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Stokes' theorem?

  1. Apr 22, 2006 #1
    For the following vector fields F , decide whether it is conservative or not by computing curl F . Type in a potential function f.

    F(x,y,z)=-3xi-2yj+k

    f(x,y,z)=?

    I'm not sure what the problem is asking. calcualting curl F needs integration and a boundary. I dont know why they ask for f(x,y,z). could someone explain what this problem as asking me to do?
     
  2. jcsd
  3. Apr 22, 2006 #2
    Why would calculating the curl require integration? The function f is the scalar potential asscoiated with the field F. hint: The second question sort of answers the first.
     
  4. Apr 22, 2006 #3
    so to calculate f, i take the partial derivative of F?
     
  5. Apr 22, 2006 #4
    You differntiate F, well not literally, to find its curl. To find f, assuiming that you've confirmed that F is conservative, you have to calculate the line integral of F from some reference point, say the origin, to some point (x,y,z).
     
  6. Apr 22, 2006 #5
    The vector function [itex] \vec F(x,y,z) [/itex] is equal to [itex] \nabla f(x,y,z) [/itex] if [itex] \nabla \times \vec F(x,y,z) = 0 [/itex] right?

    So [itex] f(x,y,z) [/itex] will exist ONLY when [itex] \vec F [/itex] is conservative. So what does the gradient ask for?

    So you must satisfy the following conditions:
    [tex] \frac{\partial}{\partial x}f(x,y,z) = F_x [/tex]

    [tex] \frac{\partial}{\partial y}f(x,y,z) = F_y [/tex]

    [tex] \frac{\partial}{\partial z}f(x,y,z) = F_z [/tex]

    Where [itex] F_x [/itex] is the x-component of the vector function.
     
  7. Apr 22, 2006 #6
    F(x,y,z)=-3xi-2yj+k

    [tex] \nabla f(x,y,z) = \frac{\partial f}{\partial x} i + \frac{\partial f}{\partial y} j +\frac{\partial f}{\partial z} k[/tex]

    [tex] \nabla f(x,y,z) = -3 i-2j [/tex]

    would this be f?
     
  8. Apr 22, 2006 #7

    nrqed

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    No. What you want is a potential f(x,y,z) such that [itex] \nabla f [/itex] gives you back the vector field [itex]{\vec F}[/itex] that you gave in your first post. This is what Frogpad was saying in his last post.
     
  9. Apr 22, 2006 #8

    nrqed

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    First thinsg first. Have you computed the curl of F and checked if F was conservative?
     
  10. Apr 22, 2006 #9
    i guess a more general question would be what IS the curl of F and why is it necessary to check if F is conserved? I have not been taught this, and my book introduces curl with integration such that:

    [tex] \int _C F dr=\int \int curlFdS[/tex]
     
  11. Apr 22, 2006 #10

    nrqed

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    Oh! So you haven't been taught the definition of the curl! I can see how things may be confusing! The above is an application of the curl (and could be taken as a definition, I suppose), but the curl of a vector field is defines to be [itex] \nabla \times {\vec F(x,y,z)} [/itex], that is, it is the cross product if you will of the nabla operator with the vector field. If your book does not define this it is because it assumes you are familiar with this. Look at a more basic math or E&M book and it will be defined there for sure.

    The key point is that if the curl of a vector field is zero then the vector field can be written as a gradient of a scalar function (your small cap f). If the vector field represents a force field, then this implies that one can define a potential energy (because the force is conservative)
     
  12. Apr 22, 2006 #11
    could you give me an example of how to find if the curl of a vector field is zero? And how would one find f if the curl is zero?
     
  13. Apr 23, 2006 #12

    nrqed

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    The curl is a vector. Its x component is [itex] \partial_y F_z(x,y,z) -\partial_z F_y(x,y,z) [/itex], the y component is [itex] - \partial_x F_z(x,y,z) +\partial_z F_x(x,y,z) [/itex] and the z component is [itex] \partial_x F_y(x,y,z) -\partial_y F_x(x,y,z) [/itex]. Calculating this for your F you should get zero for all three components.
     
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