LECTURE NOTES
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2012/2013
1.5. Cylindrical and Spherical Coordinates
1.5.1. Polar Coordinate
Given a point 𝑃(𝑥, 𝑦) with cartesian coordinate (x,y) in two dimensional
space, we may use its distance r from the origin and the angle 𝜃 measured in
the counterclockwise sense made with the x- axis to locate its position. This
gives the polar coordinate (𝑟, 𝜃)
Figure 1.5.1 The polar coordinate
The relation between Cartesian and polar coordinates are given by the
following formulas :
𝑦
𝑥
𝑥 = 𝑟𝑐𝑜𝑠 𝜃 , 𝑦 = 𝑟𝑠𝑖𝑛 𝜃
𝑟 2 = 𝑥 2 + 𝑦 2 , tan 𝜃 =
The convention is that 𝜃 is positive if it is measured in the
counterclockwise sense, and is negative otherwise. If 𝑟 < 0, the radius
is measured at the same distance |𝑟| from the origin., but on opposite
side of the origin. For example : for the polar coordinate of the point Q
𝜋
below, we may write either (−1, 4 ) or (1,
3𝜋
4
)
Figure 1.5.2. Example of point in polar coordinate
Example 1.5.1. Find the equation in polar coordinates of the curve
𝑥 2 + 𝑦 2 = 4𝑥
Lecture Notes of Calculus III by Rubono Setiawan, S.Si.,M.Sc.
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LECTURE NOTES
Solution It’s easy to proved that
coordinates is 𝑟 = 4 cos 𝜃.
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2012/2013
the equation above in polar
1.5.2. Cylindrical Coordinates
Given a point P with cartesian coordinates ( x, y, z) in three dimensional
space, we may use the polar coordinates (𝑟, 𝜃) for the position of the foot of
the perpendicular from P onto xy-plane. The the triple (𝑟, 𝜃, 𝑧) determines the
position of P it is called the cylindrical coordinates of P.
Figure 1.5.3. Cylindrical coordinates
The relation between Cartesian and Cylindrical coordinates are given in the
following formulas :
𝑦
, 𝑧=𝑧
𝑥
𝑥 = 𝑟𝑐𝑜𝑠 𝜃, 𝑦 = 𝑟𝑠𝑖𝑛 𝜃,
𝑧=𝑧
𝑟 2 = 𝑥 2 + 𝑦 2 , tan 𝜃 =
Cylindrical coordinates are useful in problems that involve symmetry about an
axis, and the z-axis is chosen to coincide with this axis of symmetry. For
instance, the axis of the circular cylinder with Cartesian equation 𝑥 2 + 𝑦 2 =
𝑐 2 in the z- axis. In cylindrical coordinates this cylinder has the very simple
equation 𝑟 = 𝑐. This is the reason for the name ‘’ cylindrical ‘’ coordinates.
Example 1.5.2.
Describe the surface whose equation in cylindrical
coordinates is 𝑧 = 𝑟.
Solution
The equation says that the z-value, or in this case is height, of each point on
the surface is the same as r. Because, 𝜃 doesn’t appear, it can vary. So any
horizontal trace in the plane 𝑧 = 𝑘, (𝑘 > 0) is a circle of radius k. Thes trace
suggest that the surface is a cone. This prediction can be confirmed by
converting the equation into rectangular ( polar ) coordinates, so we have :
𝑧2 = 𝑟2 = 𝑥2 + 𝑦2
We recognize those equation as being a circular cone whose axis in the z-axis.
Further more, to get more precious analysis, see the following figure :
Lecture Notes of Calculus III by Rubono Setiawan, S.Si.,M.Sc.
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LECTURE NOTES
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2012/2013
Figure 1.5.4. The surface 𝑧 = 𝑟 in cylindrical coordinates
Let P be a arbitrary point on this surface with cylindrical coordinates (𝑟, 𝜃, 𝑧).
Since 𝑧 = 𝑟, the triangle OPQ in which < 𝑂𝑄𝑃 is a right angle is isosceles with
𝑂𝑄 = 𝑟 = 𝑧 = 𝑃𝑄. Thus the cone opens up an angle of 450 with the 𝑧 − 𝑎𝑥𝑖𝑠.
From, previous analysis we get 𝑧 2 = 𝑟 2 = 𝑥 2 + 𝑦 2 , which is the Cartesian
equation of double cone. If we take positive squares root on both sides, the
graph of resulting equation 𝑧 = √𝑥 2 + 𝑦 2 is the inverted cone on the upper
half space 𝑧 ≥ 0.
1.5.3. Spherical Coordinates
The Spherical Coordinates (𝜌, 𝜃, ϕ) of a point P in space are shown in
Figure below, where |𝑂𝑃| is the distance from the origin to P, is the same
angle as in cylindrical coordinates, and 𝜙 is the angle between the positive zaxis and the line segment OP. Note that 𝜌 ≥ 0 and 0 ≤ 𝜙 ≤ 𝜋.
