# Curves & line integrals

In this part of the course we work on the following skills:

- Work with parametric paths
- Evaluate and work with scalar line integrals
- Evaluate and work with vector line integrals
- Work with potentials and conservative vector fields

See also the graded exercises and additional exercises associated to this part of the course.

Curves have played a part in earlier parts of the course and now we turn our attention to precisely what we mean by this notion. Up until now we relied more on an intuition, an idea of some type of 1D subset of higher dimensional space. We will also define how we can integrate scalar and vector fields along these curves. These types of integrals have a natural and important physical relevance. We will then study some of the properties of these integrals. To start let's recall a random selection of curves we have already seen:

- Circle:
- Semi-circle:
, - Ellipse:
- Line:
- Line (in 3D):
, - Parabola (in 3D):
,

In the above list the curves are written in a way where we are describing a set of points using certain constraint or constraints. In some cases in *implicit* form, in some cases in *explicit* form. For example, for the circle we formally mean the set

## Curves, paths & line integrals

Let *differentiable* if each component

Definition

We say that *piecewise differentiable* if *differentiable* on each of these intervals.

Definition

If *path*.

Note that different functions can trace out the *same* curve in different ways. Also note that a path has an inherent direction. We say that this is a *parametric representation* of a given curve. We already saw examples of paths in spiral and circular motion. A few examples of paths are as follows.

, , , , , ,

Observe how some of these paths represent the same curve, perhaps traversed in a different direction.

Let

Definition (line integral of vector field)

Let *line integral* of the vector field

Sometimes the same integral is written as

Example

Consider the vector field

Solution

We start by calculating

This means that

Now we consider the question of defining the line integral for scalar fields. Such a line integral allows us also to define the *length of a curve* in a meaningful way. Again let

Definition (line integral of scalar field)

Let *line integral* of the scalar field

Subsequently we will primarily work with the line integral of a vector field. However the analogous results hold also for this integral and the proofs are essentially the same. Namely it is linear and also respects how a path can be decomposed or joined with other paths which changing the value of the integral. Moreover, the value of the integral along a given path is independent of the choice of parametrization of the curve. In this case, even if the curve is parametrized in the opposite direction then the integral takes the same value. Consequently it makes sense to define the length of the curve as the line integral of the unit scalar field, i.e., the length of a curve parametrized by the path

## Basic properties of the line integral

Having defined the line integral, the next step is to clarify its behaviour, in particular the following key properties.

Theorem

**Linearity:** Suppose

**Joining / splitting paths:** Suppose

is a path. Then

Alternatively, if we write

As already mentioned, for a given curve there are many different choices of parametrization. For example, consider the curve

Definition (equivalent paths)

We say that two paths *equivalent* if there exists a differentiable function

Furthermore, we say that

*in the same direction*ifand , *in the opposite direction*ifand .

With this terminology we can precisely describe the dependence of the integral on the choice of parametrization.

Theorem

Let

Proof

Suppose that the paths are continuously differentiable path, decomposing if required. Since

Changing variables, adding a minus sign if path is opposite direction because we need to swap the limits of integration, completes the proof.

### Gradients & work

Let

This equality has the following intuitive interpretation if we suppose for a moment that

As a first example of work in physics let's consider gravity. The gravitational field on earth is

This coincides we what we know, work done depends only on the change in height.

As a second example of work in physics let's consider a particle moving in a force field. Let

In this case we see, as expected, the work done on the particle moving in the force field is equal to the change in kinetic energy.

## The second fundamental theorem

Recall that, if

Theorem (second fundamental theorem for line integrals)

Suppose that

Proof

Suppose that

By the 2^{nd} fundamental theorem in

Our earth has mass

We can calculate

## The first fundamental theorem

First we need to consider a basic topological property of sets. In particular we want to avoid the possibility of the set being several disconnected pieces, in other words we want to guarantee that we can get from one point to another in the set in a way without every leaving the set (see figure).

Definition

The set *connected* if, for every pair of points