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The transformation from polar coordinates (r, ... The Jacobian matrix for this coordinate change is ...
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 ...
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 ...
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
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.
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 .
The name is motivated by the importance of changes of coordinate in physics: the covariant derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Jacobian matrix of the transformation.
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]