coordinate systems - math 212bfitzpat/teaching/... · the polar coordinate map is r r 0 0 fp(r; )=...

Coordinate SystemsMath 212

Brian D. Fitzpatrick

Duke University

January 24, 2020

MATH

Overview

Rectangular CoordinatesDefinitionExamples

Polar CoordinatesDefinitionExamples

Cylindrical CoordinatesDefinitionExamples

Spherical CoordinatesDefinitionExamples

Rectangular CoordinatesDefinition

ConventionWe often measure location with rectangular coordinates.

x

y

x0

y0

(x0, y0)

x

y

z

x0

y0

z0

(x0, y0, z0)


ProblemRectangular coordinates conveniently describe lines and planes.

x+y=1

x+y+z=1

They less conveniently describe other interesting objects.

x2+y2=1 x2+y2+z2=1



x+y=1

x+y+z=1


x2+y2=1

x2+y2+z2=1



x+y=1

x+y+z=1


x2+y2=1 x2+y2+z2=1

Rectangular CoordinatesExamples

Example

Consider the “disk” in R2 described by x2 + y2 ≤ 4.

x

y

The disk is described by

−√

4− x2 ≤ y ≤√

4− x2 − 2 ≤ x ≤ 2


Example


x

y


−√

4− x2 ≤ y ≤√

4− x2

− 2 ≤ x ≤ 2


Example


x

y


−√

4− x2 ≤ y ≤√

4− x2 − 2 ≤ x ≤ 2


Example

Consider the region in R3 between these “stacked” paraboloids.

z

z =x2 + y2

2+ 2

z = x2 + y2

z = 4

This region is described by the inequalities

x2 + y2 ≤ z ≤ x2 + y2

2+ 2 −

√4− x2 ≤ y ≤

√4− x2 − 2 ≤ x ≤ x


Example


z

z =x2 + y2

2+ 2

z = x2 + y2

z = 4


x2 + y2 ≤ z ≤ x2 + y2

2+ 2

−√

4− x2 ≤ y ≤√

4− x2 − 2 ≤ x ≤ x


Example


z

z =x2 + y2

2+ 2

z = x2 + y2

z = 4


x2 + y2 ≤ z ≤ x2 + y2

2+ 2 −

√4− x2 ≤ y ≤

√4− x2

− 2 ≤ x ≤ x


Example


z

z =x2 + y2

2+ 2

z = x2 + y2

z = 4


x2 + y2 ≤ z ≤ x2 + y2

2+ 2 −

√4− x2 ≤ y ≤

√4− x2 − 2 ≤ x ≤ x


QuestionCan we describe interesting regions more easily?

Polar CoordinatesDefinition

ObservationLocation in R2 can be measured with distance and angle.

θ

r =

√ x2 +

y2

x =

r · cos(θ)

y =

r · sin(θ)

x

y



θ

r =

√ x2 +

y2

x = r · cos(θ)

y =

r · sin(θ)

x

y



θ

r =

√ x2 +

y2

x = r · cos(θ)

y = r · sin(θ)

x

y


DefinitionThe polar coordinate map is

r

θ

r0

θ0

F p(r ,θ)=

r ·cos(θ)r ·sin(θ)

−−−−−−−−−−−−→ x

y

The polar change of coordinates formulas are

x = r · cos(θ) r2 = x2 + y2

y = r · sin(θ) tan(θ) = y/x



r

θ

r0

θ0 F p(r ,θ)=


−−−−−−−−−−−−→ x

y


x = r · cos(θ) r2 = x2 + y2




r

θ

r0

θ0 F p(r ,θ)=


−−−−−−−−−−−−→ x

y


x = r · cos(θ)

r2 = x2 + y2




r

θ

r0

θ0 F p(r ,θ)=


−−−−−−−−−−−−→ x

y


x = r · cos(θ)

r2 = x2 + y2

y = r · sin(θ)

tan(θ) = y/x



r

θ

r0

θ0 F p(r ,θ)=


−−−−−−−−−−−−→ x

y


x = r · cos(θ) r2 = x2 + y2

y = r · sin(θ)

tan(θ) = y/x



r

θ

r0

θ0 F p(r ,θ)=


−−−−−−−−−−−−→ x

y


x = r · cos(θ) r2 = x2 + y2


Polar CoordinatesExamples

Example

The equation r = 3 defines a circle with radius three.

r

θ

3

F p(r ,θ)−−−−→ x

y

3

This follows from the equation

x2 + y2 = r2 = 32


Example


r

θ

3

F p(r ,θ)−−−−→ x

y

3


x2 + y2 =

r2 = 32


Example


r

θ

3

F p(r ,θ)−−−−→ x

y

3


x2 + y2 = r2 =

32


Example


r

θ

3

F p(r ,θ)−−−−→ x

y

3


x2 + y2 = r2 = 32


Example

The equation θ = π/4 defines a line.

r

θ

π/4

F p(r ,θ)−−−−→

π/4

x

y


y

x= tan(π/4) = 1

which gives y =

x .


