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Advanced Calculus (MATH 104) - Polar Coordinates: Harish Bhandari

This document discusses different coordinate systems including cylindrical and spherical coordinates. It provides definitions and explanations of what different parameters represent in each system. Examples are given to convert between rectangular and cylindrical/spherical coordinates. Key points covered include: 1. Cylindrical coordinates represent a point using (r, θ, z) where r and θ are the polar coordinates of the vertical projection on the xy-plane and z is the vertical coordinate. 2. Spherical coordinates represent a point using (ρ, θ, φ) where ρ is the distance from the origin, θ is the azimuthal angle, and φ is the polar angle. 3. Relations between the different coordinate systems are presented along with examples

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Bibhushan Gautam
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87 views10 pages

Advanced Calculus (MATH 104) - Polar Coordinates: Harish Bhandari

This document discusses different coordinate systems including cylindrical and spherical coordinates. It provides definitions and explanations of what different parameters represent in each system. Examples are given to convert between rectangular and cylindrical/spherical coordinates. Key points covered include: 1. Cylindrical coordinates represent a point using (r, θ, z) where r and θ are the polar coordinates of the vertical projection on the xy-plane and z is the vertical coordinate. 2. Spherical coordinates represent a point using (ρ, θ, φ) where ρ is the distance from the origin, θ is the azimuthal angle, and φ is the polar angle. 3. Relations between the different coordinate systems are presented along with examples

Uploaded by

Bibhushan Gautam
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© © All Rights Reserved
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Advanced Calculus(MATH 104)-Polar Coordinates

Harish Bhandari

Department of Mathematics
Kathmandu University
Dhulikhel, Nepal

Lecture 4
August 10, 2019
Cylindrical Coordinates

Definition
Cylindrical coordinate represent a
point P in space by ordered triples
(r , θ, z) in which
i. r and θ are polar coordinate for
the vertical projection of P on
xy plane.
ii. z is rectangular vertical
coordinate.
Relation between rectangular and cylindrical coordinates

Harish Bhandari Coordinates systems


Note
What does following represent in
cylindrical coordinate system?
i. r = a
ii. r = 0
iii. θ = θ0
iv. z = z0

Explanations

i. Cylinder about z-axis


ii. z-axis
iii. Plane containing z-axis, making angle θ0 with x-axis
iv. Plane perpendicular to z-axis

Harish Bhandari Coordinates systems


Harish Bhandari Coordinates systems
Examples
Find the rectangular Find the cylindrical
coordinates from the coordinates from the
following cylindrical following rectangular
coordinates: coordinates:
√
i. (4, π6 , 3) a. (1, 3, 2)
ii. (3, π3 , 5) b. (1, 1, 5)
iii. (4, 7π c. (3, −3, 7)
6 , 3) √ √
iv. (2, π, −4) d. (−2 2, 2 2, 4)
Exercise

a. Find the cartesian


form of
z = r 2 (Paraboloid)
b. Find the circular
cylinder in
cylindrical
coordinate:
4x 2 + 4y 2 = 9
Harish Bhandari Coordinates systems
Spherical Coordinates

Definition
Cylindrical coordinate represent a point P in space by ordered
triples (ρ, θ, φ) in which
i. ρ is the distance of P from origin.
~ makes with positive z-axis (0 ≤ φ ≤ π).
ii. φ is the angle OP
iii. θ is the angle from cylindrical coordinates.

Harish Bhandari Coordinates systems


Note
What does following represent in
spherical coordinate system?
i. ρ = a
ii. φ = φ0
iii. θ = θ0
iv. φ = φ0 > π2

Explanations

i. Sphere of radius a centered at origin.


ii. Cone, vertex at origin, axis along z-axis
iii. Half plane containing z-axis, making angle θ0 with positive
x-axis
iv. Cone open downwards

Harish Bhandari Coordinates systems


Relations of spherical coordinates
Spherical coordinates and rectangular coordinates

Spherical coordinates and Cylindrical coordinates

Harish Bhandari Coordinates systems


Relations of spherical coordinates

Harish Bhandari Coordinates systems


Example

1. Find the spherical coordinate equation for


a. x 2 +p
y 2 + (z − 1)2 = 1
b. z = x 2 + y 2
√
2. Find the spherical coordinates for (0, 2 3, −2)

Harish Bhandari Coordinates systems

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