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In vector calculus, the Jacobian matrix (/ dʒ ə ˈ k oʊ b i ə n /, [1] [2] [3] / dʒ ɪ-, j ɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives.
The polynomial x − x p has derivative 1 − p x p−1 which is 1 (because px is 0) but it has no inverse function. However, Kossivi Adjamagbo suggested extending the Jacobian conjecture to characteristic p > 0 by adding the hypothesis that p does not divide the degree of the field extension k(X) / k(F). [3]
The first Jacobian rotation will be on the off-diagonal cell with the highest absolute value, which by inspection is [1,4] with a value of 11, and the rotation cell will also be [1,4], =, = in the equations above. The rotation angle is the result of a quadratic solution, but it can be seen in the equation that if the matrix is symmetric, then a ...
Newton's method for solving f(x) = 0 uses the Jacobian matrix, J, at every iteration. However, computing this Jacobian can be a difficult and expensive operation; for large problems such as those involving solving the Kohn–Sham equations in quantum mechanics the number of variables can be in the hundreds of thousands.
If we condense the skew entries into a vector, (x,y,z), then we produce a 90° rotation around the x-axis for (1, 0, 0), around the y-axis for (0, 1, 0), and around the z-axis for (0, 0, 1). The 180° rotations are just out of reach; for, in the limit as x → ∞ , ( x , 0, 0) does approach a 180° rotation around the x axis, and similarly for ...
The same terminology applies. A regular solution is a solution at which the Jacobian is full rank (). A singular solution is a solution at which the Jacobian is less than full rank. A regular solution lies on a k-dimensional surface, which can be parameterized by a point in the tangent space (the null space of the Jacobian).
If the rank of this matrix is n − 1, then the polynomials define an algebraic curve (otherwise they define an algebraic variety of higher dimension). If the rank remains n − 1 when the Jacobian matrix is evaluated at a point P on the curve, then the point is a smooth or regular point; otherwise it is a singular point.
For m = 0 the generalized Jacobian J m is just the usual Jacobian J, an abelian variety of dimension g, the genus of C. For m a nonzero effective divisor the generalized Jacobian is an extension of J by a connected commutative affine algebraic group L m of dimension deg(m)−1. So we have an exact sequence 0 → L m → J m → J → 0. The ...