Search results
Results From The WOW.Com Content Network
The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
This means that the subset of points without a second derivative has measure zero in the curve. However, in other senses, the set of points with a second derivative can be small. In particular, for the graphs of generic non-smooth convex functions, it is a meager set, that is, a countable union of nowhere dense sets. [42]
The covariant derivative incorporates the 4-gradient plus spacetime curvature effects via the Christoffel symbols The strong equivalence principle can be stated as: [ 4 ] : 184 "Any physical law which can be expressed in tensor notation in SR has exactly the same form in a locally inertial frame of a curved spacetime."
In the calculus of variations and classical mechanics, the Euler–Lagrange equations [1] are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange.
L'Hôpital's rule (/ ˌ l oʊ p iː ˈ t ɑː l /, loh-pee-TAHL) or L'Hospital's rule, also known as Bernoulli's rule, is a mathematical theorem that allows evaluating limits of indeterminate forms using derivatives.
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, ′ = ′ = (′) ′.
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 ...
As an open set is a set that is a neighborhood of all its points, a function : is continuous at every point of X if and only if it is a continuous function. If X and Y are metric spaces, it is equivalent to consider the neighborhood system of open balls centered at x and f(x) instead of all neighborhoods.