polar coordinates

polar coordinates

Let

x

,

y

be

Cartesian coordinates

for

ℝ

2

.

Then

r

≥

0

,

θ

∈

[

0

,

2

⁢

π

)

related to

(

x

,

y

)

by

x

⁢

(

r

,

θ

)

=

r

⁢

cos

⁡

θ

,

y

⁢

(

r

,

θ

)

=

r

⁢

sin

⁡

θ

,

are the

polar coordinates

for

(

x

,

y

)

. It is simply written

(

r

,

θ

)

.

The polar coordinates of Cartesian coordinates

(

x

,

y

)

∈

ℝ

2

∖

{

0

}

are

r

⁢

(

x

,

y

)

=

x

2

+

y

2

,

θ

⁢

(

x

,

y

)

=

arctan

⁡

(

x

,

y

)

,

where

arctan

is defined here (

http://planetmath.org/OperatornamearcTanWithTwoArguments

).

Polar basis.

Polar coordinates are equipped with an orthonormal base

{

𝐞

𝐫

,

𝐞

θ

}

, which can be defined in terms of the standard cartesian base

{

𝐢

,

𝐣

}

in

ℝ

2

as follows.

[

𝐞

𝐫

𝐞

θ

]

=

[

cos

⁡

θ

⁢

𝐢

+

sin

⁡

θ

⁢

𝐣

-

sin

⁡

θ

⁢

𝐢

+

cos

⁡

θ

⁢

𝐣

]

,

where

𝐞

𝐫

,

𝐞

θ

are so-called

radial

and

traverse

or

angular

vector, respectively. Since these vectors are variable in

direction

, they are

differentiable

. In fact,

[

d

⁢

𝐞

𝐫

d

⁢

θ

d

⁢

𝐞

θ

d

⁢

θ

]

=

[

𝐞

θ

-

𝐞

𝐫

]

.

The geometrical action of the

derivative

operator

d

/

d

⁢

θ

is like a

rotation operator

that rotates each base vector a counter-clockwise angle equals to

π

/

2

.

Position vector

.

For an arbitrary point of polar coordinates

(

r

,

θ

)

, its position vector comes given by the single equation

𝐫

=

r

⁢

𝐞

𝐫

.

Relations

with

complex numbers

.

When the

Euclidean plane

ℝ

2

is identified with

ℂ

by

(

x

,

y

)

↔

x

+

y

⁢

i

,

it is possible to define

multiplications

on

ℝ

2

.  Via polar coordinates, the

formula

for this multiplication becomes very simple, thanks to Euler’s formula (

http://planetmath.org/EulerRelation

)

cos

⁡

θ

+

i

⁢

sin

⁡

θ

=

e

i

⁢

θ

.

Thus, we have the following identification:

(

r

,

θ

)

↔

(

x

,

y

)

↔

x

+

y

⁢

i

=

r

⁢

cos

⁡

θ

+

(

r

⁢

sin

⁡

θ

)

⁢

i

=

r

⁢

e

i

⁢

θ

.

If

P

=

(

r

1

,

θ

1

)

and

Q

=

(

r

2

,

θ

2

)

, the

product

of

P

and

Q

works out to be

(

r

1

⁢

r

2

,

θ

1

+

θ

2

)

.
(Even if one is not familiar with the complex

exponential

, this assertion may be checked
directly using the

angle sum identities

for

cos

and

sin

.)

Multiplications of polar coordinates have some simple geometric

interpretations

. For example, if

R

=

(

1

,

α

)

and

Q

=

(

r

,

β

)

,
then

Q

→

R

⁢

Q

given by

(

1

,

α

)

⁢

(

r

,

β

)

=

(

r

,

α

+

β

)

is the

rotation

of

Q

by
angle

α

. If

S

=

(

t

,

0

)

, then

(

t

,

0

)

⁢

(

r

,

β

)

=

(

t

⁢

r

,

β

)

can
be viewed as the

scaling

of

Q

along the ray

O

⁢

Q

→

by

t

. Note also that multiplication by

(

t

,

0

)

has the same effect
as multiplication by the scalar

t

.

For more on polar coordinates, including their construction and

extensions

on domain of polar coordinates

r

and

θ

, 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

∂

⁡

(

x

,

y

)

∂

⁡

(

r

,

θ

)

=

r

so we may compute the integral of a scalar field

f

as

∫

f

⁢

(

r

,

θ

)

⁢

r

⁢

𝑑

r

⁢

𝑑

θ

.

Partial derivative

operators transform as follows:

∂

∂

⁡

x

=

cos

⁡

θ

⁢

∂

∂

⁡

r

-

1

r

⁢

sin

⁡

θ

⁢

∂

∂

⁡

θ

∂

∂

⁡

y

=

sin

⁡

θ

⁢

∂

∂

⁡

r

+

1

r

⁢

cos

⁡

θ

⁢

∂

∂

⁡

θ

∂

∂

⁡

r

=

cos

⁡

θ

⁢

∂

∂

⁡

x

+

sin

⁡

θ

⁢

∂

∂

⁡

y

∂

∂

⁡

θ

=

-

r

⁢

sin

⁡

θ

⁢

∂

∂

⁡

x

+

r

⁢

cos

⁡

θ

⁢

∂

∂

⁡

y

Title

polar coordinates

Canonical name

PolarCoordinates

Date of creation

2013-03-22 15:12:16

Last modified on

2013-03-22 15:12:16

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

14

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51-01

Related topic

DerivationOfRotationMatrixUsingPolarCoordinates

Related topic

CylindricalCoordinates

Related topic

ArgumentOfProductAndQuotient

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