What is a vector line integral and how to evaluate it?

 In a recent post, we defined a line integral and showed how to evaluate it. We considered a function over its curve domain. We divided the domain curve into small pieces, took a point P inside each piece, evaluated it by the scalar function and multiply f(P) by the length of a piece Δs. We added all the products f(P)Δs, let Δs shrink to zero and took the limit of the sum. The result is the scalar line integral. Now the function considered is a vector field, can we use the same method used for the scalar line integral? Let's make the following considerations. A scalar line integral add up quantities like density or temperature along a curve. A vector field evaluates how much a field pushes in the direction of travel along a curve. We will make this consideration in evaluating the vector Mathematics and Physics entangle. Let's consider a physical problem to calculate the vector line integral. We consider the travel of a bird along a definite path and we want to evaluate how much the wind helps it move along that path.

Let 

be the vector field that represents the force of the wind that moves the bird along the chosen path. This path is represented by a parameterized curve r(t) such that a ≤ t ≤ b. The wind is moving in a positive direction which is the direction given by the curve. The opposite direction would be negative.

 Let P₀, P₁, P₂, ......... Pₙ divide the curve into n pieces. The distance from Pi-1 to Pi is represented by Δsᵢ. Let P*ᵢ be a point  between those the correspond force is F(P*ᵢ). If the distance Δsᵢ is small, when the bird is moving from Pi-1 to Pi it's moving in the direction of the tangent T(P*i). The work done by the force of the wind is given by: F(P*i).(Δsᵢ.T(P*i)). The total work done along C is given by:

Let the arc length of C be arbitrarily small as n→∞, the work done by the wind in moving the bird along its path is represented by:

 

Definition

The vector line integral of a vector field F along a smooth curve is given by:

The unit tangent vector is given by T = r'(t)/𐽕r'(t)𐽕 and ds = 𐽕r'(t)𐽕.dt. Let's calculate F.Tds by substituting T and ds:

Then the vector line integral becomes:

Example

Solution

Let's use the formula: 

Let's calculate F(r(t)) and r'(t):

Let's substitute F(r(t)) and r'(t) in the formula:

Practice

 Find the value of the integral ∫_C F · dr, where C is the semicircle of radius 2 parameterized by r(t) = ⟨2 cos t, 2 sin t⟩, 0 ≤ t ≤ π, and F = ⟨−y, x⟩.