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If m = n, then f is a function from R n to itself and the Jacobian matrix is a square matrix. We can then form its determinant, known as the Jacobian determinant. The Jacobian determinant is sometimes simply referred to as "the Jacobian". The Jacobian determinant at a given point gives important information about the behavior of f near that
In linear algebra, the trace of a square matrix A, denoted tr(A), [1] is the sum of the elements on its main diagonal, + + +.It is only defined for a square matrix (n × n).The trace of a matrix is the sum of its eigenvalues (counted with multiplicities).
In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums J ( χ , ψ ) for Dirichlet characters χ , ψ modulo a prime number p , defined by
Condition numbers can also be defined for nonlinear functions, and can be computed using calculus.The condition number varies with the point; in some cases one can use the maximum (or supremum) condition number over the domain of the function or domain of the question as an overall condition number, while in other cases the condition number at a particular point is of more interest.
It was first conjectured in 1939 by Ott-Heinrich Keller, [1] and widely publicized by Shreeram Abhyankar, as an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus. The Jacobian conjecture is notorious for the large number of attempted proofs that turned out to contain subtle ...
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [1]If A is a differentiable map from the real numbers to n × n matrices, then
For the sum of squares of consecutive integers, see Square pyramidal number; For representing an integer as a sum of squares of 4 integers, see Lagrange's four-square theorem; Legendre's three-square theorem states which numbers can be expressed as the sum of three squares; Jacobi's four-square theorem gives the number of ways that a number can ...
If the "numerator" is 1, rules 3 and 4 give a result of 1. If the "numerator" and "denominator" are not coprime, rule 3 gives a result of 0. Otherwise, the "numerator" and "denominator" are now odd positive coprime integers, so we can flip the symbol using rule 6, then return to step 1. In addition to the codes below, Riesel [4] has it in Pascal.