Line Integrals
Contents
5. Line Integrals#
If we apply the total differential expression:
and now try to integrate this:
we see that this can be thought of doing the scalar product between two vectors
We call integrals of this kind line integrals or otherwise called contour integrals.
Definition
We can define:
which we call a vector line integral. This is an integral through coordinate space along a contour, here denoted
For some vector field:
this simplifies to:
We notice that that this is a scalar, found from scalar product of
We can also think about a scalar line integral:
where
If we pick a closed path through the vector field, we call this a loop integral:
What does a line integral actually do? We can think about moving along some line which we denoted

Fig. 5.1 Ilustration of a line integral, with
5.1. Parameterising and evaluating integrals#
Whilst our expressions for line integrals are powerful, they also do not really explain how we can go about calculating integrals.
To see how we can go about this, we need to think about how to parameterise our paths (contours) through coordinate space. Once
we have a good way to think about parameterising our contours, we will aim to use a change of variable to evaluate the integrals.
5.1.1. Parameterising contours#
Lets think about different contours between the points

Fig. 5.2 Ilustration of a three contours
If we travel along contour
We can use some standard parameterisation schemes to capture all the relevant details here:
For straight lines over points
to , we can parameterise this contour using:
where
Worked example
For the contour
and for the contour
For functions
over range , i.e. over , we can parameterise using :
We note that here we can also go in the opposite orientation
Worked example
For the contour
and in order to reverse the contour direction we can just switch
For a circular path (or circular arc) between points
and such that , we can parameterise using :
For scalar line integrals, the direction we travel along the contour does not change the result:
this is because the length element
For vector line integrals, the direction we travel along the contour does change the result:
and given that for
5.1.2. Evaluating line integrals#
If we can find one variable
where the infinitesimal coordinate vector derivatives are given by:
Worked examples
1. Evaluate the scalar line integral
a. A straight line linking
Here we can use the straight line parametrisation for
Then we can rewrite the integrand in this parameterisation:
Finally lets evaluate the integral:
b. A semi-circular arc linking
Here we can use polar coordinates, with
with
Rewriting the integrand in this parameterisation:
Finally lets evaluate the integral:
We can picture this system by plotting the surface

Fig. 5.3 Ilustration of a parabolic surface
We can see that the distance for contour (a) is shorter than for contour (b) and since the scalar function is
2. Evaluate the vector line integral
Here we need a parameterise the line over
Using this we can rewrite the variables in terms of
and so we can find the dot product:
which we can then just integrate:
Further worked examples
1. Find the line integral of the vector field:
over the following contours:
a. Straight line path
The path followed can be parameterised over
Thus
and
so the line integral is found by:
b. Curved path following
So the parameterisation for
Thus
and
so the line integral is found by:
c. Curved path following
So the parameterisation for
Thus
and
so the line integral is found by:
5.2. Conservative vector fields#
In the case of a conservative vector field
If this is true, then when we calculate the line integral of a conservative vector field:
over some points
Hence this answer only depends on the starting/ending values of the path integral, not the path taken.