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Calculus 3 - How to make non conservative vector field into conservative vector field

  1. Jan 6, 2012 #1
    Hi,

    I'm studying calculus 3 and am currently learning about conservative vector fields.

    =============================
    Fundamental Theorem for Line Integrals
    =============================
    Let F be a a continuous vector field on an open connected region R in [itex]ℝ^{2}[/itex] (or D in [itex]ℝ^{3}[/itex]). There exists a potential function ψ where F = [itex]\nabla[/itex]ψ (which means that F is conservative) if and only if

    [itex]\int_{C}[/itex] F[itex]\bullet[/itex]T ds = [itex]\int_{C}[/itex] F[itex]\bullet[/itex]dr = ψ(B) - ψ(A)

    for all points A and B in R and all smooth oriented curves C from A to B.

    ==========================
    Test for Conservative Vector Fields
    ==========================

    Let F = <f,g,h> be a vector field defined on a connected and simply connected region of [itex]ℝ^{3}[/itex], where f,g, and h have continuous first partial derivatives on D. Then, F is a conservative vector filed on D (there is a potential function ψ such that F = [itex]\nabla[/itex]ψ) if and only if

    [itex]f_{y}[/itex] = [itex]g_{x}[/itex], [itex]f_{z}[/itex] = [itex]h_{x}[/itex], and [itex]g_{z}[/itex] = [itex]h_{y}[/itex].

    For vector fields in [itex]ℝ^{2}[/itex], we have the single condition [itex]f_{y}[/itex] = [itex]g_{x}[/itex].

    ==
    Q. I'm also studying elementary differential equations and it looks to me like I can take any non conservative vector field and make it into a conservative vector field by finding a integration factor and then finding ψ. Once I find ψ I can just use the fundamental theorem of line integrals and evaluate it at two points. This method, if I can do this, appears to me to be a more easier way of evaluated non conservative vector field line integrals.

    It appears to me that the following is true:

    f(x,y,z) = [itex]ψ_{x}[/itex]
    g(x,y,z) = [itex]ψ_{y}[/itex]
    h(x,y,z) = [itex]ψ_{z}[/itex]

    Given some arbitrary vector field, F(f(x,y,z),g(x,y,z),h(x,y,z)), and I find that
    [itex]f(x,y,z)_{y}[/itex] ≠ [itex]g(x,y,z)_{x}[/itex]
    or
    [itex]ψ_{xy}[/itex] ≠ [itex]ψ_{yx}[/itex]

    then shouldn't I be able to multiply by some function μ(x) to make the statement true

    f(x,y,z)μ(x) and g(x,y,z)μ(x)
    or
    [itex]ψ_{x}[/itex]μ(x) and [itex]ψ_{y}[/itex]μ(x)

    then
    [itex]f(x,y,z)_{y}[/itex]μ(x) = [itex]g(x,y,z)μ(x)_{x}[/itex]
    or
    [itex]ψ_{xy}[/itex]μ(x) = [itex](ψ_{y}μ(x))_{x}[/itex]

    Can't I then proceed to find μ(x) and then use the fact that
    f(x,y,z)μ(x) = [itex]ψ_{x}[/itex]μ(x)
    and solve for ψ

    ψ = [itex]\int[/itex] [itex]ψ_{x}[/itex]μ(x) dx = [itex]f(x,y,z)_{2}[/itex] + f(y,z)
    or
    ψ = [itex]\int[/itex] [itex]f(x,y,z)[/itex]μ(x) dx = [itex]f(x,y,z)_{2}[/itex] + f(y,z)

    and then proceed to solve for f(y,z) some how?

    I feel as if there's some way to make any non conservative vector filed equation into a conservative vector valued function and then just apply the Fundamental Theorem for Line Integrals some how but I'm not exactly sure I'm solving for it correctly. Thanks for any help. I hope to learn how to solve such a differential equation.
     
    Last edited: Jan 7, 2012
  2. jcsd
  3. Jan 7, 2012 #2

    jedishrfu

    Staff: Mentor

    Re: Calculus 3 - How to make non conservative vector field into conservative vector f

    I think there may be some non-conservative fields that could be represented by a conservative part and a non-conservative part but to say that non-conservative vector fields can be mapped to conservative ones is probably not true at all.

    From the definition the conservative vector field can be represented by a potential whose gradient is the conservative vector field right? so given a potential function of x,y,z what could I add to it to make things no longer conservative say a time varying component. physically that makes it non-conservative right?

    I'm thinking: gravity about the sun is conservative but if you make the sun's mass somehow vary over time then that would make it non-conservative. depending on the size and periodicity of the variations (if they are even periodic) so while you could represent the potential as some time varying function the grav force field wouldn't be conservative.

    I think General Relativity has that issue and that why Mercury precesses in it correctly.

    Im real rusty here someone please slap me.
     
  4. Jan 7, 2012 #3
    Re: Calculus 3 - How to make non conservative vector field into conservative vector f

    Well I don't see why it couldn't be done if you had a vector field that had only only two variables without a problem at all.

    If I had a vector field
    F = <f(x,y),g(x,y)>

    I would run into something like this

    ψ = [itex]\int[/itex] [itex]f(x,y)[/itex]μ(x) dx = [itex]f(x,y)_{2}[/itex] + f(y)

    then I could say

    [itex]ψ_{y}[/itex] = [itex]f(x,y)_{2y}[/itex] + f'(y) = g(x,y)μ(x)
    because g(x,y)μ(x) = [itex]ψ_{y}[/itex]
    and solve for f(y)
    f(y) = [itex]\int[/itex] [g(x,y) - [itex]f(x,y)_{2y}[/itex]] dy

    then
    ψ = [itex]f(x,y)_{2}[/itex] + [itex]\int[/itex] [g(x,y)μ(x) - [itex]f(x,y)_{2y}[/itex]] dy
     
    Last edited: Jan 7, 2012
  5. Jan 7, 2012 #4
    Re: Calculus 3 - How to make non conservative vector field into conservative vector f

    ok so I figured out were to go from here
    ψ = [itex]\int[/itex] [itex]f(x,y,z)[/itex]μ(x) dx = [itex]f(x,y,z)_{2}[/itex] + f(y,z)
    to solve for ψ
    it was easier than I thought.
    As far as I know this is a perfectly fine thing to do? It doesn't show up in my calculus book so I just want to make sure. I can make any non conservative vector field into a conservative vector field by multiply by a integration factor and then just evaluating it from the limits of the integral by the fundamental therom of line integrals? I just want to make sure I don't see why not.
     
  6. Jan 7, 2012 #5

    jedishrfu

    Staff: Mentor

    Re: Calculus 3 - How to make non conservative vector field into conservative vector f

    i was arguing from the physics pt where a conservative vector field was represented by the grad of a potential field and then trying to see how the pot field could be altered. Gen Rel creates this situation and thats the explanation why mercury precesses Gen Rel describes a non-conservative vector field about the sun due to the interaction of x,y,z and t the spacetime metric.

    but im glad you solved your problem.
     
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