Examples of "pushforward"
The pushforward measure is also denoted as formula_2.
The Grothendieck–Riemann–Roch theorem relates the pushforward map
where "X" is a vector on "M", and "d" denotes the pushforward.
The pushforward measure formula_8---called a distribution of "X" in this context---is a probability measure
We build the pushforward homomorphism as follows (for singular or simplicial homology):
where formula_6 is the pushforward along π of the vector "v" ∈ TUT"M".
The law of the process "X" is then defined to be the pushforward measure
A measurable function formula_76 induces a pushforward measure, – the probability measure formula_77 on formula_78 defined by
In the case of a fiber bundle, one can also define a pushforward map π
where formula_61 denotes the pushforward measure of formula_62 induced by the canonical projection map formula_63.
The pushforward along a vector function f with respect to vector v in R is given by
Conversely, if "f" is proper, for "Y" a subvariety of "X" the pushforward is defined to be
If "μ" is finite, "ν" can be taken to be the pushforward "π""μ", and then the "λ" are probabilities.
Further unwinding the definitions, the pushforward of a vector field under a map between manifolds is defined by
An -valued random vector is called "comonotonic", if its multivariate distribution (the pushforward measure) is comonotonic, this means
The pullback is naturally defined as the dual (or transpose) of the pushforward. Unraveling the definition, this means the following:
from the tangent space of "M" at "x" to the tangent space of "N" at "φ"("x"). The application of d"φ" 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 measure that associates to the set "A" the number γ("A"−"h") is the pushforward measure, denoted ("T")(γ). Here "T" : R → R refers to the translation map: "T"("x") = "x" + "h".. The above calculation shows that the Radon–Nikodym derivative of the pushforward measure with respect to the original Gaussian measure is given by
Nevertheless, it follows from this that if Φ is invertible, pullback can be defined using pushforward by the inverse function Φ. Combining these two constructions yields a pushforward operation, along an invertible linear map, for tensors of any rank ("r","s").
at each point "v" ∈ T"M"; here π : TT"M" → T"M" denotes the pushforward (differential) along the projection π : T"M" → "M" associated to the tangent bundle.