# 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. The line integral over a closed path are written with the symbol ∮{\textstyle \textstyle \oint }.

## Real-valued Line Integrale

A path γ:[a, b]→R$^n${\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 R$^n${\textstyle \mathbb {R} ^{n}}. γ(t)∈R$^n${\textstyle \gamma (t)\in \mathbb {R} ^{n}} indicates the place where the value is t∈[a, b]{\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
    f:R$^n$→R{\textstyle f\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} }
along a continuously differentiable piece path γ:[a, b]→R$^n${\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

γ$^′${\textstyle {\gamma \, }'} refers to the derivation from γ{\textstyle \gamma } to t{\textstyle t}. γ(t)∈R$^n${\textstyle \gamma (t)\in \mathbb {R} ^{n}} and γ$^′$(t)∈R$^n${\textstyle {\gamma \, }'(t)\in \mathbb {R} ^{n}} are a vectors. The derivation vector γ$^′$(t)∈R$^n${\textstyle {\gamma \, }'(t)\in \mathbb {R} ^{n}} indicates the change behavior in each component function of γ=(γ$_1$, …, γ$_n$){\textstyle \gamma =(\gamma _{1}, \dots , \gamma _{n})}.

### Remark - Component functions

The component functions γ$_i$:[a, b]→R{\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 γ{\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 γ$^′${\textstyle {\gamma }'} of the path γ{\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 γ(t)=(cos⁡(t), sin⁡(t), t)∈R$^3${\textstyle \gamma (t)=(\cos(t), \sin(t), t)\in \mathbb {R} ^{3}} and γ$^′$(t)=(−sin⁡(t), cos⁡(t), 1)∈R$^3${\textstyle {\gamma \, }'(t)=(-\sin(t), \cos(t), 1)\in \mathbb {R} ^{3}}. The derivation vector γ$^′$(t)∈R$^3${\textstyle {\gamma \, }'(t)\in \mathbb {R} ^{3}} indicates the change behavior in each component function of γ=(γ$_1$, γ$_2$, γ$_3$){\textstyle \gamma =(\gamma _{1}, \gamma _{2}, \gamma _{3})}.

## Task

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

### Vector length of the derivation vector of the path

‖γ$^′$(t)‖$_2${\textstyle \|{\gamma \, }'(t)\|_{2}} indicates the Euclidian norm of the vector γ$^′$(t)∈R$^n${\textstyle {\gamma \, }'(t)\in \mathbb {R} ^{n}}.

### Picture of the path - track

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