Example


r

θ

π/4

F p(r ,θ)−−−−→ π/4x

y


y

x= tan(π/4) = 1

which gives y =

x .


Example


r

θ

π/4

F p(r ,θ)−−−−→ π/4x

y


y

x=

tan(π/4) = 1

which gives y =

x .


Example


r

θ

π/4

F p(r ,θ)−−−−→ π/4x

y


y

x= tan(π/4) =

1

which gives y =

x .


Example


r

θ

π/4

F p(r ,θ)−−−−→ π/4x

y


y

x= tan(π/4) = 1

which gives y =

x .


Example


r

θ

π/4

F p(r ,θ)−−−−→ π/4x

y


y

x= tan(π/4) = 1

which gives y = x .


Example

Consider the polar equation r

2

=

r

cos(θ)

x2 − x + y2 = 0

.

This gives

(x − 1

2

)2

− 1

4+ y2 = 0 x

y

1/2 1

(x−1/2)2+y2=(1/2)2

This circle lives in the region where −π/2 ≤ θ ≤ π/2

.


Example

Consider the polar equation r2 = r cos(θ)

x2 − x + y2 = 0

.

This gives

(x − 1

2

)2

− 1

4+ y2 = 0 x

y

1/2 1

(x−1/2)2+y2=(1/2)2


.


Example


x2 − x + y2 = 0

. This gives

(x − 1

2

)2

− 1

4+ y2 = 0

x

y

1/2 1

(x−1/2)2+y2=(1/2)2


.


Example


x2 − x + y2 = 0

. This gives

(x − 1

2

)2

− 1

4+ y2 = 0 x

y

1/2 1

(x−1/2)2+y2=(1/2)2


.


Example


x2 − x + y2 = 0

. This gives

(x − 1

2

)2

− 1

4+ y2 = 0 x

y

1/2 1

(x−1/2)2+y2=(1/2)2

This circle lives in the region where −π/2 ≤ θ ≤ π/2.


Example

Consider the “disk” D ⊂ R2 described by x2 + y2 ≤ 4.

x

y

In polar coordinates, D is described by

0 ≤ r ≤ 2 0 ≤ θ ≤ 2π


Example


x

y


0 ≤ r ≤ 2

0 ≤ θ ≤ 2π


Example


x

y


0 ≤ r ≤ 2 0 ≤ θ ≤ 2π


Example

Consider the following region in R2.

x

y

x2 + y2 = 1 (x − 1)2 + y2 = 1

r = 1 r = 2 · cos(θ)

θ = π/3

θ = −π/3

In polar coordinates, the region is described by

1 ≤ r ≤ 2 · cos(θ) − π

3≤ θ ≤ π

3


Example


x

y

x2 + y2 = 1

(x − 1)2 + y2 = 1

r = 1 r = 2 · cos(θ)

θ = π/3

θ = −π/3


1 ≤ r ≤ 2 · cos(θ) − π

3≤ θ ≤ π

3


Example


x

y

x2 + y2 = 1 (x − 1)2 + y2 = 1

r = 1 r = 2 · cos(θ)

θ = π/3

θ = −π/3


1 ≤ r ≤ 2 · cos(θ) − π

3≤ θ ≤ π

3


Example


x

y

x2 + y2 = 1 (x − 1)2 + y2 = 1

r = 1

r = 2 · cos(θ)

θ = π/3

θ = −π/3


1 ≤ r ≤ 2 · cos(θ) − π

3≤ θ ≤ π

3


Example


x

y

x2 + y2 = 1 (x − 1)2 + y2 = 1

r = 1 r = 2 · cos(θ)

θ = π/3

θ = −π/3


1 ≤ r ≤ 2 · cos(θ) − π

3≤ θ ≤ π

3


Example


x

y

x2 + y2 = 1 (x − 1)2 + y2 = 1

r = 1 r = 2 · cos(θ)

