What does it signifies if jacobian is function of material coordinate?
Answers
Answered by
0
In vector calculus, the Jacobian matrix(/dʒəˈkoʊbiən/,[1][2][3] /dʒɪ-, jɪ-/) is the matrixof all first-order partial derivatives of a vector-valued function. When the matrix is a square matrix, both the matrix and its determinantare referred to as the Jacobian in literature.[4]
Suppose f : ℝn → ℝm is a function which takes as input the vector x ∈ ℝn and produces as output the vector f(x) ∈ ℝm. Then the Jacobian matrix J of f is an m×n matrix, usually defined and arranged as follows:
{\displaystyle \mathbf {J} ={\begin{bmatrix}{\dfrac {\partial \mathbf {f} }{\partial x_{1}}}&\cdots &{\dfrac {\partial \mathbf {f} }{\partial x_{n}}}\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial f_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{m}}{\partial x_{n}}}\end{bmatrix}}}
or, component-wise:
{\displaystyle \mathbf {J} _{ij}={\frac {\partial f_{i}}{\partial x_{j}}}.}
This matrix, whose entries are functions of x, is also denoted by Df, Jf, and ∂(f1,...,fm)/∂(x1,...,xn). (Note that some literature defines the Jacobian as the transpose of the matrix given above.)
The Jacobian matrix is important because if the function f is differentiable at a point x(this is a slightly stronger condition than merely requiring that all partial derivatives exist there), then the Jacobian matrix defines a linear map ℝn → ℝm, which is the best (pointwise) linear approximation of the function f near the point x. This linear map is thus the generalization of the usual notion of derivative, and is called the derivative or the differential of f at x.
If m = n, the Jacobian matrix is a square matrix, and its determinant, a function of x1, …, xn, is the Jacobian determinant of f. It carries important information about the local behavior of f. In particular, the function f has locally in the neighborhood of a point x an inverse function that is differentiable if and only if the Jacobian determinant is nonzero at x (see Jacobian conjecture). The Jacobian determinant also appears when changing the variables in multiple integrals (see substitution rule for multiple variables).
If m = 1, f is a scalar field and the Jacobian matrix is reduced to a row vector of partial derivatives of f—i.e. the transpose of the gradient of f.
These concepts are named after the mathematician
Suppose f : ℝn → ℝm is a function which takes as input the vector x ∈ ℝn and produces as output the vector f(x) ∈ ℝm. Then the Jacobian matrix J of f is an m×n matrix, usually defined and arranged as follows:
{\displaystyle \mathbf {J} ={\begin{bmatrix}{\dfrac {\partial \mathbf {f} }{\partial x_{1}}}&\cdots &{\dfrac {\partial \mathbf {f} }{\partial x_{n}}}\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial f_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{m}}{\partial x_{n}}}\end{bmatrix}}}
or, component-wise:
{\displaystyle \mathbf {J} _{ij}={\frac {\partial f_{i}}{\partial x_{j}}}.}
This matrix, whose entries are functions of x, is also denoted by Df, Jf, and ∂(f1,...,fm)/∂(x1,...,xn). (Note that some literature defines the Jacobian as the transpose of the matrix given above.)
The Jacobian matrix is important because if the function f is differentiable at a point x(this is a slightly stronger condition than merely requiring that all partial derivatives exist there), then the Jacobian matrix defines a linear map ℝn → ℝm, which is the best (pointwise) linear approximation of the function f near the point x. This linear map is thus the generalization of the usual notion of derivative, and is called the derivative or the differential of f at x.
If m = n, the Jacobian matrix is a square matrix, and its determinant, a function of x1, …, xn, is the Jacobian determinant of f. It carries important information about the local behavior of f. In particular, the function f has locally in the neighborhood of a point x an inverse function that is differentiable if and only if the Jacobian determinant is nonzero at x (see Jacobian conjecture). The Jacobian determinant also appears when changing the variables in multiple integrals (see substitution rule for multiple variables).
If m = 1, f is a scalar field and the Jacobian matrix is reduced to a row vector of partial derivatives of f—i.e. the transpose of the gradient of f.
These concepts are named after the mathematician
Similar questions