Let be Cartesian coordinates for .
Then , related to by
are the polar coordinates for . It is simply written .
The polar coordinates of Cartesian coordinates are
where is defined here (http://planetmath.org/OperatornamearcTanWithTwoArguments).
Polar basis. Polar coordinates are equipped with an orthonormal base , which can be defined in terms of the standard cartesian base in as follows.
Position vector. For an arbitrary point of polar coordinates , its position vector comes given by the single equation
Thus, we have the following identification:
Multiplications of polar coordinates have some simple geometric interpretations. For example, if and , then given by is the rotation of by angle . If , then can be viewed as the scaling of along the ray by . Note also that multiplication by has the same effect as multiplication by the scalar .
For more on polar coordinates, including their construction and extensions on domain of polar coordinates and , see here (http://planetmath.org/ConstructionOfPolarCoordinates).
Calculus in polar coordiantes. For reference, here are some formulae for computing integrals and derivatives in polar coordinates. The Jacobian for transforming from rectangular to polar cordinates is
so we may compute the integral of a scalar field as
Partial derivative operators transform as follows:
|Date of creation||2013-03-22 15:12:16|
|Last modified on||2013-03-22 15:12:16|
|Last modified by||CWoo (3771)|