# 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
Staff Emeritus
Homework Helper
Gold Member
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
Staff Emeritus
Homework Helper
Gold Member
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
Staff Emeritus
$$- \int_C \frac{\vec c' \cdot \vec c'}{|\vec c'|} dt.$$