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  2. Notation for differentiation - Wikipedia

    en.wikipedia.org/wiki/Notation_for_differentiation

    When taking the antiderivative, Lagrange followed Leibniz's notation: [7] = ′ = ′. However, because integration is the inverse operation of differentiation, Lagrange's notation for higher order derivatives extends to integrals as well. Repeated integrals of f may be written as

  3. Lagrangian mechanics - Wikipedia

    en.wikipedia.org/wiki/Lagrangian_mechanics

    In the Lagrangian, the position coordinates and velocity components are all independent variables, and derivatives of the Lagrangian are taken with respect to these separately according to the usual differentiation rules (e.g. the partial derivative of L with respect to the z velocity component of particle 2, defined by v z,2 = dz 2 /dt, is ...

  4. Chain rule - Wikipedia

    en.wikipedia.org/wiki/Chain_rule

    In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g.More precisely, if = is the function such that () = (()) for every x, then the chain rule is, in Lagrange's notation, ′ = ′ (()) ′ (). or, equivalently, ′ = ′ = (′) ′.

  5. Euler–Lagrange equation - Wikipedia

    en.wikipedia.org/wiki/Euler–Lagrange_equation

    The Euler–Lagrange equation was developed in connection with their studies of the tautochrone problem. The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in ...

  6. Inverse function rule - Wikipedia

    en.wikipedia.org/wiki/Inverse_function_rule

    Integrating this relationship gives = ′ (()) +.This is only useful if the integral exists. In particular we need ′ to be non-zero across the range of integration. It follows that a function that has a continuous derivative has an inverse in a neighbourhood of every point where the derivative is non-zero.

  7. Calculus - Wikipedia

    en.wikipedia.org/wiki/Calculus

    In Lagrange's notation, the symbol for a derivative is an apostrophe-like mark called a prime. Thus, the derivative of a function called f is denoted by f′, pronounced "f prime" or "f dash". For instance, if f(x) = x 2 is the squaring function, then f′(x) = 2x is its derivative (the doubling function g from above).

  8. Lagrangian (field theory) - Wikipedia

    en.wikipedia.org/wiki/Lagrangian_(field_theory)

    In field theory, the independent variable is replaced by an event in spacetime (x, y, z, t), or more generally still by a point s on a Riemannian manifold.The dependent variables are replaced by the value of a field at that point in spacetime (,,,) so that the equations of motion are obtained by means of an action principle, written as: =, where the action, , is a functional of the dependent ...

  9. Calculus of variations - Wikipedia

    en.wikipedia.org/wiki/Calculus_of_Variations

    The calculus of variations began with the work of Isaac Newton, such as with Newton's minimal resistance problem, which he formulated and solved in 1685, and later published in his Principia in 1687, [2] which was the first problem in the field to be formulated and correctly solved, [2] and was also one of the most difficult problems tackled by variational methods prior to the twentieth century.