Line integrals are the natural generalization of the integral into the sum of some function over any arbitrary curve instead of just over an interval on the real line.

Line Integral of a Scalar Function

Formally the line integral of scalar function over the curve is defined as

Where is some parameterization of on the interval .

Derivation

We can use the same intuition that arises from approximating integrals with Riemann sum's to arise on our the definition of line integrals of scalar functions. Riemann sum's state that the sum of some function over a region can be written as an infinite sum of rectangles with the height of at some point in each infinitely small subinterval with width :

For a line integral, the key difference is that instead of summing over an interval on the real line, we sum over an arbitrary curve . Because of this difference we generalize to instead of being some small subinterval of the [Real Number|real] line to generally just represent some small segment of the curve we are summing over. We can represent this small segment of the curve as where is the arc length along the curve. This gives us the following definition of a line integral:

To evaluate this line integral we need to express in terms of the parameterization of the curve. We can do this by noting that the arc length along a curve is given by the integral of the magnitude of the derivative of the parameterization:

Differentiating both sides with respect to gives us:

This means that can be expressed as . Substituting this back into our definition of the line integral gives us:

Line Integral of a Vector Field

The line integral of a vector field over a curve is defined as

Where is some parameterization of on the interval .

Independence of Path

TO-WRITE

Derivation

Just as with the scalar case, we build intuition from a Riemann sum. This time however, the object we are summing is a vector field and we want to capture how much the field acts along the curve at each point, that is, we want to accumulate the component of in the direction of travel. This aim originates from the physical definitions of work.

At each point on the curve, the direction of travel is given by the unit tangent vector:

so the component of along the curve at each point is given by the dot product of with . Summing this scalar quantity over the curve with respect to arc length gives us:

Using the same substitution as in the scalar case, we can express in terms of the parameterization of the curve and we can express in terms of to get: