Path Integral Applications

Topics: Path Integral - Calculus

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Path integrals can be used for many applications. Apart from the obvious use for calculating the area of a surface, there are many others such as calculating the average of a function over a curve or calculating the centre of mass given a density function.

Average of a Function §

Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be a function and let $c$ be a curve in ${\mathbb{R}}^n$ parametrised by $r:[a,b]\to {\mathbb{R}}^n$.

The average of $f$ over $c$, denoted ${\mu }_f$ is given by:

$${\mu }_f=\frac{{\int }_{c}fds}{{\int }_a^b\Vert c^{\prime }(t)\Vert dt}$$

Note that the denominator is just the arc length of $c$.

Centre of Mass §

Let $\rho :{\mathbb{R}}^3\to \mathbb{R}$ be a function that defines the density of an object in any one of its points. Let $c$ be a curve that defines the points where the object exists.

The total mass of the object defined by $c$ is given by:

$$M_t={\int }_c\rho (x,y,z)ds$$

The coordinates of its centre of mass are given by:

$${\mathbf{C}}_{\mathbf{m}}=\frac{1}{M_t}\left({\int }_c\rho (x,y,z)xds,{\int }_c\rho (x,y,z)yds,{\int }_c\rho (x,y,z)zds\right)$$