Path Integral

Topics: Calculus - Integral

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Naming

In all my notes, path integral refers to the integral of a scalar function over a curve, while line integral refers to the integral of a vector field over a curve.

Some authors call both line integrals (of a scalar field and of a vector field, respectively).

(definition)

Let $f:{\mathbb{R}}^3\to \mathbb{R}$ be a function and let $c$ be a curve in ${\mathbb{R}}^3$. $f$ returns a value for every point of $c$.

We define the path integral of $f$ over $c$ as:

$${\int }_{c}fds:={\int }_I(f\circ r)(s)ds={\int }_a^b(f\circ r)(s)ds$$

…where $s$ is the arc length parametrisation of $c$.

Area of a Surface

Path integrals have many applications. We can use them to calculate the average of $f$ over the curve, the centre of mass of an object, among others, but its use for calculating the area of a surface is especially useful for understanding them.

Let’s first go back to the concept of traditional integrals. They allow us to calculate the area under a curve generated by a specific function. The “base” of that surface was a straight line, the $X$-axis, while the “height” of the surface was given by the function.

Path integrals aren’t much different: they allow us to calculate the area of the surface between $c$ and $f$. $c$, the curve, is the “base” of the surface, which is now not necessarily a straight line, while $f$ gives us the height of every point in that curve.

Path integrals have many applications.

(theorem)

Let $f:{\mathbb{R}}^3\to \mathbb{R}$ be a function and let $c$ be a curve in ${\mathbb{R}}^3$:

$${\int }_{c}fds={\int }_a^b(f\circ r)(t)\Vert r^{\prime }(t)\Vert dt$$

…where $r:[a,b]\to {\mathbb{R}}^3$ is any parametrisation of the curve $c$.