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The Jacobian of the gradient of a scalar function of several variables has a special name: the Hessian matrix, which in a sense is the "second derivative" of the function in question. Jacobian determinant
The curl of the gradient of any continuously twice-differentiable scalar field (i.e., differentiability class) is always the zero vector: =. It can be easily proved by expressing ∇ × ( ∇ φ ) {\displaystyle \nabla \times (\nabla \varphi )} in a Cartesian coordinate system with Schwarz's theorem (also called Clairaut's theorem on equality ...
The Jacobian matrix is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. [ 9 ] [ 10 ] A further generalization for a function between Banach spaces is the Fréchet derivative .
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.
In mathematics, the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ. Let O ( x 1 , … , x n ) {\displaystyle {\mathcal {O}}(x_{1},\ldots ,x_{n})} denote the ring of smooth functions in n {\displaystyle n} variables and f {\displaystyle f} a function in the ring.
In vector calculus, the del operator is used to define the concepts of gradient, divergence, and curl in terms of partial derivatives. A matrix of partial derivatives, the Jacobian matrix, may be used to represent the derivative of a function between two spaces of arbitrary dimension.
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
The problem of computing a full Jacobian of f : R n → R m with a minimum number of arithmetic operations is known as the optimal Jacobian accumulation (OJA) problem, which is NP-complete. [20] Central to this proof is the idea that algebraic dependencies may exist between the local partials that label the edges of the graph.