Line integral

Introduction

The Curve, Line, Path or Contour integral expands the standard integral term for the Integration in the complex plane (Complex Analysis) or in the multidimensional space $\mathbb {R} ^{n}$ or $\mathbb {C} ^{n}$ . The path, the line or the curve, via which is integrated, is called the integration path^[1]. The line integral over a closed path are written with the symbol ${\textstyle \textstyle \oint }$ .

Real-valued Line Integrale

A path ${\textstyle \gamma \colon [a,b]\to \mathbb {R} ^{n}}$ is given which is imaged from an interval (e.g. interpreted as a time interval) into the vector space ${\textstyle \mathbb {R} ^{n}}$ . ${\textstyle \gamma (t)\in \mathbb {R} ^{n}}$ indicates the place where the value is ${\textstyle t\in [a,b]}$ . The difference is

Line integral first type and
Line integral second type.

Pathintegral first type

The path integral of a continuous Function

{\textstyle f\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} }

along a continuously differentiable piece path ${\textstyle \gamma \colon [a,b]\to \mathbb {R} ^{n}}$ is defined as

\int \limits _{\gamma }\!f\,\mathrm {d} s:=\int \limits _{a}^{b}\!f(\gamma (t))\,\|{\gamma \,}'(t)\|_{2}\,\mathrm {d} t.

Deduction of the path

${\textstyle {\gamma \,}'}$ refers to the derivation from ${\textstyle \gamma }$ to ${\textstyle t}$ . ${\textstyle \gamma (t)\in \mathbb {R} ^{n}}$ and ${\textstyle {\gamma \,}'(t)\in \mathbb {R} ^{n}}$ are a vectors. The derivation vector ${\textstyle {\gamma \,}'(t)\in \mathbb {R} ^{n}}$ indicates the change behavior in each component function of ${\textstyle \gamma =(\gamma _{1},\dots ,\gamma _{n})}$ .

Remark - Component functions

The component functions ${\textstyle \gamma _{i}:[a,b]\to \mathbb {R} }$ are illustrations for which the derivation with the knowledge from the real analysis can be calculated.

Example of a path and its derivation

A differentiable path is defined first ${\textstyle \gamma }$ with

{\begin{array}{rrcl}\gamma :&[0,2\pi ]&\rightarrow &\mathbb {R} ^{2}\\&t&\mapsto &\gamma \left(t\right)={\begin{pmatrix}5\cdot \cos(t)\\3\cdot \sin(t)\end{pmatrix}}\end{array}}

The track of the path forms an ellipse with the half axes 5 and 3.

Derivation of the path in the two-dimensional space

The derivation ${\textstyle {\gamma }'}$ of the path ${\textstyle \gamma }$ results directly from the derivation of the component functions

{\begin{array}{rrcl}{\gamma }':&[0,2\pi ]&\rightarrow &\mathbb {R} ^{2}\\&t&\mapsto &{\gamma }'\left(t\right)={\begin{pmatrix}-5\cdot \sin(t)\\3\cdot \cos(t)\end{pmatrix}}\end{array}}

Example - Deduction of the Way in the Three-dimensional Space

Now a vector is ${\textstyle \gamma (t)=(\cos(t),\sin(t),t)\in \mathbb {R} ^{3}}$ and ${\textstyle {\gamma \,}'(t)=(-\sin(t),\cos(t),1)\in \mathbb {R} ^{3}}$ . The derivation vector ${\textstyle {\gamma \,}'(t)\in \mathbb {R} ^{3}}$ indicates the change behavior in each component function of ${\textstyle \gamma =(\gamma _{1},\gamma _{2},\gamma _{3})}$ .

Task

Draw the trail of the path in ${\textstyle \mathbb {R} ^{2}}$ (Ellipse) and plotted the trail of the path in ${\textstyle \mathbb {R} ^{3}}$ with CAS4Wiki plots.

