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Proving that if F is conservative, F can be written as the gradient of a scalar field

  1. Aug 15, 2012 #1
    1. The problem statement, all variables and given/known data

    The proof begins: Suppose that F is conservative. Then a scalar field ε(r) can be defined as the line integral of F from the origin to the point r. So ∫F dot dr = ε(r), where the limits of integration are from 0 to r.

    The next step, however, eludes me: From the definition of an integral, it then follows that an infinitesimal change in ε is given by dε = F dot dr.




    2. Relevant equations



    3. The attempt at a solution

    Usually total differentials are related to partial derivatives, tangent planes, and Taylor expansions. I'm failing to fill in the intermediate steps in deriving dε = F dot dr from ∫F dot dr = ε(r) using the "definition of integral". Any insight?
     
  2. jcsd
  3. Aug 15, 2012 #2
    Re: Proving that if F is conservative, F can be written as the gradient of a scalar f

    Upon second thought:


    d/dr [∫F dot dr] = d/dr [∫F dot ndr] , where n is a unit vector in the direction of dr. So,

    d/dr [∫F dot ndr] = d/dr [ε(r)]. Thus,

    F dot n = d/dr [ε(r)]

    and so, multiplying by dr:

    dε(r) = F dot ndr = F dot dr




    Is this valid?
     
  4. Aug 15, 2012 #3

    gabbagabbahey

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    Re: Proving that if F is conservative, F can be written as the gradient of a scalar f

    Close, but [itex]\mathbf{r}[/itex] is a vector, and so an infinitesimal change in [itex]\epsilon ( \mathbf{r} )[/itex] is really

    [tex]d\epsilon = \frac{ \partial \epsilon}{ \partial x} dx + \frac{ \partial \epsilon}{ \partial y} dy + \frac{ \partial \epsilon}{ \partial z} dz = \mathbf{ \nabla } \epsilon \cdot d\mathbf{r}[/tex]
     
  5. Aug 15, 2012 #4
    Re: Proving that if F is conservative, F can be written as the gradient of a scalar f

    Hi gabbagabbahey! I understand that what you posted is the equation for the total differential, I am just struggling to understand how from ∫F dot dr = ε(r) (where the limits of integration are from 0 to r) one deduces the result- in particular, using the "definition of integral".

    That is, how can we derive dε = F dot dr from what we have above?
     
  6. Aug 16, 2012 #5

    gabbagabbahey

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    Re: Proving that if F is conservative, F can be written as the gradient of a scalar f

    Well, to me, the statement "From the definition of an integral" means using the fundamental theorem of calculus (FTC). For a simple one-dimensional integral, FTC tells you that if [itex]F(b)-F(a) = \int_a^b f(x) dx[/itex], then [itex]F'(x)=f(x)[/itex] (or [itex]dF = f(x)dx[/itex]).

    For line integrals, this generalizes to the statement that if [itex]F(\mathbf{b}) - F(\mathbf{a}) = \int_{ \mathbf{a} }^{ \mathbf{b} } \mathbf{f} ( \mathbf{r} ) \cdot d \mathbf{r}[/itex] regardless of which path you choose for the integration, then [itex]\mathbf{\nabla} F = \mathbf{f} ( \mathbf{r} )[/itex] (or, equivalently [itex]dF=\mathbf{f} ( \mathbf{r} ) \cdot d \mathbf{r}[/itex])
     
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