Definition:Primitive (Calculus)

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Definition

Primitive of Real Function

Let $F$ be a real function which is continuous on the closed interval $\left[{a \,.\,.\, b}\right]$ and differentiable on the open interval $\left({a \,.\,.\, b}\right)$.

Let $f$ be a real function which is continuous on the open interval $\left({a \,.\,.\, b}\right)$.


Let:

$\forall x \in \left({a \,.\,.\, b}\right): F' \left({x}\right) = f \left({x}\right)$

where $F'$ denotes the derivative of $F$ with respect to $x$.


Then $F$ is a primitive of $f$, and is denoted:

$\displaystyle F = \int f \left({x}\right) \, \mathrm d x$


Primitive of Complex Function

Let $F: D \to \C$ be a complex function which is complex-differentiable on a connected domain $D$.

Let $f: D \to \C$ be a continuous complex function.


Let:

$\forall z \in D: \map {F'} z = \map f z$

where $F'$ denotes the derivative of $F$ with respect to $z$.


Then $F$ is a primitive of $f$, and is denoted:

$\displaystyle F = \int \map f z \rd z$


Primitive of Vector-Valued Function

Let $U \subset \R$ be an open set in $\R$.

Let $\mathbf f: U \to \R^n$ be a vector-valued function on $U$:

$\forall x \in U: \mathbf f \left({x}\right) = \displaystyle \sum_{k \mathop = 1}^n f_k \left({x}\right) \mathbf e_k$

where:

$f_1, f_2, \ldots, f_n$ are real functions from $U$ to $\R$
$\left({e_1, e_2, \ldots, e_k}\right)$ denotes the standard ordered basis on $\R^n$.

Let $\mathbf f$ be differentiable on $U$.


Let $\mathbf g \left({x}\right) := \dfrac \d {\d x} \mathbf f \left({x}\right)$ be the derivative of $\mathbf f$ with respect to $x$.


The primitive of $\mathbf g$ with respect to $x$ is defined as:

$\displaystyle \int \mathbf g \left({x}\right) \rd x := \mathbf f \left({x}\right) + \mathbf c$

where $\mathbf c$ is an arbitrary constant vector.


Also known as

A primitive is also known as an antiderivative.

The term indefinite integral is also popular.


Integration

The process of finding a primitive for a function is known as integration.


Arbitrary Constant

From the language in which it is couched, it is apparent that the primitive of a function may not be unique, otherwise we would be referring to $F$ as the primitive of $f$.

This point is made apparent in Primitives which Differ by Constant: if a function has a primitive, there is an infinite number of them, all differing by a constant.

That is, if $F$ is a primitive for $f$, then so is $F + C$, where $C$ is a constant.

This constant is known as an arbitrary constant.


Indefinite Integral

From the Fundamental Theorem of Calculus, it is apparent that to find the value of a definite integral for a function between two points, one can find the value of the primitive of the function at those points and subtract one from the other.

Thus arises the notation:

$\displaystyle \int f \left({x}\right) \ \mathrm d x = F \left({x}\right) + C$

where $C$ is the arbitrary constant.


In this context, the expression $\displaystyle \int f \left({x}\right) \ \mathrm d x$ is known as the indefinite integral of $f$.


Also see

  • Results about primitives can be found here.


Sources