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When m = n, the Jacobian matrix is square, so its determinant is a well-defined function of x, known as the Jacobian determinant of f. It carries important information about the local behavior of f .
The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose ...
In the theory of many-particle systems, Jacobi coordinates often are used to simplify the mathematical formulation. These coordinates are particularly common in treating polyatomic molecules and chemical reactions, [3] and in celestial mechanics. [4] An algorithm for generating the Jacobi coordinates for N bodies may be based upon binary trees. [5]
Thus the differential is a linear transformation, between tangent spaces, associated to the smooth map at each point. Therefore, in some chosen local coordinates, it is represented by the Jacobian matrix of the corresponding smooth map from R m {\displaystyle \mathbb {R} ^{m}} to R n {\displaystyle \mathbb {R} ^{n}} .
Here det g is the determinant of the matrix formed by the components of the metric tensor in the coordinate chart. That Λ is well-defined on functions supported in coordinate neighborhoods is justified by Jacobian change of variables.
Each entry of this matrix represents a partial derivative, specifying the rate of change of one range coordinate with respect to a change in a domain coordinate. Of course, the Jacobian matrix of the composition g ° f is a product of corresponding Jacobian matrices: J x (g ° f) =J ƒ(x) (g)J x (ƒ).
In each local chart a Riemannian metric is given by smoothly assigning a 2×2 positive definite matrix to each point; when a different chart is taken, the matrix is transformed according to the Jacobian matrix of the coordinate change. The manifold then has the structure of a 2-dimensional Riemannian manifold.
The transformation from a reference frame 1 to a reference frame 2 can be described with three translations Δx, Δy, Δz, three rotations Rx, Ry, Rz and a scale parameter μ. The Helmert transformation (named after Friedrich Robert Helmert, 1843–1917) is a geometric transformation method within a three-dimensional space.