For a single variable we have(adsbygoogle = window.adsbygoogle || []).push({});

[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?

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# Multivariable analog to the total derivative?

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