Search results
Results From The WOW.Com Content Network
Furthermore, if the Jacobian determinant at p is positive, then f preserves orientation near p; if it is negative, f reverses orientation. The absolute value of the Jacobian determinant at p gives us the factor by which the function f expands or shrinks volumes near p ; this is why it occurs in the general substitution rule .
The strong real Jacobian conjecture was that a real polynomial map with a nowhere vanishing Jacobian determinant has a smooth global inverse. That is equivalent to asking whether such a map is topologically a proper map , in which case it is a covering map of a simply connected manifold , hence invertible.
Any non-linear differentiable function, (,), of two variables, and , can be expanded as + +. If we take the variance on both sides and use the formula [11] for the variance of a linear combination of variables (+) = + + (,), then we obtain | | + | | +, where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and = is the ...
If it is true, the Jacobian conjecture would be a variant of the inverse function theorem for polynomials. It states that if a vector-valued polynomial function has a Jacobian determinant that is an invertible polynomial (that is a nonzero constant), then it has an inverse that is also a polynomial function. It is unknown whether this is true ...
[9] [10] [11] Such examples are easy to create using homoclinic connections.) If the Jacobian of the dynamical system at an equilibrium happens to be a stability matrix (i.e., if the real part of each eigenvalue is strictly negative), then the equilibrium is asymptotically stable.
where is the Jacobian of the sensor measurement residual equations (_, _) =, with respect to the unknowns, _; is the Jacobian of the sensor measurement residual equations with respect to the measured quantities _, and is the correlation matrix for noise in the measured quantities.
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.
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.