Post on 12-Feb-2017
Cylindrical and Cylindrical and Spherical CoordinatesSpherical Coordinates
Representation and Conversions
By: Shalini Singh Assistant Professor
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CONTENTSCONTENTS
Cylindrical Co-ordinatesCo-ordinates formation in 3-DCylindrical FormationSpherical Co-ordinatesSpherical Representation in 3-DConversion of Cylindrical Into SphericalRectangular Co-ordinatesIntegralConclusion
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Introduction Introduction The laws of electromagnetic are invariant with coordinate system. When we solve problems in electromagnetic. For Eg. Finding Electric Field at a point, we desire the fields. The position is expressed in terms of Co-ordinates. The co-ordinates are specified by the co-ordinate System.
Most Common Co-ordinate systems are:
•Cartesian Or Rectangular Coordinate system• Cylindrical Coordinate system•Spherical Coordinate system
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Representing 3D points in Representing 3D points in Cylindrical Coordinates. Cylindrical Coordinates.
r
Recall polar representations in the plane
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Representing 3D points in Representing 3D points in Cylindrical Coordinates. Cylindrical Coordinates.
r
Cylindrical coordinates just adds a z-coordinate to the polar coordinates (r,).
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r
Representing 3D points in Representing 3D points in Cylindrical Coordinates. Cylindrical Coordinates.
r
(r, , z)
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Cylindrical coordinates just adds a z-coordinate to the polar coordinates (r,).
cos( )sin( )
x ry rz z
r
r
(r,,z)
Rectangular to Cylindrical
2 2 2
tan( )
r x yyx
z z
Cylindrical to rectangular
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Spherical Coordinates are the 3D analog of polar representations in the plane.
We divide 3-dimensional space into1. a set of concentric spheres centered at the
origin.2. rays emitting outwards from the origin
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Representing 3D points in Representing 3D points in Spherical CoordinatesSpherical Coordinates
(x,y,z)
We start with a point (x,y,z) given in rectangular coordinates.
Then, measuring its distance from the origin, we locate it on a sphere of radius centered at the origin.
Next, we have to find a way to describe its location on the sphere.
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Representing 3D points in Representing 3D points in Spherical CoordinatesSpherical Coordinates
We use a method similar to the method used to measure latitude and longitude on the surface of the Earth.
We find the great circle that goes through the “north pole,” the “south pole,” and the point.
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Representing 3D points in Representing 3D points in Spherical CoordinatesSpherical Coordinates
We measure the latitude or polar angle starting at the “north pole” in the plane given by the great circle.
This angle is called . The range of this angle is
Note:
all angles are measured in radians, as always.
0 .
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Representing 3D points in Representing 3D points in Spherical CoordinatesSpherical Coordinates
We use a method similar to the method used to measure latitude and longitude on the surface of the Earth.
Next, we draw a horizontal circle on the sphere that passes through the point.
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Representing 3D points in Representing 3D points in Spherical CoordinatesSpherical Coordinates
And “drop it down” onto the xy-plane.
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Representing 3D points in Representing 3D points in Spherical CoordinatesSpherical Coordinates
We measure the latitude or azimuthal angle on the latitude circle, starting at the positive x-axis and rotating toward the positive y-axis.
The range of the angle is
Angle is called .
0 2 .
Note that this is the same angle as the in cylindrical coordinates!
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Finally, a Point in Spherical Finally, a Point in Spherical Coordinates!Coordinates!
( , ,)Our designated point on the sphere is indicated by the three spherical coordinates ( , , ) ---(radial distance, azimuthal angle, polar angle).
Please note that this notation is not at all standard and varies from author to author and discipline to discipline. (In particular, physicists often use to refer to the azimuthal angle and refer to the polar angle.)
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(x,y,z)
z
r
First note that if r is the usual cylindrical coordinate for (x,y,z)
we have a right triangle with •acute angle , •hypotenuse , and •legs r and z.
It follows that
sin( ) cos( ) tan( )r z rz
What happens if is not acute?
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Converting Spherical to Converting Spherical to Rectangular CoordinatesRectangular Coordinates
2 2 2
2 2
2 2 2
tan( )
tan( )
cos( )
x y zyx
x yrz zz z
x y z
(x,y,z)
z
r
Rectangular to Spherical
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Cylindrical and Cylindrical and Spherical CoordinatesSpherical Coordinates
Integration
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Rectangular CoordinatesRectangular Coordinates
We know that in a Riemann Sum approximation for a triple integral, a summand
This computes the function value at some point in the little “sub-cube” and multiplies it by the volume of the little cube of length , width , and height .
* * *( , , ) .i i i i i if x y z x y z
* * *
function value volume of the small at a sampling point cube
( , , ) .i i i i i if x y z x y z
ix iy iz
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Integration Elements: Integration Elements: Cylindrical Coordinates Cylindrical CoordinatesWhat happens when we consider small changes
in the cylindrical coordinates r, and z?
, , andr z
We no longer get a cube, and (similarly to the 2D case with polar coordinates) this affects integration.
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Integration Elements: Integration Elements: Cylindrical Coordinates Cylindrical CoordinatesWhat happens when we consider small changes
in the cylindrical coordinates r, z?
, , andr z
Start with our previous picture of cylindrical coordinates:
r
r
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What happens when we consider small changes
in the cylindrical coordinates r, and z?
, , andr z
Start with our previous picture of cylindrical coordinates:
Expand the radius by a small amount:
r
r
r+r
r
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This give us a “wedge.”
Combining this with the cylindrical shell created by the change in r, we get
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This give us a “wedge.”
Intersecting this wedge with the cylindrical shell created by the change in r, we get
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Finally , we look at a small vertical change z .
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Integration in Cylindrical Integration in Cylindrical Coordinates.Coordinates.
dA r dr d
We need to find the volume of this little solid.As in polar coordinates, we have the area of a horizontal cross section is. . .
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Integration in Cylindrical Integration in Cylindrical Coordinates.Coordinates.
dV r dr d dz
We need to find the volume of this little solid.Since the volume is just the base times the height. . .
So . . .( , , )
S
f r z r dr d dz
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CONCLUSION
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