Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Multivariable analog to the total derivative?

  1. Jun 17, 2010 #1
    For a single variable we have

    [tex]\int_{x_1}^{x_2} f(x) dx = F(x_2)-F(x_1)[/tex]

    if f(x) = dF/dx. f(x) is then a total derivative. What is the analog in 3D so that

    [tex]\int_V f(\vec{x}) d^3x[/tex]

    does not depend on the values of f in the interior of V?

    In case there is not a single answer, let me give the context. In the calculus of variations two Lagrangians are equivalent if

    [tex]L_2(q(t),\dot{q}(t),t)=\lambda L_1(q(t),\dot{q}(t),t) + \frac{d}{dt}F(q(t),\dot{q}(t),t)[/tex]

    where lambda is a constant and F is any function. (That is, their actions are extremized for the same function q(t).) What replaces dF/dt in this equivalency if we have a multi-parameter action

    [tex]S=\int L(q(\vec{x}),\partial q(\vec{x}),\vec{x}) d^3x[/tex]

    (where [tex]\partial q[/tex] stands for the various partial derivatives of q)?

    Is it [tex]\nabla \cdot \vec{F}[/tex] for some vector function F? Or is there more to it than that?
    Last edited: Jun 17, 2010
  2. jcsd
  3. Jun 17, 2010 #2


    Staff: Mentor

    I think you have your terminology wrong. In your first example F is an antiderivative of f and f is the derivative of F.

    The total derivative refers to a function of two or more variables, for example f(x, y). The total differential of f in this case is
    [tex]df = \frac{\partial f}{\partial x}~dx + \frac{\partial f}{\partial y}~dy[/tex]

    If it turns out that x and y are differentiable functions of t, then the total derivative of f looks like this:
    [tex]\frac{df}{dt} = \frac{\partial f}{\partial x}~\frac{dx}{dt} + \frac{\partial f}{\partial y}~\frac{dy}{dt}[/tex]
  4. Jun 17, 2010 #3
  5. Jun 18, 2010 #4
    Thanks to both .
  6. Jun 18, 2010 #5
  7. Jun 19, 2010 #6
    Thanks, Studiot
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Multivariable analog total Date
A Maximization Problem Jan 31, 2018
A Time differentiation of fluid line integrals Apr 7, 2017
I Multi-dimensional Integral by Change of Variables Feb 12, 2017
I Help needed; problematic integral Feb 6, 2017
Analogy for Curl with Torque: Correct? Mar 31, 2012