Polar coordinates

The numbers $\rho$ and $\phi$ related to rectangular Cartesian coordinates $x$ and $y$ by the formulas:

$$ x = \rho \cos \phi ,\ \ y = \rho \sin \phi , $$

where $ 0 \leq \rho < \infty $, $ 0 \leq \phi < 2 \pi $. The coordinate lines are: concentric circles ($\rho = \textrm{ const } $) and rays ( $ \phi = \textrm{ const } $).

Figure: p073410a

The system of polar coordinates is an orthogonal system. To each point in the $ Oxy $- plane (except the point $ O $ for which $ \rho = 0 $ and $ \phi $ is undefined, i.e. can be any number $ 0 \leq \phi < 2 \pi $) corresponds a pair of numbers $ ( \rho , \phi ) $ and vice versa. The distance $ \rho $ between a point $ P $ and $ ( 0 , 0 ) $( the pole) is called the polar radius, and the angle $ \phi $ is called the polar angle. The Lamé coefficients (scale factors) are:

$$ L _ \rho = 1 ,\ L _ \phi = \rho . $$

The surface element is:

$$ d \sigma = \rho d \rho d \phi . $$

The fundamental operations of vector analysis are:

$$ \mathop{\rm grad} _ \rho f = \frac{\partial f }{\partial \rho } ,\ \ \mathop{\rm grad} _ \phi f = \frac{1} \rho \frac{\partial f }{\partial \phi } ; $$

$$ \mathop{\rm div} \mathbf a = \frac{1} \rho a _ \rho + \frac{\partial a _ \rho }{\partial \rho } + \frac{1} \rho \frac{\partial a _ \phi }{\partial \phi } ,\ \mathbf a = ( a _ \rho , a _ \phi ) ; $$

$$ \Delta f = \frac{1} \rho \frac \partial {\partial \rho } \left ( \rho \frac{\partial f }{\partial \rho } \right ) + \frac{1}{\rho ^ {2} } \frac{ \partial ^ {2} f }{\partial \phi ^ {2} } = \frac{\partial ^ {2} f }{\partial \rho ^ {2} } + \frac{1} \rho \frac{\partial f }{\partial \rho } + \frac{1}{\rho ^ {2} } \frac{\partial ^ {2} f }{\partial \phi ^ {2} } . $$

The numbers $ r $ and $ \psi $ related to Cartesian rectangular coordinates $ x $ and $ y $ by the formulas:

$$ x = a r \cos \psi ,\ \ y = b r \sin \psi , $$

where $ 0 \leq r < \infty $, $ 0 \leq \psi < 2 \pi $, $ a, b > 0 $, $ a \neq b $, are called generalized polar coordinates. The coordinate lines are: ellipses ( $ r = \textrm{ const } $) and rays ( $ \psi = \textrm{ const } $).

Comments

The generalization of polar coordinates to 3 dimensions are the spherical coordinates.

By viewing a point $ ( x, y) $ as a complex number $ z = x+ iy $, the polar coordinates $ ( \rho , \phi ) $ correspond to the representation of $ z $ as $ z = \rho e ^ {i \phi } $.

See also Complex number.

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References

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[1]	G.A. Korn, T.M. Korn, "Mathematical handbook for scientists and engineers" , McGraw-Hill (1961)
[a1]	H. Triebel, "Analysis and mathematical physics" , Reidel (1986) pp. 103
[a2]	K. Rektorys (ed.) , Applicable mathematics , Iliffe (1969) pp. 216