Proved that if the Jacobian conjecture is false, then it has a counterexample with integer coefficients and Jacobian determinant 1.
12.
When the Jacobian determinant is positive, the transition function is said to be "'orientation preserving " '.
13.
The strong real Jacobian conjecture was that a real polynomial map with a nowhere vanishing Jacobian determinant has a smooth global inverse.
14.
The use of determinants in calculus includes the Jacobian determinant in the change of variables rule for integrals of functions of several variables.
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When f has a Jacobian determinant | J |, then g is usually taken to be g = 1 / | J |.
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In particular, the Jacobian determinant of a bi-Lipschitz mapping det D " ? " is well-defined almost everywhere.
17.
The Jacobian determinant is used when making a change of variables when evaluating a multiple integral of a function over a region within its domain.
18.
Under changes of coordinates, the volume element changes by the absolute value of the Jacobian determinant of the coordinate transformation ( by the 1-density.
19.
If I were to evaluate the whole thing using one set of substitutions, then the Jacobian determinant would have to be of order 6 ( ! ).
20.
If =, the Jacobian matrix is a square matrix, and its determinant, a function of, is the "'Jacobian determinant "'of.
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