Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )=...
Transcript of Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )=...
![Page 1: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/1.jpg)
Coordinate SystemsMath 212
Brian D. Fitzpatrick
Duke University
January 24, 2020
MATH
![Page 2: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/2.jpg)
Overview
Rectangular CoordinatesDefinitionExamples
Polar CoordinatesDefinitionExamples
Cylindrical CoordinatesDefinitionExamples
Spherical CoordinatesDefinitionExamples
![Page 3: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/3.jpg)
Rectangular CoordinatesDefinition
ConventionWe often measure location with rectangular coordinates.
x
y
x0
y0
(x0, y0)
x
y
z
x0
y0
z0
(x0, y0, z0)
![Page 4: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/4.jpg)
Rectangular CoordinatesDefinition
ConventionWe often measure location with rectangular coordinates.
x
y
x0
y0
(x0, y0)
x
y
z
x0
y0
z0
(x0, y0, z0)
![Page 5: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/5.jpg)
Rectangular CoordinatesDefinition
ConventionWe often measure location with rectangular coordinates.
x
y
x0
y0
(x0, y0)
x
y
z
x0
y0
z0
(x0, y0, z0)
![Page 6: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/6.jpg)
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
![Page 7: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/7.jpg)
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
![Page 8: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/8.jpg)
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
![Page 9: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/9.jpg)
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
![Page 10: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/10.jpg)
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
![Page 11: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/11.jpg)
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
![Page 12: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/12.jpg)
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
![Page 13: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/13.jpg)
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
![Page 14: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/14.jpg)
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
![Page 15: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/15.jpg)
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
![Page 16: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/16.jpg)
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
![Page 17: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/17.jpg)
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
![Page 18: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/18.jpg)
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
![Page 19: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/19.jpg)
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
![Page 20: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/20.jpg)
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
![Page 21: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/21.jpg)
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
![Page 22: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/22.jpg)
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
![Page 23: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/23.jpg)
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
![Page 24: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/24.jpg)
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
![Page 25: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/25.jpg)
Rectangular CoordinatesExamples
QuestionCan we describe interesting regions more easily?
![Page 26: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/26.jpg)
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
![Page 27: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/27.jpg)
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
![Page 28: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/28.jpg)
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
![Page 29: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/29.jpg)
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
![Page 30: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/30.jpg)
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
![Page 31: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/31.jpg)
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
![Page 32: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/32.jpg)
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
![Page 33: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/33.jpg)
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
![Page 34: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/34.jpg)
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
![Page 35: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/35.jpg)
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
![Page 36: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/36.jpg)
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
![Page 37: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/37.jpg)
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
![Page 38: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/38.jpg)
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
![Page 39: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/39.jpg)
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
![Page 40: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/40.jpg)
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
![Page 41: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/41.jpg)
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
![Page 42: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/42.jpg)
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
![Page 43: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/43.jpg)
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
![Page 44: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/44.jpg)
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
![Page 45: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/45.jpg)
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
![Page 46: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/46.jpg)
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
![Page 47: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/47.jpg)
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
![Page 48: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/48.jpg)
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 .
![Page 49: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/49.jpg)
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 .
![Page 50: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/50.jpg)
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 .
![Page 51: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/51.jpg)
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 .
![Page 52: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/52.jpg)
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 .
![Page 53: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/53.jpg)
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 .
![Page 54: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/54.jpg)
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 .
![Page 55: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/55.jpg)
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 .
![Page 56: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/56.jpg)
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
.
![Page 57: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/57.jpg)
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
.
![Page 58: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/58.jpg)
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
.
![Page 59: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/59.jpg)
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
.
![Page 60: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/60.jpg)
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
.
![Page 61: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/61.jpg)
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.
