Search results
Results From The WOW.Com Content Network
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.
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 ...
That is, the Jacobian of a composite function is the product of the Jacobians of the composed functions (evaluated at the appropriate points). The higher-dimensional chain rule is a generalization of the one-dimensional chain rule. If k, m, and n are 1, so that f : R → R and g : R → R, then the Jacobian matrices of f and g are 1 × 1.
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
Main page; Contents; Current events; Random article; About Wikipedia; Contact us; Help; Learn to edit; Community portal; Recent changes; Upload file
In the special case is defined by a homogeneous degree polynomial (+, ()) this Hodge structure can be understood completely from the Jacobian ideal. For its graded-pieces, this is given by the map [1] [, …,] ((+) (+)) (,) + (,) which is surjective on the primitive cohomology, denoted , and has the kernel .
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.
The unit circle can be specified as the level curve f(x, y) = 1 of the function f(x, y) = x 2 + y 2.Around point A, y can be expressed as a function y(x).In this example this function can be written explicitly as () =; in many cases no such explicit expression exists, but one can still refer to the implicit function y(x).