path integral

The path integral is a generalization of the integral that is very useful in theoretical and applied physics. Consider a vector field $\vec{F}\!:\mathbb{R}^n\rightarrow\mathbb{R}^m$ and a path $\gamma\subset\mathbb{R}^n$ . The path integral of $\vec{F}$ along the path $\gamma$ is defined as a definite integral. It can be constructed to be the Riemann sum of the values of $\vec{F}$ along the curve $\gamma$ . Thusly, it is defined in terms of the parametrization of $\gamma$ , mapped into the domain $\mathbb{R}^n$ of $\vec{F}$ . Analytically, $$\int_\gamma \vec{F}\cdot d\vec{x} = \int_a^b \vec{F}(\vec{\gamma}(t))\cdot d\vec{x}$$ where $\vec{\gamma}(a), \vec{\gamma}(b)$ are elements of $\mathbb{R}^n$ , and $d\vec{x}=\langle\frac{dx_1}{dt},\cdots,\frac{dx_n}{dt}\rangle dt$ where each $x_i$ is parametrized into a function of $t$ .

Proof and existence of path integral:
Assume we have a parametrized curve $\vec{\gamma}(t)$ with $t\in[a,b]$ . We want to construct a sum of $\vec{F}$ over this interval on the curve $\gamma$ . Split the interval $[a,\,b]$ into $n$ subintervals of size $\Delta t=(b-a)/n$ . 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 $|\vec{\gamma}'(t_i)|$ gives the average magnitude of the vector tangent to the curve at a point in the interval $\Delta t$ . $|\vec{\gamma}'(t_i)|\Delta t$ is then the approximate arc length of the curve segment produced by the subinterval $\Delta t$ . Since we want to sum $\vec{F}$ over our curve $\vec{\gamma}$ , we let the range of our curve equal the domain of $\vec{F}$ . We can then dot this vector with our tangent vector to get the approximation to $\vec{F}$ at the point $\vec{\gamma}(t_i)$ . Thus, to get the sum we want, we can take the limit as $\Delta t$ approaches 0. $$\lim_{\Delta t\rightarrow 0}\sum_a^b \vec{F}(\vec{\gamma}(t_i))\cdot\vec{\gamma}'(t_i)\Delta t$$ 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). $$\int_\gamma \vec{F}\cdot d\vec{x} = \int_a^b \vec{F}(\vec{\gamma}(t))\cdot\vec{\gamma}'(t)dt$$ 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 $\Gamma$ 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 $\vec{F}$ . This is a visualization of what we are doing when we take the integral under the curve $S:P\rightarrow\vec{F}$ .