List of definitions of terms and concepts commonly used in calculus
Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.
This glossary of calculus is a list of definitions about calculus, its sub-disciplines, and related fields.
E
- e (mathematical constant)
- The number e is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one. It is approximately equal to 2.71828,^{[34]} and is the limit of (1 + 1/n)^{n} as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series^{[35]}
- $e=\displaystyle \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!))={\frac {1}{1))+{\frac {1}{1))+{\frac {1}{1\cdot 2))+{\frac {1}{1\cdot 2\cdot 3))+\cdots$
elliptic integralIn integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler (c. 1750). Modern mathematics defines an "elliptic integral" as any function f which can be expressed in the form
- $f(x)=\int _{c}^{x}R\left(t,{\sqrt {P(t)))\right)\,dt,$
where R is a rational function of its two arguments, P is a polynomial of degree 3 or 4 with no repeated roots, and c is a constant..essential discontinuityFor an essential discontinuity, only one of the two one-sided limits needs not exist or be infinite.
Consider the function
- $f(x)={\begin{cases}\sin {\frac {5}{x-1))&{\mbox{ for ))x<1\\0&{\mbox{ for ))x=1\\{\frac {1}{x-1))&{\mbox{ for ))x>1\end{cases))$
Then, the point $\scriptstyle x_{0}\;=\;1$ is an essential discontinuity.
In this case, $\scriptstyle L^{-))$ doesn't exist and $\scriptstyle L^{+))$ is infinite – thus satisfying twice the conditions of essential discontinuity. So x_{0} is an essential discontinuity, infinite discontinuity, or discontinuity of the second kind. (This is distinct from the term essential singularity which is often used when studying functions of complex variables.Euler methodEuler's method is a numerical method to solve first order first degree differential equation with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who treated it in his book Institutionum calculi integralis (published 1768–1870).^{[36]}exponential functionIn mathematics, an exponential function is a function of the form
$f(x)=ab^{x},$
where b is a positive real number, and in which the argument x occurs as an exponent. For real numbers c and d, a function of the form $f(x)=ab^{cx+d))$ is also an exponential function, as it can be rewritten as
- $ab^{cx+d}=\left(ab^{d}\right)\left(b^{c}\right)^{x}.$
extreme value theoremStates that if a real-valued function f is continuous on the closed interval [a,b], then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in [a,b] such that:
- $f(c)\geq f(x)\geq f(d)\quad {\text{for all ))x\in [a,b].$
A related theorem is the boundedness theorem which states that a continuous function f in the closed interval [a,b] is bounded on that interval. That is, there exist real numbers m and M such that:
- $m<f(x)<M\quad {\text{for all ))x\in [a,b].$
The extreme value theorem enriches the boundedness theorem by saying that not only is the function bounded, but it also attains its least upper bound as its maximum and its greatest lower bound as its minimum.extremumIn mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema).^{[37]}^{[38]}^{[39]} Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.
As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum.