Line Integral
A line integral (also known as a curvilinear integral) is a scalar quantity obtained by summing the dot products of the vectors v along a curve L within a vector field C. $$ \int_A^B \vec{v} \cdot dl $$
Here, dl represents an infinitesimal displacement along the curve, from point A to point B.
The vector VP is the tangent vector to the curve at a given point P.
The sum of all these dot products, V·dl, yields a scalar value known as the line integral or curvilinear integral.
Applications In physics, line integrals are particularly useful for computing the work done in moving an object through a force field.
A practical example
Consider a constant vector field C, where all vectors have identical magnitude, direction, and orientation.
The gravitational field near the Earth's surface is a typical example of such a constant vector field.
The vectors represent the direction, orientation, and magnitude of the gravitational force (g).
To move an object in a straight line from point A to point B at constant velocity, one must perform a certain amount of work W to counteract gravity.
The line integral provides a way to compute the scalar value of the force F involved.
The path from A to B is divided into infinitesimal segments dl.
Note. In this example, the curve L is a straight line. As a result, the tangent vectors dl are all aligned with the direction of the line. $$ W = F·dl $$
At each point P, the dot product between the tangent vector to the path and the gravitational field vector originating at P is computed.
$$ \vec{v_1} \cdot d_l = |v_1| \cdot |dl| \cdot \cos \ θ $$
$$ \vec{v_2} \cdot d_l = |v_2| \cdot |dl| \cdot \cos \ θ $$
$$ \vec{v_3} \cdot d_l = |v_3| \cdot |dl| \cdot \cos \ θ $$
The sum of these dot products represents the line integral - namely, the total work W required to move the object along the path from A to B.
$$ \int_A^B \vec{v} \ dl = \sum_p \vec{v}_p \cdot dl $$
Example 2
The same principle applies even when the vector field is not constant.
For instance, the gravitational force weakens as one moves farther from Earth.
In this case, the vectors decrease in magnitude (length) as altitude increases.
Consequently, the dot product varies along the path.
Example 3
When the path is not straight, the trajectory follows a curve.
In this situation, the tangent vector dl must be computed at each point along the curve.
Now, the tangent vectors vary in direction, magnitude, and orientation.
The angle θ between the tangent vector and the field vector also varies continuously along the curve.
As a result, the dot product changes from point to point.
Note. Alternatively, to approximate the value of the line integral, one can compute the dot products using the vectors of a polygonal approximation of the curve, rather than the exact tangent vectors at each point.
Each dot product represents a partial (or elementary) amount of work. The sum of these partial works approximates the total work required to move the object. The smaller the segment dl in the polygonal path, the smaller the approximation error in the line integral.
Example 4
The same approach can be applied to more complex vector fields.
For example, when moving through a region with varying air or water currents.
The dot product varies continuously along the curve in this case as well.
Properties of Line Integrals
Line integrals possess the same properties as definite integrals.
- If an intermediate point C is chosen within the interval [A,B], the sum of the line integrals over the two sub-arcs [A,C] and [C,B] equals the line integral from A to B (additivity). $$ \int_A^B \vec{v} \ \cdot dl = \int_A^C \vec{v} \ \cdot dl + \int_C^B \vec{v} \cdot \ dl $$
- Reversing the direction of traversal along the path results in a line integral of opposite sign. $$ \int_A^B \vec{v} \ \cdot dl = - \int_B^A \vec{v} \cdot \ dl $$
Circulation
Circulation refers to the case where the endpoints of the curve L coincide - that is, A=B, and the curve is closed. $$ \oint_{L} \vec{v} \cdot dl $$
In this scenario, the initial and final points of the curve L are the same.
A special symbol, ∮, is used to denote the line integral in the case of circulation.
$$ \oint_{L} \vec{v} \cdot dl = \int_A^A \vec{v} \cdot dl $$
Reversing the direction along the closed curve L changes the sign of the line integral.
And so on.