Vector length of the derivation vector of the path

${\textstyle \|{\gamma \,}'(t)\|_{2}}$ indicates the Euclidian norm of the vector ${\textstyle {\gamma \,}'(t)\in \mathbb {R} ^{n}}$ .

Picture of the path - track

The image set ${\textstyle {\mbox{Spur}}(\gamma ):={\mathcal {C}}:=\gamma ([a,b])}$ of one piece differentiable curve in ${\textstyle \mathbb {R} ^{n}}$ should not be confused with the graph of a curve which is a part of the ${\textstyle [a,b]\times \mathbb {R} ^{n}}$ .

Notes

An example of such a function ${\textstyle f}$ is a scalar field with cartesischen coordinaten.
A path ${\textstyle \gamma }$ can pass through a curve ${\textstyle {\mathcal {C}}}$ either as a whole or only in sections several times.
For ${\textstyle f\equiv 1}$ , the path integral of the first type gives the length of the path ${\textstyle \gamma }$ .
The path ${\textstyle \gamma }$ forms, inter alia ${\textstyle a\in \mathbb {R} }$ on the starting point of the curve and ${\textstyle b\in \mathbb {R} }$ on its end point.
${\textstyle t\in [a,b]}$ is an element of the definition set of ${\textstyle \gamma }$ and is generally not' for time. ${\textstyle \mathrm {d} t}$ is the corresponding Differential.

Pathintegral second type

The line integral over a continuous gradient vector field

{\textstyle \mathbf {f} \colon \mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}}

with a curve also parameterized in this way is defined as the integral over the scalar product of ${\textstyle \mathbf {f} \circ \gamma }$ and ${\textstyle \gamma \,'}$ :

{\textstyle \int \limits _{\gamma }\!\mathbf {f} (\mathbf {x} )\cdot \mathrm {d} \mathbf {x} :=\int \limits _{a}^{b}\!\langle \mathbf {f} (\gamma (t)),{{\gamma }\,}'(t)\rangle \,\mathrm {d} t}

Influence of parameterization

If ${\textstyle \gamma \colon [a,b]\to \mathbb {R} ^{n}}$ and ${\textstyle \eta \colon [c,d]\to \mathbb {R} ^{n}}$ 'simplified' (d. h, ${\textstyle \gamma _{|(a,b)}}$ and ${\textstyle \eta _{|(c,d)}}$ are identical This justifies the name curve integral; if the direction of integration is visible or irrelevant, the path in the notation can be suppressed.

Curve integrals

Since a curve ${\textstyle {\mathcal {C}}}$ is the image of a path ${\textstyle \gamma }$ , the definitions of the curve integrals essentially correspond to the path integrals.

Curve integral 1. type

{\textstyle \int \limits _{\mathcal {C}}\!f\,\mathrm {d} s:=\int \limits _{a}^{b}\!f(\gamma (t))\cdot \|{{\gamma }\,}'(t)\|_{2}\,\mathrm {d} t}

Curve integral 2. type

{\textstyle \int \limits _{\mathcal {C}}\mathbf {f} (\mathbf {x} )\cdot \mathrm {d} \mathbf {x} :=\int \limits _{a}^{b}\langle \mathbf {f} (\gamma (t)),{{\gamma }\,}'(t)\rangle \,\mathrm {d} t}

Length of curve

A special case is again the length of the curve ${\textstyle {\mathcal {C}}}$ parameterized by ${\textstyle \gamma }$ :

{\textstyle {\mathcal {L}}({\mathcal {C}})=\int \limits _{\mathcal {C}}1\,\mathrm {d} s=\int \limits _{a}^{b}\|{\gamma \,}'(t)\|_{2}\,\mathrm {d} t}

Displacement element and length element

The expression occurring in the first type of curves

{\textstyle \mathrm {d} s=\|{\gamma \,}'(t)\|_{2}\,\mathrm {d} t}

is called scalar path element' or 'length element.The expression occurring in the second type of curve integrals

\mathrm {d} \mathbf {x} ={\gamma \,}'(t)\,\mathrm {d} t

is called 'vectorial path element'.

