Linear Integration of a Vector Field over a Parametric Path

In summary, the problem asks to find the correct way to evaluate an integral involving a given vector field and a dot product. The solution involves finding a scalar function that is the gradient of the vector field and using the Fundamental Theorem of Calculus. This may not be obvious at first, but reading ahead to the next chapter on Div, Grad, Curl can provide insight. Other potential methods of solving the problem include realizing that F.t ds = \psi or using other mathematical techniques.
  • #1
sriracha
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Problem statement attached. The correct way to do this seems to plug in your given x, y, z into F then integrate the dot product of F and <x',y',z'> dp from 0 to 1, however, this results in way too messy of an integral. Answer is 3/e.

<e^-(sin(pi*p/2))-((1-e^p)/(1-e))e^-(ln(1+p)/ln(2)),e^-((1-e^p)/(1-e))-(ln(1+p)/ln(2))e^-(sin(pi*p/2)),e^-(ln(1+p)/ln(2))-(sin(pi*p/2))e^-((1-e^p)/(1-e))>.<2^(x-1)ln(2),2/(pi*sqrt(1-y^2),(e-1)/((e-1)y+1)>
 

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  • #2
sriracha said:
Problem statement attached. The correct way to do this seems to plug in your given x, y, z into F then integrate the dot product of F and <x',y',z'> dp from 0 to 1, however, this results in way too messy of an integral. Answer is 3/e.

<e^-(sin(pi*p/2))-((1-e^p)/(1-e))e^-(ln(1+p)/ln(2)),e^-((1-e^p)/(1-e))-(ln(1+p)/ln(2))e^-(sin(pi*p/2)),e^-(ln(1+p)/ln(2))-(sin(pi*p/2))e^-((1-e^p)/(1-e))>.<2^(x-1)ln(2),2/(pi*sqrt(1-y^2),(e-1)/((e-1)y+1)>

It would be awfully nice if the vector field F were a gradient of some scalar function, wouldn't it? Can you guess one that works?
 
  • #3
Okay so I figured this out, but I had to read into the next chapter to do so. This is from Div, Grad, Curl, which really should be called Div, Curl, Grad. Why would Schey ask this problem before you get to gradient and how would he expect you to find it otherwise? Can anyone think of another possible path to solving this? I mean it's certainly possible that someone would realize the F.t ds = [itex]\psi[/itex], but I imagine even for a quite brilliant person that would take some time.
 

1. What is linear integration of a vector field over a parametric path?

Linear integration of a vector field over a parametric path is a mathematical process that involves calculating the total effect of a vector field along a specific path. It is used to find the net displacement or work done by a vector field on a moving object.

2. How is linear integration different from regular integration?

Linear integration differs from regular integration in that it involves integrating a vector field along a specific path, rather than over a defined area. It takes into account the direction and magnitude of the vector field along the path, rather than just the magnitude as in regular integration.

3. What is a parametric path?

A parametric path is a mathematical representation of a path in terms of one or more variables. It is typically represented by a set of parametric equations, where each variable represents a different dimension or coordinate.

4. Why is linear integration of a vector field over a parametric path useful?

Linear integration of a vector field over a parametric path is useful in many scientific fields, such as physics and engineering. It allows for the calculation of work done by a force, displacement of an object, and other important quantities.

5. What are some real-world applications of linear integration of a vector field over a parametric path?

Some real-world applications of linear integration of a vector field over a parametric path include calculating the work done by a force on an object, determining the displacement of a moving object, and finding the net force acting on a system. It is also used in fluid mechanics to calculate the flow of a fluid along a specific path.

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