![Page 62: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/62.jpg)
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
![Page 63: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/63.jpg)
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
![Page 64: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/64.jpg)
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
![Page 65: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/65.jpg)
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
![Page 66: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/66.jpg)
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
![Page 67: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/67.jpg)
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
![Page 68: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/68.jpg)
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
![Page 69: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/69.jpg)
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
![Page 70: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/70.jpg)
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
![Page 71: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/71.jpg)
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
![Page 72: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/72.jpg)
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
![Page 73: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/73.jpg)
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
![Page 74: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/74.jpg)
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
![Page 75: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/75.jpg)
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
![Page 76: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/76.jpg)
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
![Page 77: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/77.jpg)
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
![Page 78: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/78.jpg)
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
![Page 79: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/79.jpg)
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
![Page 80: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/80.jpg)
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
![Page 81: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/81.jpg)
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
![Page 82: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/82.jpg)
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
![Page 83: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/83.jpg)
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
![Page 84: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/84.jpg)
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
![Page 85: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/85.jpg)
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
![Page 86: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/86.jpg)
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
![Page 87: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/87.jpg)
Cylindrical CoordinatesDefinition
ObservationLocation in R3 can be measured with polar coordinates.
x
y
z
θ
z
r
![Page 88: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/88.jpg)
Cylindrical CoordinatesDefinition
ObservationLocation in R3 can be measured with polar coordinates.
x
y
z
θ
z
r
![Page 89: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/89.jpg)
Cylindrical CoordinatesDefinition
ObservationLocation in R3 can be measured with polar coordinates.
x
y
z
θ
z
r
![Page 90: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/90.jpg)
Cylindrical CoordinatesDefinition
ObservationLocation in R3 can be measured with polar coordinates.
x
y
z
θ
z
r
![Page 91: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/91.jpg)
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
![Page 92: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/92.jpg)
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
![Page 93: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/93.jpg)
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
![Page 94: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/94.jpg)
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
![Page 95: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/95.jpg)
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
![Page 96: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/96.jpg)
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
![Page 97: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/97.jpg)
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
![Page 98: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/98.jpg)
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
![Page 99: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/99.jpg)
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
![Page 100: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/100.jpg)
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
![Page 101: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/101.jpg)
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
![Page 102: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/102.jpg)
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
![Page 103: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/103.jpg)
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
![Page 104: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/104.jpg)
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
![Page 105: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/105.jpg)
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
![Page 106: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/106.jpg)
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
![Page 107: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/107.jpg)
Cylindrical CoordinatesExamples
ConventionTo acheive unique cylindrical representations, we often use
0 ≤ θ < 2π 0 ≤ r −∞ ≤ z ≤ ∞
![Page 108: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/108.jpg)
Spherical CoordinatesDefinition
ObservationLocation in R3 can be measured using multiple angles.
x
y
z
θ
ϕ
ρ
![Page 109: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/109.jpg)
Spherical CoordinatesDefinition
ObservationLocation in R3 can be measured using multiple angles.
x
y
z
θ
ϕ
ρ
![Page 110: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/110.jpg)
Spherical CoordinatesDefinition
ObservationLocation in R3 can be measured using multiple angles.
x
y
z
θ
ϕρ
![Page 111: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/111.jpg)
Spherical CoordinatesDefinition
ObservationLocation in R3 can be measured using multiple angles.
x
y
z
θ
ϕρ
![Page 112: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/112.jpg)
Spherical CoordinatesDefinition
ObservationLocation in R3 can be measured using multiple angles.
x
y
z
θ
ϕρ
![Page 113: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/113.jpg)
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
![Page 114: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/114.jpg)
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
![Page 115: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/115.jpg)
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
![Page 116: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/116.jpg)
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
![Page 117: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/117.jpg)
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
![Page 118: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/118.jpg)
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
![Page 119: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/119.jpg)
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
![Page 120: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/120.jpg)
Spherical CoordinatesExamples
Example
Spherical coordinates conveniently describe spheres.
ρ = 1
![Page 121: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/121.jpg)
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)
![Page 122: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/122.jpg)
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)
![Page 123: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/123.jpg)
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)
![Page 124: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/124.jpg)
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)
![Page 125: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/125.jpg)
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)
![Page 126: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/126.jpg)
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)
![Page 127: Coordinate Systems - Math 212bfitzpat/teaching/... · The polar coordinate map is r r 0 0 Fp(r; )= 2 4 r cos( ) r sin( ) 3 5! x y The polar change of coordinates formulas are x =](https://reader035.fdocuments.in/reader035/viewer/2022071101/5fda5bdd41b9f1020426d303/html5/thumbnails/127.jpg)
Spherical CoordinatesExamples
ConventionTo acheive unique spherical representations, we often use
0 ≤ θ < 2π 0 ≤ ϕ ≤ π 0 ≤ ρ