Question about Vector Fields and Line Integrals

  • #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)
 

Answers and Replies

  • #2
Orodruin
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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.
 
  • #3
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?
 
  • #4
Orodruin
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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.
 
  • #5
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.
 
  • #6
Orodruin
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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?
 
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