Evaluating Line Integral: Curl of F and its Relation

In summary, the conversation discusses evaluating a line integral with a given vector field and two different paths. The first path is a straight line from (a, 0, 0) to (a, 0, 2πb) and the second path is a circular helix parametrized by r = (a cosλ, a sinλ, bλ). The conversation also mentions computing the curl of the vector field and how it relates to the two integral calculations. The speaker is struggling with understanding the relationship between the two and asks for help. They are directed to read the notes provided and attempt to solve the problem on their own before seeking further assistance.
  • #1
Frenchy
1
0
1. Evaluate the line integral∫F . dr with F = 3(-y,x,0) from (a,0,0) to (a,0,2πb) along a straight line.

2. Do the same along a circular helix between the two points, parameterised as r = (a cosλ, a sinλ, bλ)

3. Compute the curl of F. How does this relate to the two integral calculations above?

I know Curl = \nabla x F

My notes on this don't seem to be that great, and I'm just completely lost, tbh.

Any help?
 
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  • #2
It should seem obvious that, since there is no force applied in the z in the first one, no work is done. From experience, I know that it is again zero in part 2. What do you get for the curl? Solving for it should show you tell you something important.
 

1. What is a line integral?

A line integral is a type of integral that is used to calculate the total value of a function over a specific curve or path. It is represented by the symbol ∫ and is often used in physics and engineering to calculate quantities such as work or electric flux.

2. What is curl of a vector field?

Curl is a vector operator that measures the tendency of a vector field to rotate about a point. It is represented by the symbol ∇ × F, where F is the vector field. The curl of a vector field is a vector quantity that describes the magnitude and direction of rotation at each point in the field.

3. How is the line integral related to the curl of a vector field?

The line integral of a vector field F along a curve C is equal to the surface integral of the curl of F over the surface enclosed by C. This relationship is known as Stokes' theorem and is an important tool in vector calculus.

4. What is the significance of evaluating the line integral of curl of F?

Evaluating the line integral of curl of F is useful in understanding the flow of a vector field and can help in solving problems related to fluid dynamics, electromagnetism, and other areas of physics and engineering. It also allows us to calculate physical quantities such as work and circulation along a specific path.

5. How is the calculation of line integral of curl of F done?

To calculate the line integral of curl of F, we first need to parameterize the curve C and express the vector field F in terms of its components. Then, using the properties of line integrals and Stokes' theorem, we can evaluate the integral by finding the dot product between the curl of F and the tangent vector of the curve at each point. Finally, we integrate over the curve to obtain the total value of the line integral.

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