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The argument of is a differentiable function :, and its Jacobian is identified with a -vector. When deriving the Euler–Lagrange equation, the common approach is to assume Ω {\displaystyle \Omega } has a C 2 {\displaystyle C^{2}} boundary and let the domain of definition for J {\displaystyle J} be C 2 ( Ω , R m ) {\displaystyle C^{2}(\Omega ...
If a continuous function on an open interval (,) satisfies the equality () =for all compactly supported smooth functions on (,), then is identically zero. [1] [2]Here "smooth" may be interpreted as "infinitely differentiable", [1] but often is interpreted as "twice continuously differentiable" or "continuously differentiable" or even just "continuous", [2] since these weaker statements may be ...
The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale (also known as a time-set), which may be an arbitrary closed subset of the reals. In this way, results apply not only to the set of real numbers or set of integers but to more general time scales such as a Cantor set.
The intuition behind this result is that, if the variable is actually time, then the statement = implies that the Lagrangian is time-independent. By Noether's theorem, there is an associated conserved quantity. In this case, this quantity is the Hamiltonian, the Legendre transform of the Lagrangian, which (often) coincides with the energy of ...
In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.
In multivariable calculus, the directional derivative measures the rate at which a function changes in a particular direction at a given point. [citation needed]The directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a direction ...
Let and be two linear operators, with domains () and (), acting in a Banach space. [4] [5] [6] A function (): [,] is said to have strong derivative (or to be Frechet differentiable or simply differentiable) at the point if there exists an element such that
The function is differentiable if and only if each of its components : is differentiable, so when studying total derivatives, it is often possible to work one coordinate at a time in the codomain. However, the same is not true of the coordinates in the domain.