Jacobian (Jacobian) matrix

In vector analysis, the Jacobian matrix is a matrix of first-order partial derivatives arranged in a certain way . Its determinant is called the Jacobian determinant. Also, in algebraic geometry, the Jacobian proportion of an algebraic curve represents an Jacobian cluster: The curve can be embedded in an algebraic group accompanying the curve. All of them are named after the mathematician Carl Jacobi (Carl Jacob, October 4, 1804 – February 18, 1851).

First, Jacobian matrix

The importance of Jacobian matrix is that it embodies the optimal linear approximation of a micro-equation and a given point. Thus, the Jacobian matrix is similar to the derivative of a multivariate function.

Suppose f:rn→rm is a function that transforms from a European n-dimensional space to a European m-dimensional space. This function F consists of M real functions: y1 (x1,..., xn), ..., ym (X1,..., xn). The partial derivative of these functions (if present) can form a matrix of M row n columns, which is called the Jacobian matrix:

Expressed as:

If P is a point in Rn and F is differentiable at p Point, then the derivative at this point is given by the JF (p), which is the easiest way to find the derivative. In this case, the linear operator, described by F (P), is the optimal linear approximation of the f near the point P, and X approximates to P:

Ii. Examples

The conversion from the spherical coordinate system (spherical coordinate system) to the Cartesian coordinate system is given by the F function:

The Jacobian matrix for this coordinate transformation is: