Question about Vector Fields and Line Integrals

[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)

 

[Orodruin]

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 $C$.

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.

 

[Mohamed Abdul]

$\vec F$ 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?

 

[Orodruin]

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.

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.

 

[Mohamed Abdul]

$\vec c(t)$ (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.

 

[Orodruin]

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?