Implicit function theoremOn converting relations to functions of several real variables In multivariable calculus, the implicit function theorem is a theorem that provides sufficient conditions under which a planar curve specified by F ( x , y ) = 0 {\displaystyle F(x,y)=0} can also be specified as the graph of a function f {\displaystyle f} , so that for each point ( x , y ) {\displaystyle (x,y)} on part of the curve, one has y = f ( x ) {\displaystyle y=f(x)} .
Closed-form expressionIn mathematics, a closed formexpression or formula is one that is formed with constants, variables, and a set of functions considered as basic and connected by arithmetic operations (+, −, ×, /, and integer powers) and function composition. Commonly, the basic functions that are allowed in closed forms are nth root, exponential function, logarithm, and trigonometric functions. However, the set of basic functions depends on the context.Function (mathematics)In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time.PolynomialType of mathematical expressionIn mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate x{\displaystyle x} is x2−4x+7{\displaystyle x^{2}-4x+7}.
Unit circleIn mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as S because it is a one-dimensional unit n-sphere.Argument of a functionIn mathematics, an argument of a function is a value provided to obtain the function's result. It is also called an independent variable. For example, the binary function f ( x , y ) = x 2 + y 2 {\displaystyle f(x,y)=x^{2}+y^{2}} has two arguments, x {\displaystyle x} and y {\displaystyle y} , in an ordered pair ( x , y ) {\displaystyle (x,y)} . The hypergeometric function is an example of a four-argument function.
Relation (mathematics)In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. As an example, "is less than" is a relation on the set of natural numbers; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3), and likewise between 3 and 4 (denoted as 3 < 4), but not between the values 3 and 1 nor between 4 and 4, that is, 3 < 1 and 4 < 4 both evaluate to false.Value (mathematics)In mathematics, a mathematical value is a broad term that refers to any definite entity that can be manipulated with operators according to the well-defined rules of its mathematical system. Certain values can correspond to the real world, although most values in mathematics generally exist purely as abstract objects with no connection to the real world. Numerical valuesNumbers (specifically the reals) are values that represent quantities.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.
SmoothnessIn mathematical analysis, the smoothness describes the number of times a function can be differentiated without producing discontinuities. The smoothness, or differentiability class, is an integer k{\displaystyle k} such that a function has all derivatives up to order k{\displaystyle k}, and such that all of these derivatives are continuous. One says that such a function has class Ck{\displaystyle C^{k}}.
Lambert W functionIn mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function f ( w ) = w e w {\displaystyle f(w)=we^{w}} , where w {\displaystyle w} is any complex number and e w {\displaystyle e^{w}} is the exponential function. The function is named after Johann Lambert, who considered a related problem in 1758.*As an Amazon Associate I earn from qualifying purchases.