Rules of Procedure

Be ${\textstyle \int \limits _{\gamma }\mathbf {f} (\mathbf {x} )}$ , ${\textstyle \int \limits _{\gamma }\mathbf {g} (\mathbf {x} )}$ Curve integrals of the same type (i.e. either both first or second type), be the original image of the two functions ${\textstyle \mathbf {f} }$ and ${\textstyle \mathbf {g} }$ of the same dimension and be (698104789). The following rules apply to ${\textstyle \alpha }$ , ${\textstyle \beta \in \mathbb {R} }$ and ${\textstyle c\in \mathbb {[} a,b]}$ :

${\textstyle \alpha \int \limits _{\gamma }\mathbf {f} (\mathbf {x} )+\beta \int \limits _{\gamma }\mathbf {g} (\mathbf {x} )=\int \limits _{\gamma }(\alpha \mathbf {f} (\mathbf {x} )+\beta \mathbf {g} (\mathbf {x} ))}$
${\textstyle \int \limits _{\gamma }\mathbf {f} (\mathbf {x} )=\int \limits _{\gamma |_{[a,c]}}\mathbf {f} (\mathbf {x} )+\int \limits _{\gamma |_{[c,b]}}\mathbf {f} (\mathbf {x} )}$

Notation for curve integrals of closed curves

If ${\textstyle \gamma }$ is a closed way, you write

instead of

{\textstyle \displaystyle \int \limits _{\gamma }}

also

{\textstyle \displaystyle \oint \limits _{\gamma }}

and similar for closed curves ${\textstyle {\mathcal {C}}}$

instead of

{\textstyle \displaystyle \int \limits _{\mathcal {C}}}

also

{\textstyle \displaystyle \oint \limits _{\mathcal {C}}}

.

With the circle in the Integral one would like to make clear that ${\textstyle \gamma }$ is closed. The only difference is in the notation.

Examples

If ${\textstyle {\mathcal {C}}}$ is the graph of a function ${\textstyle f\colon [a,b]\to \mathbb {R} }$ , this curve will be passed through the path

{\textstyle \gamma \colon [a,b]\to \mathbb {R} ^{2},\quad t\mapsto (t,f(t))}

parametrized with

(t,f(t))\in \mathbb {R} ^{3}

. About

{\textstyle \|\gamma {\,}'(t)\|_{2}={\sqrt {1+f'(t)^{2}}}}

the length of the curve is equal

{\textstyle \int \limits _{\mathcal {C}}\mathrm {d} s=\int \limits _{a}^{b}{\sqrt {1+f'(t)^{2}}}\,\mathrm {d} t.}

A ellipse with large half-axis ${\textstyle a}$ and small half-axis ${\textstyle b}$ is parameterized by ${\textstyle (a\cos t,\,b\sin t)}$ for ${\textstyle t\in [0,2\pi ]}$ . Your scope is therefore

{\textstyle \int \limits _{0}^{2\pi }{\sqrt {a^{2}\sin ^{2}t+b^{2}\cos ^{2}t}}\,\mathrm {d} t=4a\int \limits _{0}^{\frac {\pi }{2}}{\sqrt {1-\varepsilon ^{2}\cos ^{2}t}}\;\mathrm {d} t}

.

In this case

{\textstyle \varepsilon }

refers to the numerical eccenttricity

{\textstyle {\sqrt {1-b^{2}/a^{2}}}}

of the ellipse. The integral on the right is referred to as elliptic tntegral due to this connection.

Path Independency on Integral

If a vector field ${\textstyle \mathbf {F} }$ is a Gradient field, i.e.

