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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)
Rectangular CoordinatesDefinition
ConventionWe often measure location with rectangular coordinates.
x
y
x0
y0
(x0, y0)
x
y
z
x0
y0
z0
(x0, y0, z0)
Rectangular CoordinatesDefinition
ConventionWe often measure location with rectangular coordinates.
x
y
x0
y0
(x0, y0)
x
y
z
x0
y0
z0
(x0, y0, z0)
Rectangular CoordinatesDefinition
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
Rectangular CoordinatesDefinition
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
Rectangular CoordinatesDefinition
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
Rectangular CoordinatesDefinition
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
Rectangular CoordinatesDefinition
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
Rectangular CoordinatesDefinition
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
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
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
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
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
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
Rectangular CoordinatesExamples
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
Rectangular CoordinatesExamples
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
Rectangular CoordinatesExamples
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
Rectangular CoordinatesExamples
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
Rectangular CoordinatesExamples
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
Rectangular CoordinatesExamples
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
Rectangular CoordinatesExamples
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
Rectangular CoordinatesExamples
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
Rectangular CoordinatesExamples
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
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
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
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
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
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
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
Polar CoordinatesDefinition
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
Polar CoordinatesDefinition
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
Polar CoordinatesDefinition
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
Polar CoordinatesDefinition
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
Polar CoordinatesDefinition
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
Polar CoordinatesDefinition
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
Polar CoordinatesDefinition
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
Polar CoordinatesDefinition
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
Polar CoordinatesDefinition
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
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
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
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
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
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
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
Polar CoordinatesExamples
Example
The equation θ = π/4 defines a line.
r
θ
π/4
F p(r ,θ)−−−−→
π/4
x
y
This follows from the equation
y
x= tan(π/4) = 1
which gives y =
x .
Polar CoordinatesExamples
Example
The equation θ = π/4 defines a line.
r
θ
π/4
F p(r ,θ)−−−−→
π/4
x
y
This follows from the equation
y
x= tan(π/4) = 1
which gives y =
x .
Polar CoordinatesExamples
Example
The equation θ = π/4 defines a line.
r
θ
π/4
F p(r ,θ)−−−−→ π/4x
y
This follows from the equation
y
x= tan(π/4) = 1
which gives y =
x .
Polar CoordinatesExamples
Example
The equation θ = π/4 defines a line.
r
θ
π/4
F p(r ,θ)−−−−→ π/4x
y
This follows from the equation
y
x=
tan(π/4) = 1
which gives y =
x .
Polar CoordinatesExamples
Example
The equation θ = π/4 defines a line.
r
θ
π/4
F p(r ,θ)−−−−→ π/4x
y
This follows from the equation
y
x= tan(π/4) =
1
which gives y =
x .
Polar CoordinatesExamples
Example
The equation θ = π/4 defines a line.
r
θ
π/4
F p(r ,θ)−−−−→ π/4x
y
This follows from the equation
y
x= tan(π/4) = 1
which gives y =
x .
Polar CoordinatesExamples
Example
The equation θ = π/4 defines a line.
r
θ
π/4
F p(r ,θ)−−−−→ π/4x
y
This follows from the equation
y
x= tan(π/4) = 1
which gives y =
x .
Polar CoordinatesExamples
Example
The equation θ = π/4 defines a line.
r
θ
π/4
F p(r ,θ)−−−−→ π/4x
y
This follows from the equation
y
x= tan(π/4) = 1
which gives y = x .
Polar CoordinatesExamples
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
.
Polar CoordinatesExamples
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
This circle lives in the region where −π/2 ≤ θ ≤ π/2
.
Polar CoordinatesExamples
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
This circle lives in the region where −π/2 ≤ θ ≤ π/2
.
Polar CoordinatesExamples
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
This circle lives in the region where −π/2 ≤ θ ≤ π/2
.
Polar CoordinatesExamples
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
This circle lives in the region where −π/2 ≤ θ ≤ π/2
.
Polar CoordinatesExamples
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
This circle lives in the region where −π/2 ≤ θ ≤ π/2.
