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The Jacobian can be understood by considering a unit area in the new coordinate space; and examining how that unit area transforms when mapped into xy coordinate space in which the integral is visually understood.
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as , see conventions in spherical coordinates). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range ...
Difficult integrals may also be solved by simplifying the integral using a change of variables given by the corresponding Jacobian matrix and determinant. [1] Using the Jacobian determinant and the corresponding change of variable that it gives is the basis of coordinate systems such as polar, cylindrical, and spherical coordinate systems.
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]
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 ...
If the transformation is bijective then we call the image of the transformation, namely , a set of admissible coordinates for ´. If T {\displaystyle T} is linear the coordinate system Z i {\displaystyle Z^{i}} will be called an affine coordinate system , otherwise Z i {\displaystyle Z^{i}} is called a curvilinear coordinate system .
Jacobian matrix and determinant – Matrix of all first-order partial derivatives of a vector-valued function; List of canonical coordinate transformations; Sphere – Set of points equidistant from a center; Spherical harmonic – Special mathematical functions defined on the surface of a sphere
This is the same matrix as defines a Givens rotation, but for Jacobi rotations the choice of angle is different (very roughly half as large), since the rotation is applied on both sides simultaneously. It is not necessary to calculate the angle itself to apply the rotation. Using Kronecker delta notation, the matrix entries can be written: