Search results
Results From The WOW.Com Content Network
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 point. For instance, the continuously differentiable function f is invertible near a point p ∈ R n if the Jacobian determinant at p is non-zero.
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
In mathematics, the Jacobian conjecture is a famous unsolved problem concerning polynomials in several variables. It states that if a polynomial function from an n -dimensional space to itself has Jacobian determinant which is a non-zero constant, then the function has a polynomial inverse.
If the Euler's criterion formula is used modulo a composite number, the result may or may not be the value of the Jacobi symbol, and in fact may not even be −1 or 1. For example, For example, ( 19 45 ) = 1 and 19 45 − 1 2 ≡ 1 ( mod 45 ) .
The formula for the ... Sylvester's criterion asserts that this is equivalent to the ... The Jacobian at a point gives the best linear approximation of the distorted ...
Input: initial guess x (0) to the solution, (diagonal dominant) matrix A, right-hand side vector b, convergence criterion Output: solution when convergence is reached Comments: pseudocode based on the element-based formula above k = 0 while convergence not reached do for i := 1 step until n do σ = 0 for j := 1 step until n do if j ≠ i then ...
For example, by the Jacobian criterion for regularity, a generic point of a variety over a field of characteristic zero is smooth. (This statement is known as generic smoothness .) This is true because the Jacobian criterion can be used to find equations for the points which are not smooth: They are exactly the points where the Jacobian matrix ...
In mathematics, the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ. Let (, ...