The Weitzenböck identity is a mathematical identity in the context of Riemannian geometry, particularly involving the Laplace-Beltrami operator. Named after the mathematician Roland Weitzenböck, it relates the curvature of a Riemannian manifold to certain differential operators. In general terms, the Weitzenböck identity can express a relationship involving the Laplacian of a differential form and the curvature of the manifold.

Vector calculus identities are mathematical expressions that relate different operations in vector calculus, such as differentiation, integration, and the operations associated with vector fields—specifically the gradient, divergence, and curl. These identities are essential in physics and engineering, particularly in electromagnetism, fluid dynamics, and other fields where vector fields are prominent.

Vector algebra, also known as vector analysis or vector mathematics, comprises the mathematical rules and operations used to manipulate and combine vectors in both two-dimensional and three-dimensional space. Vectors are quantities that possess both magnitude and direction, and they are often represented graphically as arrows or numerically as ordered pairs or triples. Here are some fundamental relations and operations in vector algebra: ### 1.

Vaughan's identity is an important result in analytic number theory, particularly in the context of additive number theory and the study of sums of arithmetic functions. The identity provides a way to express the sum of a function over a set of integers in terms of more manageable sums and is often used in the context of problems involving the distribution of prime numbers.

Vandermonde's identity is a combinatorial identity that relates the sums of binomial coefficients.

The tangent half-angle formulas relate the tangent of half of an angle to the sine and cosine of the angle itself. The tangent half-angle formulas are given by: 1. In terms of sine and cosine: \[ \tan\left(\frac{\theta}{2}\right) = \frac{\sin(\theta)}{1 + \cos(\theta)} \] 2.

Sun's curious identity is a mathematical formula related to the sum of the powers of integers or specific sequences.

A squared triangular number is a special type of number that is both a triangular number and a perfect square. A triangular number is a number that can form an equilateral triangle. The \( n \)-th triangular number is given by the formula: \[ T_n = \frac{n(n + 1)}{2} \] A perfect square is an integer that is the square of an integer.

Sophie Germain's identity is a mathematical identity that relates to the sum of two cubes.

The Sommerfeld identity is a mathematical expression related to the theory of partial differential equations and applies particularly in the context of potentials in electrostatics, scattering problems, and other areas in physics. It often relates to the Green's function solutions of these equations.

The Siegel identity is a mathematical identity related to quadratic forms and the theory of modular forms in number theory. It is named after Carl Ludwig Siegel, who contributed significantly to the field. In general, the Siegel identity expresses a relationship between the values of certain quadratic forms evaluated at integer points and the values of these forms evaluated at their associated characters or modular forms. It can be considered a specific case of more general identities found within the framework of representation theory and arithmetic geometry.

Selberg's identity is a mathematical result pertaining to the theory of special functions and number theory, specifically related to the Riemann zeta function and the distribution of prime numbers. The identity is named after the Norwegian mathematician Atle Selberg. One of the most common formulations of Selberg's identity involves the relation between sums and products over integers.

The Rothe–Hagen identity is a mathematical identity related to the theory of partitions, specifically concerning the representations of integers as sums of parts. While detailed references specific to the identity might be scarce, it is often discussed in the context of combinatorial mathematics or number theory. The identity is named after mathematicians who have contributed to partition theory and can be expressed in various forms. Generally, it can relate different ways of summing integers or the coefficients of generating functions.

The Rogers–Ramanujan identities are two famous identities in the theory of partitions discovered by the mathematicians Charles Rogers and Srinivasa Ramanujan. They relate to the summation of series involving partitions of integers and have significant applications in combinatorics and number theory.

The Rogers–Ramanujan continued fraction is a famous infinite continued fraction introduced by mathematicians Leonard J. Rogers and Srinivasa Ramanujan. It is notable for its deep connections to combinatorial identities, number theory, and the theory of partitions.

The quintuple product identity is a mathematical identity related to the theory of partitions and q-series, often involving generating functions in combinatorial contexts. It is a specific case of the more general product identities that arise in the theory of modular forms and q-series.

The Q-Vandermonde identity is a generalization of the classical Vandermonde identity, which relates sums of binomial coefficients to the coefficients of a polynomial expansion. The Q-Vandermonde identity specifically introduces the concept of q-binomial coefficients (also known as Gaussian coefficients) and q-series.

The Pythagorean trigonometric identities are fundamental relationships between the sine and cosine functions that stem from the Pythagorean theorem. They are derived from the fact that for a right triangle with an angle \( \theta \), the following equation holds: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] This is the most basic Pythagorean identity.

The power rule is a fundamental principle in calculus used to differentiate functions of the form \( f(x) = x^n \), where \( n \) is any real number.

Pokhozhaev's identity is a mathematical result related to the study of certain partial differential equations, particularly in the context of nonlinear analysis and the theory of elliptic equations. It provides a relationship that can be used to derive energy estimates and to study the qualitative properties of solutions to nonlinear equations. The identity is often stated in the context of solutions to the boundary value problems for nonlinear elliptic equations and is used to establish properties such as symmetry, monotonicity, or the uniqueness of solutions.