Linearity of integration

related topics
{math, number, function}

In calculus, linearity is a fundamental property of the integral that follows from the sum rule in integration and the constant factor rule in integration. Linearity of integration is related to the linearity of summation, since integrals are thought of as infinite sums.

Let ƒ and g be functions. Now consider:

By the sum rule in integration, this is

By the constant factor rule in integration, this reduces to

Hence we have

Operator notation

The differential operator is linear — if we use the Heaviside D notation to denote this, we may extend D−1 to mean the first integral. To say that D−1 is therefore linear requires a moment to discuss the arbitrary constant of integration; D−1 would be straightforward to show linear if the arbitrary constant of integration could be set to zero.

Abstractly, we can say that D is a linear transformation from some vector space V to another one, W. We know that D(c) = 0 for any constant function c. We can by general theory (mean value theorem)identify the subspace C of V, consisting of all constant functions as the whole kernel of D. Then by linear algebra we can establish that D−1 is a well-defined linear transformation that is bijective on Im D and takes values in V/C.

That is, we treat the arbitrary constant of integration as a notation for a coset f + C; and all is well with the argument.

Full article ▸

related documents
Euler's theorem
Hilbert's Nullstellensatz
List of Fourier-related transforms
Z notation
Constant folding
Product of group subsets
Surjective function
Group object
Online algorithm
Discrete mathematics
The Third Manifesto
Sigmoid function
De Bruijn-Newman constant
Hurwitz polynomial
Derivative of a constant
Direct sum of groups
Essential singularity
Context-free language
Conjugate closure
Short five lemma
Dense set
Greibach normal form
Location parameter
Recursive language
Data element