θ = π/3

θ = −π/3


1 ≤ r ≤ 2 · cos(θ)

− π

3≤ θ ≤ π

3


Example


x

y

x2 + y2 = 1 (x − 1)2 + y2 = 1

r = 1 r = 2 · cos(θ)

θ = π/3

θ = −π/3


1 ≤ r ≤ 2 · cos(θ) − π

3≤ θ ≤ π

3


Warning

Polar coordinates are not unique.

x

y

θ

r

(−1/√

2,−1/√

2)

r θ

1 5π/4

− 1 π/41 − 3π/4− 1 9π/4

To acheive unique polar representations, we often use

0 ≤ θ < 2π 0 ≤ r


Warning


x

y

θ

r

(−1/√

2,−1/√2)

r θ

1 5π/4

− 1 π/41 − 3π/4− 1 9π/4


0 ≤ θ < 2π 0 ≤ r


Warning


x

y

θ

r

(−1/√

2,−1/√2)

r θ

1 5π/4− 1 π/4

1 − 3π/4− 1 9π/4


0 ≤ θ < 2π 0 ≤ r


Warning


x

y

θ

r

(−1/√

2,−1/√2)

r θ

1 5π/4− 1 π/4

1 − 3π/4

− 1 9π/4


0 ≤ θ < 2π 0 ≤ r


Warning


x

y

θ

r

(−1/√

2,−1/√2)

r θ

1 5π/4− 1 π/4

1 − 3π/4− 1 9π/4


0 ≤ θ < 2π 0 ≤ r


Warning


x

y

θ

r

(−1/√

2,−1/√2)

r θ

1 5π/4− 1 π/4

1 − 3π/4− 1 9π/4


0 ≤ θ < 2π

0 ≤ r


Warning


x

y

θ

r

(−1/√

2,−1/√2)

r θ

1 5π/4− 1 π/4

1 − 3π/4− 1 9π/4


0 ≤ θ < 2π 0 ≤ r

Cylindrical CoordinatesDefinition

ObservationLocation in R3 can be measured with polar coordinates.

x

y

z

θ

z

r

Cylindrical CoordinatesDefinition

DefinitionThe cylindrical coordinate map is

r

θ

z

F c (r ,θ,z)=


z

−−−−−−−−−−−−−→

x

y

z

The cylindrical change of coordinates formulas are

x = r · cos(θ) r2 = x2 + y2


Cylindrical CoordinatesExamples

Example

Quadric surfaces are easily described with cylindrical coordinates.

r2+z2=1 r2−z=0 r2−z2=1

r2−z2=−1r2−z2=0 r=1


Example


r2+z2=1

r2−z=0 r2−z2=1

r2−z2=−1r2−z2=0 r=1


Example


r2+z2=1 r2−z=0

r2−z2=1

r2−z2=−1r2−z2=0 r=1


Example


r2+z2=1 r2−z=0 r2−z2=1

r2−z2=−1r2−z2=0 r=1


Example


r2+z2=1 r2−z=0 r2−z2=1

r2−z2=−1

r2−z2=0 r=1


Example


r2+z2=1 r2−z=0 r2−z2=1

r2−z2=−1r2−z2=0

r=1


Example


r2+z2=1 r2−z=0 r2−z2=1

r2−z2=−1r2−z2=0 r=1


Example

Consider again the region between these “stacked” paraboloids.

z

z =x2 + y2

2+ 2

z = x2 + y2

z = 4

z =r2

2+ 2

z = r2

In cylindrical coordinates, this region is given by

r2 ≤ z ≤ r2

2+ 2 0 ≤ r ≤ 2 0 ≤ θ ≤ 2π


Example


z

z =x2 + y2

2+ 2

z = x2 + y2

z = 4

z =r2

2+ 2

z = r2


r2 ≤ z ≤ r2

2+ 2

0 ≤ r ≤ 2 0 ≤ θ ≤ 2π


Example


z

z =x2 + y2

2+ 2

z = x2 + y2

z = 4

z =r2

2+ 2

z = r2


r2 ≤ z ≤ r2

2+ 2 0 ≤ r ≤ 2

0 ≤ θ ≤ 2π


Example


z

z =x2 + y2

2+ 2

z = x2 + y2

z = 4

z =r2

2+ 2

z = r2


r2 ≤ z ≤ r2

2+ 2 0 ≤ r ≤ 2 0 ≤ θ ≤ 2π


ConventionTo acheive unique cylindrical representations, we often use

0 ≤ θ < 2π 0 ≤ r −∞ ≤ z ≤ ∞

Spherical CoordinatesDefinition

ObservationLocation in R3 can be measured using multiple angles.

