Path integrals generalize the notion of integration along a line into multiple dimensions. Unlike 1D functions, we are not restricted to one dimension, but multiple, so the path we integrate over is in general a path in a $n-$dimensional space.

In general we can parameterize a path in $n$-dimensional space with a single parameter, $s$

$$ \vec r(s)=x(s)\hat x+y(s)\hat y+z(s)\hat z $$

A tiny step along the path corresponds to a tiny change in $s$ given by $ds$

$$ \vec r(s+ds)=\vec r(s)+d\vec r\\ d\vec r=\frac{d\vec r}{ds}ds $$

Our $d\vec r$ vectors points tangent to the curve at a given point in the direction of the displacement.

There are two many types of path integrals

1. Path integral of a scalar field. Scalar field take $n$ inputs and gives only one output. We denote them by $f(\vec r)$. For a path we can parameterize $\vec r=\vec r(s)$ and thus $f(\vec r(s))=f(s)$

$$ I=\int_{s_1}^{s_2}f(~\vec r(s)~)ds $$

1. Path integral of a vector field. A vector field takes $n$ outputs but gives multiple outputs. We can represent these multiple outputs as a vector with direction and magnitude. We’ll denote it by $\vec V(\vec r)$. Every point is thus associated with a vector with a direction and magnitude. The path intagral will calculate the amount of $\vec V$ along the path $C$ and thus it measures how much $\vec V$ points along $C$

   $$ dI=\vec V\cdot d\vec r=V_{||}|d\vec r| $$

   

   The path integral sums the amount of $\vec V$ parallel at each point along the path

   $$ I=\int_C\vec V(\vec r)\cdot d\vec r $$

   To calculate we can use the parameterization of $\vec r$

   $$ I=\int_{s_i}^{s_f}\vec V(\vec r(s))\cdot \frac{d\vec r}{ds} ds $$

   The general procedure

   → Parameterize the path by writing a position vector in terms of one variable, $s$ and determine the start and end points

   → Calculate $\frac{d\vec r}{ds}$

   → Plug in parameterization relationships into $\vec V(\vec r)$

   → Integrate like a normal 1D integral

Path integrals and Averages

We can view path integrals in terms of averages

$$ \int_C\vec f\cdot d\vec r=\braket{f_{||}}_CL $$

Where $\braket{f_{||}}_C$ is the average amount of $\vec f$ projected onto $C$ and $L$ the total length of the path.

Conservative fields

Four Equivalent Statements for Conservative Fields

<aside> 💡 The vector field $\vec F$ is conservative$\iff$

1. $\vec F$ can be written as the gradient of a scalar function

$$ \vec F=\vec\nabla\phi $$

1. The path integral of $\vec F$ for any closed path is $0$

$$ \oint\vec F\cdot d\vec r=0 $$

1. The path integral of $\vec F$ along any path is only dependent on its endpoints, i.e. the path integral is path independent

$$ \int_C\vec F\cdot d\vec r=\phi_f-\phi_i $$

1. $\vec F$ has a curl of $0$ by Stoke’s Theorem

$$ \vec\nabla\times\vec F=0 $$

</aside>

• Conservative fields are particuraly ideal because of the fundamental theorem of calculus for line integrals is present, which follows a similar form to the familiar fundamental theorem of calculus part 2 in intro calculus.

  We can avoid integration by finding the scalar function $\phi$ and instead evaluate at the end points of the path taken.

• The easiest ways to test for a conservative field is through #4. Once confirmed then it is common to find the potential function described by #1

Finding the potential function $\phi$ for a conservative field