Figure 1.5.3.1. Spherical Coordinates
The comparison between Cartesian, Cylindrical and Spherical Coordinates :
Lecture Notes of Calculus III by Rubono Setiawan, S.Si.,M.Sc.
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LECTURE NOTES
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2012/2013
The spherical coordinate system is espesially useful in problems where
there is a symmetry about a point, and the origin is placed at this point. For
example, the sphere with center the origin and radius c has the simple
equation; this is the reason for the name ‘’spherical” coordinates.
The graph of the equation 𝜃 = 𝑐 is a vertical half plane, and the
equation 𝜙 = 𝑐 represents a half – cone with the z – axis as its axis.
Figure 1.5.3.1. Equation 𝜃 = 𝑐
Figure 1.5.3.2. Equation 𝜙 = 𝑐
The relationship between rectangular and spherical coordinates can be seen
from Figure 1.5.1. From triangles OPQ and OPP’ we have
𝑧 = 𝜌 cos 𝜙 and 𝑟 = 𝜌 sin 𝜙
But 𝑥 = 𝑟 cos 𝜃 and 𝑦 = 𝑟 𝑠𝑖𝑛 𝜃, so to convert from spherical to rectangular
coordinates we use equations :
Lecture Notes of Calculus III by Rubono Setiawan, S.Si.,M.Sc.
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LECTURE NOTES
𝑥 = 𝜌 sin 𝜙 cos 𝜃
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(1.5.3.1)
𝑦 = 𝜌 sin ϕ sin 𝜃, 𝑧 = 𝜌 cos 𝜃
Also, the distance formula shows that :
𝜌2 = 𝑥 2 + 𝑦 2 + 𝑧 2
Example 1.5.3.1.
𝜋 𝜋
4 3
The point (2, , ) is given in spherical coordinates. Plot the point and find its
rectangular coordinates
Solution : we plot the point in figure below , from equation 1.5.3.1., we have :
𝜋
𝜋
1
3
√3
𝑥 = 𝜌 sin 𝜙 cos 𝜃 = 2 sin cos = 2 ( ) ( ) = √
3
4
2
2
√2
𝜋
𝜋
1
3
√3
𝑦 = 𝜌 sin 𝜙 sin 𝜃 = 2 sin sin = 2 ( ) ( ) = √
3
4
2
2
√2
𝑧 = 𝜌 cos 𝜙 = 2 cos
𝜋 𝜋
3
𝜋
1
= 2. ( ) = 1
3
2
3
Thus the point (2, 4 , 3 ) is (√2 , √2 , 1) in rectangular coordinates
Example 1.5.3.2.
The point (0, √3, 2) is given in rectangular coordinates. Find spherical
coordinates for this point
Solution
From equation 1.3.5.2 we have :
𝜌 = √𝑥 2 + 𝑦 2 + 𝑧 2 = √0 + 3 + 4 = √7
and so equation 1.3.5.1 give
𝑧
2
2
=
, 𝜙 = 𝑎𝑟𝑐 cos
𝜌 √7
√7
𝑥
𝜋
cos 𝜃 =
= 0, 𝜃 =
𝜌 sin 𝜙
2
cos 𝜙 =
𝜋
Therefore, spherical coordinates of the given point are (√7, 2 , 𝑎𝑟𝑐 cos
2
)
√7
Example 1.5.3.3.
Find an equation in spherical coordinates for the hyperboloid of two sheets
with equation 𝑥 2 + 𝑦 2 + 𝑧 2 = 1
Solution
Subtituting the expression 1.5.3.1. into the given equations, we have :
Lecture Notes of Calculus III by Rubono Setiawan, S.Si.,M.Sc.
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LECTURE NOTES
SEMESTER 1
2012/2013
𝜌2 𝑠𝑖𝑛 2 𝜙𝑐𝑜𝑠 2 𝜃 − 𝜌2 𝑠𝑖𝑛2 𝜙𝑠𝑖𝑛2 𝜃 − 𝜌2 𝑐𝑜𝑠 2 𝜙 = 1
𝜌2 [𝑠𝑖𝑛 2 𝜙(𝑐𝑜𝑠 2 𝜃 − 𝑠𝑖𝑛2 𝜃) − 𝑐𝑜𝑠 2 𝜙] = 1
𝜌2 (𝑠𝑖𝑛 2 𝜙𝑐𝑜𝑠2𝜃 − 𝑐𝑜𝑠 2 𝜙) = 1
Example 1.5.3.4.
Find the of the paraboloid 𝑧 = 𝑥 2 + 𝑦 2 in spherical coordinates ?
Lecture Notes of Calculus III by Rubono Setiawan, S.Si.,M.Sc.
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