x

y

z

θ

ϕ

ρ


ObservationLocation in R3 can be measured using multiple angles.

x

y

z

θ

ϕρ


DefinitionThe spherical coordinate map is

ρ

ϕ

θF s(ρ,ϕ,θ)=

ρ·sin(ϕ)·cos(θ)ρ·sin(ϕ)·sin(θ)ρ·cos(ϕ)

−−−−−−−−−−−−−−−−−→

x

y

z

The spherical change of coordinates formulas are

x = ρ · sin(ϕ) · cos(θ) ρ2 = x2 + y2 + z2

y = ρ · sin(ϕ) · sin(θ) tan(ϕ) =√

x2 + y 2/z

z = ρ · cos(ϕ) tan(θ) = y/x



ρ

ϕ

θF s(ρ,ϕ,θ)=


−−−−−−−−−−−−−−−−−→

x

y

z


x = ρ · sin(ϕ) · cos(θ)

ρ2 = x2 + y2 + z2


x2 + y 2/z




ρ

ϕ

θF s(ρ,ϕ,θ)=


−−−−−−−−−−−−−−−−−→

x

y

z



ρ2 = x2 + y2 + z2

y = ρ · sin(ϕ) · sin(θ)

tan(ϕ) =√

x2 + y 2/z




ρ

ϕ

θF s(ρ,ϕ,θ)=


−−−−−−−−−−−−−−−−−→

x

y

z



ρ2 = x2 + y2 + z2


tan(ϕ) =√

x2 + y 2/z

z = ρ · cos(ϕ)

tan(θ) = y/x



ρ

ϕ

θF s(ρ,ϕ,θ)=


−−−−−−−−−−−−−−−−−→

x

y

z


x = ρ · sin(ϕ) · cos(θ) ρ2 = x2 + y2 + z2


tan(ϕ) =√

x2 + y 2/z

z = ρ · cos(ϕ)

tan(θ) = y/x



ρ

ϕ

θF s(ρ,ϕ,θ)=


−−−−−−−−−−−−−−−−−→

x

y

z


x = ρ · sin(ϕ) · cos(θ) ρ2 = x2 + y2 + z2


x2 + y 2/z

z = ρ · cos(ϕ)

tan(θ) = y/x



ρ

ϕ

θF s(ρ,ϕ,θ)=


−−−−−−−−−−−−−−−−−→

x

y

z


x = ρ · sin(ϕ) · cos(θ) ρ2 = x2 + y2 + z2


x2 + y 2/z


Spherical CoordinatesExamples

Example

Spherical coordinates conveniently describe spheres.

ρ = 1


Example

Consider the region “above” z2 = x2 + y2 and “below” z = 1.

z

z2 = x2 + y2

z = 1

ϕ = π/4

ρ · cos(ϕ) = 1

In spherical coordinates, the region is described by

0 ≤ θ ≤ 2π 0 ≤ ϕ ≤ π

40 ≤ ρ ≤ 1

cos(ϕ)


Example


z

z2 = x2 + y2

z = 1

ϕ = π/4

ρ · cos(ϕ) = 1


0 ≤ θ ≤ 2π

0 ≤ ϕ ≤ π

40 ≤ ρ ≤ 1

cos(ϕ)


Example


z

z2 = x2 + y2

z = 1

ϕ = π/4

ρ · cos(ϕ) = 1


0 ≤ θ ≤ 2π 0 ≤ ϕ ≤ π

4

0 ≤ ρ ≤ 1

cos(ϕ)


Example


z

z2 = x2 + y2

z = 1

ϕ = π/4

ρ · cos(ϕ) = 1


0 ≤ θ ≤ 2π 0 ≤ ϕ ≤ π

40 ≤ ρ ≤ 1

cos(ϕ)


ConventionTo acheive unique spherical representations, we often use

0 ≤ θ < 2π 0 ≤ ϕ ≤ π 0 ≤ ρ

coordinate systems - math 212bfitzpat/teaching/... · the polar coordinate map is r r 0 0 fp(r; )=...

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