To better understand line integral, we will first review the following two cases, the dot product of two vectors and the difference of two vectors.
Image 1: Difference Between two Vectors
Consider the vector
that traces a curve C. This is called position vector.
At the time
this vector has value
and at the time
it has value
The difference between these two vectors is:
is so small, we can assume that is tangent to the curve C.
Image 2: Dot Product of Vectors
The dot product of vector
by the vector
is equal to the product of the projection of the magnitude of the first vector on the second vector, by the magnitude of the second vector.
We actually multiply the part of
that has the same direction with
In other words:
You are on a hill and you are pulling a box. You apply a force
and you move the box from point A to point B. How can we calculate the work done from this force over the distance AB?
Image 3:Line Integral
The force, which will be called vector field, can be described by the function:
The path AB is a curve and can be described by the position vector:
The work done W is equal to the product of force
by the displacement D. Since work is produced only by the part of the force that is at the direction of the displacement, a tiny piece of work can be found by the dot product of force
by a tiny displacement
Integrating across the whole displacement AB will supply us with the whole work done:
The integration of a vector across a curve is called line integral.
Making some substitutions to the line integral, we get:
We have a space vector function and the position vector in parametric equations. So:
We have the dot product calculated and instead of parametric equations for the position vector, we have y as a function of x. So:
We will examine the line integral for a space vector and position vector that are only xy functions, because it easier to be shown.
Image 4: Geometric Interpretation
So we have the space vector and the position vector respectively:
We can imagine the space vector as a surface or a “blanket” over the XY plane and the position vector as a curve set on the XY plane.
Just like the usual integral is the area under the curve of the function G(x) when we integrate over x,
the line integral
is the area of the “curtain” that connects the curve of the position vector and the projection of the curve on the “blanket” of the space vector, since we integrate over the position vector r.