Properties of vector line integrals

 Reversing orientation 

We continue with the same example as previously where we consider the work done by the wind in helping a bird fly along a certain path in a definite direction. Let's say that the bird flies in the opposite direction. What happens to the work done by the wind when the bird flies in the opposite direction. Let's compare the 2 works by comparing two vector line integrals.

In the previous example the value of the integral ∫꜀ F·dr where C is a semicircle parameterized by r(t) = <cost, sint> where 0 ≤ t  ≤ ℼ and F = <-y, x> is ℼ. In this example the direction of travel and the wind is illustrated by the following figure:

The direction of travel is the same as the direction of the field.

Let's consider a second example ∫꜀ F·dr where C is a semicircle parameterized by r(t) = <cos (t + ℼ), sint> where 0 ≤ t  ≤ ℼ and F = <-y, x> is ℼ.

The direction of travel and the wind are illustrated in the following figure:

The direction of travel is the opposite of the direction of the field

Let's evaluate this integral:

This is the same integral as previously except the direction of travel reverses. The parameterization starts r(0) = <-1,0> and ends at r(ℼ) = <1,0>. Let's calculate the integral:

The sign of the integral ∫꜀ F·dr changes when the orientation is reversed. Let C be an oriented curve. Let -C be the same curve with the orientation reversed. Based on the previous 2 examples, then:

The reverse orientation of the curve changes the sign of a line integral. This is one of the properties of of the vector line integrals. Other properties of the vector line integrals are included in the following:

 

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⟩. 

Scalar line integral: Independence of parameterization

 Objective: Show that the value of a line integral doesn't change if the parameterization of the curve changes. 

In the previous post we calculated the value of a line integral. What if the parameterization of the curve changes? Let's use the same example and change the parameterization of the curve?

Example

  Solution  

Let's apply the formula to evaluate a line integral:

Let's calculate f(r(t)) by substituting x, y and z in the function:

Let's calculate the radical expression:

Let's substitute f(x,y.z), f(r(t)) and the radical expression in the formula:

We obtain the same result as in the previous example. Scalar line integrals are independent of parameterization. 

Practice

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

 Objective: Evaluate a scalar line integral

Let's consider a simple integral of the form: 

The domain of integration [a b] is a line segment that can be considered as a curve in the x-axis even though it doesn't have a curvature. Let's say that we want to integrate a function over a domain of integration that is not a straight line. The resulting integral is a line integral. A line integral is a type of integral with a domain of integration that can be a line or a curve. The line integral can be the integral of a multivariable function or the the integral of a vector field over a domain of integration that is a curve. In the case of a multivariable function the line integral is called a scalar line integral. If the function is a vector field the line integral is called a vector line integral.

How to evaluate a scalar line integral?   

We divide the curve C into small pieces of curve C₁, C₂, C₃,......Cₙ by the points P₀, P₁, P₂....Pₙ. Each piece has a length Δs₁, Δs₂, Δs₃.....Δsₙ.  Let choose a point P*ᵢ inside the piece Δsᵢ and evaluate f at P we have f(P). Multiplying f(P*i) by Δsᵢ we have the product f(P*i)Δs. Add all the products and let the arc length of the pieces shrink to zero by taking a limit we have:

Calculating the integral using this formula would be cumbersome. The following theorems provide better ways to evaluate a scalar line integral.

Theorem I

Theorem II

Let f be a continuous function with a domain that includes a smooth curve C with parameterization:

Let's substitute: 

in the previous formula. Then we have a definite expression to calculate the line integral:

Example

Solution

Let's use the formula in theorem II : 

Let's calculate f(r,t) and the radical expression:

Let's substitute the given scalar line integral and these 2 expressions in the formula:

Evaluating the integral on the right we have:

Finally the value of the given integral is:

Practice

Vector field of a fluid

 Objective: Define the vector field of a fluid

Let's observe the movement of a fluid: air, water, oil. Each point of the fluid moves in a direction and at a certain speed at any given time. The speed and the direction of the movement of the fluid together represent the velocity of the fluid at that point. The concept of velocity and speed can be both understood mathematically. The velocity at a point is represented by a vector. The magnitude of the vector represents the speed of the fluid at that point. The set of all vectors drawn at different points represent the vector field of the fluid.

Example

Solution

Practice

Using vector field to compare two physical phenomenas : a storm and a whirlpool

 Objectives:

1) Compare two physical phenomenas: a storm and a whirlpool

2) Use the vector field to show the physical similarities and one distinction

When scientists study a storm, they can see the movement inside and the direction of the movement. They can also locate the center of the storm. As the movement goes further from the center, the intensity of the movement increases. In the previous post, we have used the vector field to model the shapes, the direction and the relative intensity of these movements.

 A whirlpool is a body of rotating water in which the flow moves in a circular pattern around a central point. The defining physical characteristic of a whirlpool is that the water spins faster near the centre and slower farther away. This is quite different in the case of a storm. We are going to use the same vector field method to study the physical characteristics of the whirlpool. We will also show that the vector field of the storm and that of the whirlpool confirm the physical characteristics of of these two phenomena. Let's start by drawing the vector field of the whirlpool.

Let's consider a whirlpool that can be modeled by the following function: 

Choosing the base points, we obtain the following table of values:

Let's draw the vector field:

This field confirms the physical characteristics of the whirlpool. It moves in a circular clockwise direction around a central point. 

Let's calculate the norm of any vector F(a,b) of the field:

As the vectors approach to the origin, the magnitude goes to infinity or become very large. This confirms the physical characteristic of the whirlpool where the intensity of the movement of the water grows as we approach to the center. 

If we compare the vector field of the whirlpool to that of the storm, we can see that they are both rotational. The only difference is that in the case of the vector field of the storm the magnitude of the vectors increase as we goes farther from the origin. Therefore the intensity of the movement of the storm increases as we move away from the center. In the case of the vector field of the  whirlpool the magnitude of the vectors increase as we approach to the center. Therefore the intensity of the movement increases as we approach to the center.

Drawing a rotational field

 Objective: Draw a rotational field

Rotational fields are used to study major storm systems including cyclones and hurricanes. In the northern hemisphere, storms rotate counterclockwise. In the southern hemisphere they rotate clockwise. This is an effect of the earth's rotation around its axis called Coriolis effect.

Example

 Draw the rotational field F<x,y> = (y, -x)

Solution

 

 

The vector field is now complete. We can show now that the vectors are tangent to the circles of radius r = 1, r -⎷ 2, r = 2.

We can also compare the rotational field to the radial field and show that the vectors of the rotational field are perpendicular to the ones in the radial field. The following figure shows that.

We can confirm  the perpendicularity of the vectors of the radial field to the vectors of the rotational by showing that their dot product is equal to zero.