path integral

The path integral is a generalization of the integral that is very useful in theoretical and applied physics. Consider a vector field $\overrightarrow{F}:{ℝ}^n\to {ℝ}^m$ and a path (http://planetmath.org/PathConnected) $\gamma \subset {ℝ}^n$. The path integral of $\overrightarrow{F}$ along the path $\gamma$ is defined as a definite integral. It can be constructed to be the Riemann sum of the values of $\overrightarrow{F}$ along the curve $\gamma$. Thusly, it is defined in terms of the parametrization of $\gamma$, mapped into the domain ${ℝ}^n$ of $\overrightarrow{F}$. Analytically,

$${\int }_{\gamma }\overrightarrow{F}\cdot 𝑑\overrightarrow{x}={\int }_a^b\overrightarrow{F}(\overrightarrow{\gamma }(t))\cdot 𝑑\overrightarrow{x}$$

where $\overrightarrow{\gamma }(a),\overrightarrow{\gamma }(b)$ are elements of ${ℝ}^n$, and  $d\overrightarrow{x}=⟨\frac{d{x}_1}{dt},\mathrm{\cdots },\frac{d{x}_n}{dt}⟩dt$  where each $x_i$ is parametrized into a function of $t$.

Proof and existence of path integral:
Assume we have a parametrized curve $\overrightarrow{\gamma }(t)$ with $t\in [a,b]$. We want to construct a sum of $\overrightarrow{F}$ over this interval on the curve $\gamma$. Split the interval $[a,b]$ into $n$ subintervals of size $\mathrm{\Delta }t=(b-a)/n$. Note that the arc lengths need not be of equal length, though the intervals are of equal size. Let $t_i$ be an element of the $i$th subinterval. The quantity $|{\overrightarrow{\gamma }}^{\prime }(t_i)|$ gives the average magnitude of the vector tangent to the curve at a point in the interval $\mathrm{\Delta }t$. $|{\overrightarrow{\gamma }}^{\prime }(t_i)|\mathrm{\Delta }t$ is then the approximate arc length of the curve segment produced by the subinterval $\mathrm{\Delta }t$. Since we want to sum $\overrightarrow{F}$ over our curve $\overrightarrow{\gamma }$, we let the range of our curve equal the domain of $\overrightarrow{F}$. We can then dot this vector with our tangent vector to get the approximation to $\overrightarrow{F}$ at the point $\overrightarrow{\gamma }(t_i)$. Thus, to get the sum we want, we can take the limit as $\mathrm{\Delta }t$ approaches 0.

$$\underset{\mathrm{\Delta }t\to 0}{lim}\sum _a^b\overrightarrow{F}(\overrightarrow{\gamma }(t_i))\cdot {\overrightarrow{\gamma }}^{\prime }(t_i)\mathrm{\Delta }t$$

This is a Riemann sum, and thus we can write it in integral form. This integral is known as a path or line integral (the older name).

$${\int }_{\gamma }\overrightarrow{F}\cdot 𝑑\overrightarrow{x}={\int }_a^b\overrightarrow{F}(\overrightarrow{\gamma }(t))\cdot {\overrightarrow{\gamma }}^{\prime }(t)𝑑t$$

Note that the path integral only exists if the definite integral exists on the interval $[a,b]$.

Properties:
A path integral that begins and ends at the same point is called a closed path integral, and is denoted with the summa symbol with a centered circle: $\oint$. These types of path integrals can also be evaluated using Green’s theorem.
Another property of path integrals is that the directed path integral on a path $\mathrm{\Gamma }$ in a vector field is equal to the negative of the path integral in the opposite direction along the same path. A directed path integral on a closed path is denoted by summa and a circle with an arrow denoting direction.

Visualization Aids:

This is an image of a path $\gamma$ superimposed on a vector field $\overrightarrow{F}$.

This is a visualization of what we are doing when we take the integral under the curve $S:P\to \overrightarrow{F}$.