{\textstyle \mathbf {\nabla } V=\mathbf {F} }

,

This applies to derivation of function composition of ${\textstyle V}$ and ${\textstyle \mathbf {r} (t)}$

{\textstyle {\frac {\mathrm {d} }{\mathrm {d} t}}V(\mathbf {r} (t))=\mathbf {\nabla } V(\mathbf {r} (t))\cdot {\dot {\mathbf {r} }}(t)=\mathbf {F} (\mathbf {r} (t))\cdot {\dot {\mathbf {r} }}(t)}

,

which exactly corresponds to the integral of the path integral over ${\textstyle \mathbf {F} }$ to ${\textstyle \mathbf {r} (t)}$ .

Dependence of integral boundaries 1

This follows for a given curve ${\textstyle {\mathcal {S}}}$

{\textstyle \int \limits _{\mathcal {S}}\mathbf {F} (\mathbf {x} )\cdot \,\mathrm {d} \mathbf {x} =\int \limits _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot {\dot {\mathbf {r} }}(t)\,\mathrm {d} t=\int \limits _{a}^{b}{\frac {\mathrm {d} }{\mathrm {d} t}}V(\mathbf {r} (t))\,\mathrm {d} t=V(\mathbf {r} (b))-V(\mathbf {r} (a)).}

Visualization

The following image show two arbitrary curves $S1$ and $S2$ in a Gradient vector field connecting point $1$ with point $2$ .

Dependence of integral boundaries 2

This means that the integral of ${\textstyle \mathbf {F} }$ over ${\textstyle {\mathcal {S}}}$ depends solely on points ${\textstyle \mathbf {r} (b)}$ and ${\textstyle \mathbf {r} (a)}$ and the path between them is irrelevant to the result. For this reason, the integral of a gradient field is referred to as “displaced”.

Remark - closed paths - Ringintegral

In particular, the ring integral applies to the closed curve ${\textstyle {\mathcal {S}}}$ with two arbitrary paths ${\textstyle {\mathcal {S}}_{1}}$ and ${\textstyle {\mathcal {S}}_{2}}$ :

{\textstyle \oint \limits _{\mathcal {S}}\mathbf {F} (\mathbf {x} )\,\mathrm {d} \mathbf {x} =\int \limits _{1,{\mathcal {S}}_{1}}^{2}\mathbf {F} (\mathbf {x} )\,\mathrm {d} \mathbf {x} +\int \limits _{2,{\mathcal {S}}_{2}}^{1}\mathbf {F} (\mathbf {x} )\,\mathrm {d} \mathbf {x} =0}

Application in Physics

This is particularly important in Physics, since, for example, the Gravitation has these properties. Since the energy in these force fields is always a conservation variable, they are referred to in physics as conservative force.

Scaler fields - Potential energy

The scalar field ${\textstyle V}$ is the Potential or the potential Energy. Conservative force fields receive the mechanical energy, i.e. the sum of kinetic Energy and potential energy. According to the above integral, a work of 0 J is applied on a closed curve overall.

Number of revolutions

Path independence can also be shown with the application of the Integrability condition.

if the vector field is not possible as a gradient field only in a (small) environment ${\textstyle U}$ of a point, the closed path integral of curves outside ${\textstyle U}$ is proportional to the number of turns around this point and otherwise independent of the exact curve (see Algebraic Topology: Methodology).

Remark - Complex pathintegrale

If ${\textstyle \mathbb {R} ^{n}}$ is replaced by ${\textstyle \mathbb {C} }$ , complex path integrals are treated which are treated in the Complex Analysis.

Literature

Harro Heuser: Lehrbuch der Analysis – Teil 2. 1981, 5. Auflage, Teubner 1990, ISBN 3-519-42222-0. p. 369, Theorem 180.1; p. 391, Theorem 184.1; p. 393, Theorem 185.1.

References

↑ Klaus Knothe, Heribert Wessels: Finite Elemente. Eine Einführung für Ingenieure. 3. Auflage. 1999, ISBN 3-540-64491-1, S. 524.

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[cite2-1] Klaus Knothe, Heribert Wessels: Finite Elemente. Eine Einführung für Ingenieure. 3. Auflage. 1999, ISBN 3-540-64491-1, S. 524.

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