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Visual proof of the Pythagorean identity: for any angle , the point (,) = (, ) lies on the unit circle, which satisfies the equation + =.Thus, + =. In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables ...
[9] [10] As the first representation in European mathematics of a number as the result of an infinite process rather than of a finite calculation, [11] Eli Maor highlights Viète's formula as marking the beginning of mathematical analysis [2] and Jonathan Borwein calls its appearance "the dawn of modern mathematics". [12]
A left identity element that is also a right identity element if called an identity element. The empty set ∅ {\displaystyle \varnothing } is an identity element of binary union ∪ {\displaystyle \cup } and symmetric difference , {\displaystyle \triangle ,} and it is also a right identity element of set subtraction ∖ : {\displaystyle ...
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate the corresponding electrostatic or gravitational ...
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
The Beltrami identity, named after Eugenio Beltrami, is a special case of the Euler–Lagrange equation in the calculus of variations. The Euler–Lagrange equation serves to extremize action functionals of the form
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then the polarization identity can be used to express this inner product entirely in terms of the norm.
Pokhozhaev's identity is an integral relation satisfied by stationary localized solutions to a nonlinear Schrödinger equation or nonlinear Klein–Gordon equation. It was obtained by S.I. Pokhozhaev [1] and is similar to the virial theorem. This relation is also known as G.H. Derrick's theorem. Similar identities can be derived for other ...