Question about Vector Fields and Line Integrals

In other words$$- \int_C \frac{\vec c' \cdot \vec c'}{|\vec c'|} dt.$$What can you say about the numerator here and its relation to the denominator?The numerator is the dot product of the tangent vector with itself, which is the magnitude of the tangent vector squared. This is equivalent to the denominator, so the integral simplifies to -1.
  • #1
Mohamed Abdul

Homework Statement


(a) Consider the line integral I = The integral of Fdr along the curve C

i) Suppose that the length of the path C is L. What is the value of I if the vector field F is normal to C at every point of C?
ii) What is the value of I if the vector field F is is a unit vector pointing in the negative direction along the curve at every point of C?

Homework Equations


Integral of Fdr along c is F(r(t)) * r'(t)

The Attempt at a Solution


I understand the process of computing a line integral, but am unsure of these two parts.

I know for i) that a vector times its normal vector is 0, but F isn't r, it is a vector field of r.

For ii) I'm not even sure as to how to proceed. I do not know what my F(r(t)) would be nor my r'(t)
 
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  • #2
Mohamed Abdul said:
I know for i) that a vector times its normal vector is 0, but F isn't r, it is a vector field of r.
##\vec F## is not normal to ##\vec r##, it is normal to the curve ##C##.

Mohamed Abdul said:
For ii) I'm not even sure as to how to proceed. I do not know what my F(r(t)) would be nor my r'(t)
You do not need to know this. Write down an expression for the integral and work with what you have.
 
  • #3
Orodruin said:
##\vec F## is not normal to ##\vec r##, it is normal to the curve ##C##.You do not need to know this. Write down an expression for the integral and work with what you have.
So then for the first problem I would parametrize c and so then I would have the integral of the vector field of that c(t) times the derivative of c. And since the vector field is normal to c at every point would that mean that F(c(t)) * c'(t) would be 0?

As for the second one, I got that it would be the integral of c(t)/The magnitude of c(t) times c'(t). Am I on the right track with that?
 
  • #4
Mohamed Abdul said:
As for the second one, I got that it would be the integral of c(t)/The magnitude of c(t) times c'(t). Am I on the right track with that?
##\vec c(t)## (the position vector in your notation) is not the relevant vector according to the problem statement. The field is supposed to be anti-parallel to the tangent vector, not the position vector.
Mohamed Abdul said:
So then for the first problem I would parametrize c and so then I would have the integral of the vector field of that c(t) times the derivative of c. And since the vector field is normal to c at every point would that mean that F(c(t)) * c'(t) would be 0?
Yes.
 
  • #5
Orodruin said:
##\vec c(t)## (the position vector in your notation) is not the relevant vector according to the problem statement. The field is supposed to be anti-parallel to the tangent vector, not the position vector.

Yes.
So then my vector field wouldn't be the unit vector of c but the negative derivative of that unit vector since unlike in the first part, F is pointing in a certain direction.

So I would get the integral of -c'(t)/the magnitude of c'(t) multiplied by c'(t). Looking at the solution, the answer was supposed to be -c, but I am unsure of how this integral resolves to that value.
 
  • #6
Mohamed Abdul said:
integral of -c'(t)/the magnitude of c'(t) multiplied by c'(t)
In other words
$$
- \int_C \frac{\vec c' \cdot \vec c'}{|\vec c'|} dt.
$$
What can you say about the numerator here and its relation to the denominator?
 

Related to Question about Vector Fields and Line Integrals

1. What is a vector field?

A vector field is a mathematical concept that represents the assignment of a vector to each point in a given space or region. It can be visualized as arrows or lines that indicate the direction and magnitude of a vector at every point.

2. What is a line integral?

A line integral is a mathematical concept that calculates the total effect of a vector field along a given curve. It involves integrating the dot product of the vector field and the tangent vector of the curve over a specific interval.

3. How are vector fields and line integrals related?

Vector fields and line integrals are closely related because a line integral calculates the total effect of a vector field along a specific curve. In other words, a line integral is a way to measure the behavior of a vector field along a particular path.

4. What is the significance of vector fields and line integrals in physics?

In physics, vector fields and line integrals are essential tools for understanding the behavior of physical quantities, such as force and velocity. They are used to analyze and calculate the effects of these quantities in various systems and scenarios.

5. How are vector fields and line integrals used in real-world applications?

Vector fields and line integrals have a wide range of applications, including in engineering, fluid dynamics, and electromagnetism. They are used to model and analyze the behavior of complex systems and to solve practical problems in various fields.

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