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The function () is defined on the interval [,].For a given , the difference () takes the maximum at ′.Thus, the Legendre transformation of () is () = ′ (′).. In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, [1] is an involutive transformation on real-valued functions that are ...
valid for any vector fields X and Y and any tensor field T.. Considering vector fields as infinitesimal generators of flows (i.e. one-dimensional groups of diffeomorphisms) on M, the Lie derivative is the differential of the representation of the diffeomorphism group on tensor fields, analogous to Lie algebra representations as infinitesimal representations associated to group representation ...
where T is any tensor, is the Levi-Civita connection associated to the metric, and the trace is taken with respect to the metric. Recall that the second covariant derivative of T is defined as ∇ X , Y 2 T = ∇ X ∇ Y T − ∇ ∇ X Y T . {\displaystyle \nabla _{X,Y}^{2}T=\nabla _{X}\nabla _{Y}T-\nabla _{\nabla _{X}Y}T.}
The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel.Levi-Civita, [1] along with Gregorio Ricci-Curbastro, used Christoffel's symbols [2] to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.
Once a value of y is chosen, say a, then f(x,y) determines a function f a which traces a curve x 2 + ax + a 2 on the xz-plane: = + +. In this expression, a is a constant, not a variable, so f a is a function of only one real variable, that being x. Consequently, the definition of the derivative for a function of one variable applies:
Note that, for general Ehresmann connections, the horizontal lift is path-dependent. When two smooth curves in M, coinciding at γ 1 (0) = γ 2 (0) = x 0 and also intersecting at another point x 1 ∈ M, are lifted horizontally to E through the same e ∈ π −1 (x 0), they will generally pass through different points of π −1 (x 1).
The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on R n. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor algebra of a manifold
This is evidently an alternative definition of Wirtinger derivative respect to the complex conjugate variable: [10] it is a more general one, since, as noted a by Henrici (1993, p. 294), the limit may exist for functions that are not even differentiable at =. [11] According to Fichera (1969, p. 28), the first to identify the areolar derivative ...