In differential geometry, **pushforward** is a linear approximation of smooth maps on tangent spaces. 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 a generalization of the total derivative of ordinary calculus. Explicitly, the differential 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* tangent vectors on *M**forward* to tangent vectors on *N*. The differential of a map *φ* is also called, by various authors, the **derivative** or **total derivative** of *φ*.

Let *φ* : *U* → *V* be a smooth map from an open subset *U* of to an open subset *V* of . 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

We wish to generalize this to the case that *φ* is a smooth function between *any* smooth manifolds *M* and *N*.

Let *φ* : *M* → *N* be a smooth map of smooth manifolds. Given some *x* ∈ *M*, the **differential** 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).

If tangent vectors are defined as equivalence classes of curves through *x* then the differential is given by

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

Alternatively, if tangent vectors are defined as derivations acting on smooth real-valued functions, then the differential is given by

for an arbitrary function and an arbitrary derivation at point (a derivation is defined as a linear map that satisfies the Leibniz rule, see: definition of tangent space via derivations). By definition, the pushforward of is in and therefore itself is a derivation, .

After choosing two charts around *x* and around *φ*(*x*), *φ* is locally determined by a smooth map

between open sets of and , 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.

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 the Jacobian matrix of the corresponding smooth map from to . 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*.

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.

Also, the differential of a local diffeomorphism is a linear isomorphism of tangent spaces.

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:

where *π*_{M} and *π*_{N} denote the bundle projections of the tangent bundles of *M* and *N* respectively.

induces a bundle map from *TM* to the pullback bundle *φ*^{∗}*TN* over *M* via

where and The latter map may in turn be viewed as a section of the vector bundle Hom(*TM*, *φ*^{∗}*TN*) over *M*. The bundle map *dφ* is also denoted by *Tφ* and called the **tangent map**. In this way, *T* is a functor.

Given a smooth map *φ* : *M* → *N* and a vector field *X* on *M*, it is not usually possible to identify a pushforward of *X* by φ with some vector field *Y* 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.

A section of *φ*^{∗}*TN* over *M* is called a **vector field along φ**. For example, if

Suppose that *X* is a vector field on *M*, i.e., a section of *TM*. Then, yields, in the above sense, 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

In some situations, given a *X* vector field on *M*, there is a unique vector field *Y* on *N* 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

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

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In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, and geometry.

In mathematics, a **vector-valued differential form** on a manifold *M* is a differential form on *M* with values in a vector space *V*. More generally, it is a differential form with values in some vector bundle *E* over *M*. Ordinary differential forms can be viewed as **R**-valued differential forms.

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In mathematics, the **differential geometry of surfaces** deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: *extrinsically*, relating to their embedding in Euclidean space and *intrinsically*, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

In mathematics, particularly differential topology, the **double tangent bundle** or the **second tangent bundle** refers to the tangent bundle (*TTM*,*π*_{TTM},*TM*) of the total space *TM* of the tangent bundle (*TM*,*π*_{TM},*M*) of a smooth manifold *M* . A note on notation: in this article, we denote projection maps by their domains, e.g., *π*_{TTM} : *TTM* → *TM*. Some authors index these maps by their ranges instead, so for them, that map would be written *π*_{TM}.

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

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