomorphism is a smooth map f : M → N between smooth manifolds such that for every point p of M there exists an open neighbourhood U of p such that f(U) is open in N and f|U : U → f(U) is a diffeomorphism. ... Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is...

The differential is frequently expressed using a variety of other notations such as

It follows from the definition that the differential of a composite is the composite of the differentials (i.e., functorial behaviour). This is the chain rule for smooth maps. In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ...

Also, the differential of a local diffeomorphism is an linear isomorphism of tangent spaces. In mathematics, a local diffeomorphism is a smooth map f : M → N between smooth manifolds such that for every point p of M there exists an open neighbourhood U of p such that f(U) is open in N and f|U : U → f(U) is a diffeomorphism. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

The differential on the tangent bundle

The differential of a smooth map φ induces, in an obvious manner, a bundle map (in fact a vector bundle homomorphism) from the tangent bundle of M to the tangent bundle of N, denoted by dφ or φ_*, which fits into the following commutative diagram: In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

where π_M and π_N denote the bundle projections of the tangent bundles of M and N respectively. Wikipedia does not have an article with this exact name. ...

Equivalently (see bundle map), φ_* = dφ is a bundle map from TM to the pullback bundle φ^*TN over M, which may in turn be viewed as a section of the vector bundle Hom(TM,φ^*TN) over M. In mathematics, a pullback bundle or induced bundle is a common construction in the theory of fiber bundles. ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ...

Pushforward of vector fields

Given a smooth map φ:M→N and a vector field X on M, it is not usually possible to define a pushforward of X by φ as vector field on N. For example, if the map φ is not surjective, there is no natural way to define such a pushforward outside of the image of φ. Also, if φ is not injective there may be more than one choice of pushforward at a given point. Nevertheless, one can make this difficulty precise, using the notion of a vector field along a map. Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

A section of φ^*TN over M is called a vector field along φ. For example, if M is a submanifold of N and φ is the inclusion, then a vector field along φ is just a section of the tangent bundle of N along M; in particular, a vector field on M defines such a section via the inclusion of TM inside TN. This idea generalizes to arbitrary smooth maps. In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ...

Suppose that X is a vector field on M, i.e., a section of TM. Then, applying the differential pointwise to X yields the pushforward φ_*X, which is a vector field along φ, i.e., a section of φ^*TN over M.

Any vector field Y on N defines a pullback section φ^*Y of φ^*TN with (φ^*Y)_x = Y_φ(x). A vector field X on M and a vector field Y on N are said to be φ-related if φ_*X = φ^*Y as vector fields along φ. In other words, for all x in M, dφ_x(X)=Y_φ(x). In mathematics, a pullback bundle or induced bundle is a common construction in the theory of fiber bundles. ...

In some situations, given a X vector field on M, there is a unique vector field Y on M which is φ-related to X. This is true in particular when φ is a diffeomorphism. In this case, the pushforward defines a vector field Y on N, given by In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...

A more general situation arises when φ is surjective (for example the bundle projection of a fiber bundle). Then a vector field X on M is said to be projectable if for all y in N, dφ_x(X_x) is independent of the choice of x in φ^-1({y}). This is precisely the condition that guarantees that a pushforward of X, as a vector field on N, is well defined. In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ...

See also

• Pullback

Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is...

References

• John M. Lee, Introduction to Smooth Manifolds, (2003) Springer Graduate Texts in Mathematics 218.
• Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-42627-2 See section 1.6.
• Ralph Abraham and Jarrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 1.7 and 2.3.