Linearity of integration
Encyclopedia
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
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.
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
, linearity is a fundamental property of the integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
that follows from the sum rule in integration
Sum rule in integration
In calculus, the sum rule in integration states that the integral of a sum of two functions is equal to the sum of their integrals. It is of particular use for the integration of sums, and is one part of the linearity of integration....
and the constant factor rule in integration. Linearity of integration is related to the linearity of summation
Summation
Summation is the operation of adding a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation. The numbers to be summed may be integers, rational numbers,...
, since integrals are thought of as infinite sums.
Let ƒ and g be functions. Now consider:
By the sum rule in integration
Sum rule in integration
In calculus, the sum rule in integration states that the integral of a sum of two functions is equal to the sum of their integrals. It is of particular use for the integration of sums, and is one part of the linearity of integration....
, this is
By the constant factor rule in integration, this reduces to
Hence we have
Operator notation
The differential operatorDifferential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
is linear — if we use the Heaviside D notation to denote this, we may extend D−1 to mean the first integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
. To say that D−1 is therefore linear requires a moment to discuss the arbitrary constant of integration
Arbitrary constant of integration
In calculus, the indefinite integral of a given function is only defined up to an additive constant, the constant of integration. This constant expresses an ambiguity inherent in the construction of antiderivatives...
; 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
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
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
Mean value theorem
In calculus, the mean value theorem states, roughly, that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc. Briefly, a suitable infinitesimal element of the arc is parallel to the...
)identify the subspace C of V, consisting of all constant functions as the whole kernel of D. Then by linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
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
Coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
f + C; and all is well with the argument.