The geometrical meaning of the value of the determinant of the matrix: is the area ratio before the corresponding linear transformation matrix.
Concept: In vector analysis, is a Jacobian matrix of partial derivatives in a manner arranged in a matrix, which is referred to as Jacobian determinant
The importance of the Jacobian matrix is that it embodies a micro-optimal linear approximation equation and the given point. Thus, the derivative of the Jacobian matrix of the function similar polyol.
In summary, the Jacobian matrix can be understood as:
If a vector in the n-dimensional Euclidean space is mapped to the corresponding rule of another vector m dimensional Euclidean space is F, F from the real function of m, namely:
Then the Jacobian matrix is a m × n matrix:
Wherein the input vector x = (x1, ..., xn), the output vector y = (y1, ..., ym),
If p is 
is at this point the number of guide
In this case, the linear mapping F that is in the vicinity of the point p optimal linear approximation, that is to say when x is close enough to the point p, we have
When m = n, the matrix becomes a square, F becomes a mapping from the n-dimensional Euclidean space to the n-dimensional Euclidean space, square Jacobian determinant is
The type transpose get:
which is:
And 
is a vector, the vector n-dimensional space, then the differential, then, is the equivalent of:
among them:
, 
which is:
The above-described vector written in the form of an orthogonal unit vectors:
Represents the absolute value of the determinant of the volume of infinitesimal left determinant must be positive, the volume can not be negative, the absolute value of the right side plus determinant
The common factor is proposed
That is, the volume of this micro elements:
Jacobian