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
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
Polar CoordinatesExamples
Warning
Polar coordinates are not unique.
x
y
θ
r
(−1/√
2,−1/√2)
r θ
1 5π/4− 1 π/4
1 − 3π/4− 1 9π/4
To acheive unique polar representations, we often use
0 ≤ θ < 2π 0 ≤ r
Polar CoordinatesExamples
Warning
Polar coordinates are not unique.
x
y
θ
r
(−1/√
2,−1/√2)
r θ
1 5π/4− 1 π/4
1 − 3π/4
− 1 9π/4
To acheive unique polar representations, we often use
0 ≤ θ < 2π 0 ≤ r
Polar CoordinatesExamples
Warning
Polar coordinates are not unique.
x
y
θ
r
(−1/√
2,−1/√2)
r θ
1 5π/4− 1 π/4
1 − 3π/4− 1 9π/4
To acheive unique polar representations, we often use
0 ≤ θ < 2π 0 ≤ r
Polar CoordinatesExamples
Warning
Polar coordinates are not unique.
x
y
θ
r
(−1/√
2,−1/√2)
r θ
1 5π/4− 1 π/4
1 − 3π/4− 1 9π/4
To acheive unique polar representations, we often use
0 ≤ θ < 2π
0 ≤ r
Polar CoordinatesExamples
Warning
Polar coordinates are not unique.
x
y
θ
r
(−1/√
2,−1/√2)
r θ
1 5π/4− 1 π/4
1 − 3π/4− 1 9π/4
To acheive unique polar representations, we often use
0 ≤ θ < 2π 0 ≤ r
Cylindrical CoordinatesDefinition
ObservationLocation in R3 can be measured with polar coordinates.
x
y
z
θ
z
r
Cylindrical CoordinatesDefinition
ObservationLocation in R3 can be measured with polar coordinates.
x
y
z
θ
z
r
Cylindrical CoordinatesDefinition
ObservationLocation in R3 can be measured with polar coordinates.
x
y
z
θ
z
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)=
r ·cos(θ)r ·sin(θ)
z
−−−−−−−−−−−−−→
x
y
z
The cylindrical change of coordinates formulas are
x = r · cos(θ) r2 = x2 + y2
y = r · sin(θ) tan(θ) = y/x
Cylindrical CoordinatesDefinition
DefinitionThe cylindrical coordinate map is
r
θ
z
F c (r ,θ,z)=
r ·cos(θ)r ·sin(θ)
z
−−−−−−−−−−−−−→
x
y
z
The cylindrical change of coordinates formulas are
x = r · cos(θ) r2 = x2 + y2
y = r · sin(θ) tan(θ) = y/x
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
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
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
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
Cylindrical CoordinatesExamples
Example
Quadric surfaces are easily described with cylindrical coordinates.
r2+z2=1 r2−z=0 r2−z2=1
r2−z2=−1
r2−z2=0 r=1
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
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
Cylindrical CoordinatesExamples
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
Cylindrical CoordinatesExamples
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
Cylindrical CoordinatesExamples
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
Cylindrical CoordinatesExamples
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
Cylindrical CoordinatesExamples
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
Cylindrical CoordinatesExamples
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
Cylindrical CoordinatesExamples
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
Cylindrical CoordinatesExamples
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
Spherical CoordinatesDefinition
ObservationLocation in R3 can be measured using multiple angles.
x
y
z
θ
ϕ
ρ
Spherical CoordinatesDefinition
ObservationLocation in R3 can be measured using multiple angles.
x
y
z
θ
ϕρ
Spherical CoordinatesDefinition
ObservationLocation in R3 can be measured using multiple angles.
x
y
z
θ
ϕρ
Spherical CoordinatesDefinition
ObservationLocation in R3 can be measured using multiple angles.
x
y
z
θ
ϕρ
Spherical CoordinatesDefinition
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
Spherical CoordinatesDefinition
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
Spherical CoordinatesDefinition
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
Spherical CoordinatesDefinition
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
Spherical CoordinatesDefinition
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
Spherical CoordinatesDefinition
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
Spherical CoordinatesDefinition
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
Spherical CoordinatesExamples
Example
Spherical coordinates conveniently describe spheres.
ρ = 1
Spherical CoordinatesExamples
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)
Spherical CoordinatesExamples
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)
Spherical CoordinatesExamples
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)
Spherical CoordinatesExamples
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)
Spherical CoordinatesExamples
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
4
0 ≤ ρ ≤ 1
cos(ϕ)
Spherical CoordinatesExamples
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)
Spherical CoordinatesExamples
ConventionTo acheive unique spherical representations, we often use
0 ≤ θ < 2π 0 ≤ ϕ ≤ π 0 ≤ ρ
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