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).
We define the path integral of over as:
…where is the arc length parametrisation of .
Area of a Surface
Path integrals have many applications. We can use them to calculate the average of 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 -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 and . , the curve, is the “base” of the surface, which is now not necessarily a straight line, while gives us the height of every point in that curve.
Path integrals have many applications.
…where is any parametrisation of the curve .