Spherical Co-ordinate System...

Geographers specify a location on the Earth’s surface using three scalar values:
longitude, latitude, and altitude.

Both longitude and latitude are angular measures, while altitude is a measure of distance.

Latitude, longitude, and altitude are similar to Spherical Co-ordinates.

Spherical coordinates consist of one scalar value (r), with units of distance, while the other two scalar values (θ, φ) have angular units (degrees or radians).

A Vector in Spherical System is represented as
(A_r A_θ, A_φ)
or
A = A_ra_r + A_θa_θ + A_φa_φ

Where a_r, a_θ and a_φ are the unit vectors in r, θ and φ direction respectively.

The physical significance of each parameter of spherical coordinates:



- The value r expresses the distance of the point from origin (i.e. similar to altitude). It is the radius of the sphere.

- The angle θ is the angle formed with the z- axis (i.e. similar to latitude). It is also called the co-latitude angle. It is measured clockwise.

- The angle φ, also called the azimuthal angle, indicates the rotation angle around the z-axis (i.e. similar to longitude). It is basically measured from the x axis in the x-y plane. It is measured counter-clockwise.


Range of the variables:

It defines the minimum and the maximum value that r, θ and φ can have in spherical co-ordinate system.

- 0 ≤ r ≤ ∞
- 0 ≤ θ ≤ π
- 0 ≤ φ ≤ 2π

Spherical System - Unit Vectors:

Since the co-ordinate system is orthogonal, the unit vectors  
a_r, a_θ and a_φ are mutually perpendicular to each other.

- a_r points in the direction of increasing r i.e.
    a_r points away from the z-axis.
- _aθ points in the direction of increasing θ.
- a_φ points in the direction of increasing φ.

Consider the parallelogram ABOC,

- X = ρcos φ.
- Y = ρsin φ.
- Z = Z.

Consider the second parallelogram OCPQ, we have

- ρ = r sin θ
- z = r cos θ

So from the above data available we can say,

- x = r sin θ cos φ.
- Y = r sin θ sin φ.
- Z = r cos θ.

Similarly relationship between spherical and cylindrical coordinates can be derived as:

Relationship between (a_x, a_y, a_z) and (a_r, a_θ, a_φ)

From the cylindrical coordinate system we know that,

a_x = cosφ a_ρ – sinφ a_φ
a_y = sinφ a_ρ + cosφ a_φ
a_z = a_z

From the above figure, we can write a_ρ in terms of a_r and a_θ as

a_ρ = cos (90^o – θ) a_r + cos θ a_θ
a_z = cos θ a_r - sin θ a_θ  

Hence the unit vectors of cartesian and spherical co-ordinate system are related as:



a_x = sin θ cos φ a_r + cos θ cos φ a_θ - sin φ a_φ
a_y = sin θ sin φ a_r + cos θ sin φ a_θ + cos φ a_φ
a_z = cos θ a_r - sin θ a_θ

Transformation of vector A from (A_r, A_θ, A_Ф) to (A_x, A_y, A_z) i.e. transformation of Vector A from Spherical to Cartesian can be obtained as    

Transformation of vector A from (A_x, A_y, A_z) to (A_r, A_θ, A_Ф) i.e. transformation of Vector A from Cartesian to Spherical can be obtained as    

For more Information check out the following links:

- Spherical coordinate system - Wikepedia.

- Spherical coordinate system - Wolfram Mathworld.

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- Differential Analysis (dl, ds, dv)- Spherical Coordinate System.

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Field Theory - Differential Analysis - Cylindrical Co-ordinate System...

Differential Length(dl):
 

In General the differential length is given as

dl = dρ a_ρ + ρdφ a_φ+ dz a_z




Differential Length for a surface is given as:


- dl = dρ a_ρ+ ρdφ a_φ ---(For ρ-φ Plane or Z constant Plane)

- dl = ρdΦ a_φ+ dz a_z ---(For φ-z Plane or ρ Constant Plane)

- dl = dρ a_ρ+ dz a_z ---(For ρ-z Plane or φ Constant Plane)

Differential length for a line parallel to ρ, φ and z axis are respectively given as:


- dl = dρ a_ρ ---(For a line parallel to ρ axis)

- dl = ρdφ a_φ ---(For a line parallel to φ axis)

∫ dl = ∫_o^2π ρdφ = ρ( 2π - 1) = 2πρ

This resembles the circumference of a circle. Hence if φ varies with ρ and z constant, then the length is the circumference of the circle.

dl = dza_z ---(For a line parallel to z axis)





Differential Surface (ds):

- ds = ρdρ dφ a_z
 

This surface describes a circular disc. Always remember- To define a circular disk we need two parameter one distance measure and one angular measure. An angular parameter will always give a curved line or an arc.

In this case dΦ is measured in terms of change in arc.

Arc is given as:
Arc= radius * angle


Therefore, whenever there is a change in angle the radius always remains constant. Hence ρ always assist dφ.


∫ ds = ∫_o^ρ∫_o^2π ρ dρ dφ = (ρ²/2) (2π -1) = π ρ<sup>2

This answer describes the surface area of a circle. Hence the surface is a circular disc.</sup>






- ds = ρdφ dz a_ρ
 
This surface describes the curved surface of the cylinder. We can also say that this surface defines a hollow cylinder.

Suppose the height of the cylinder varies from 0 to h.

∫ ds = ∫_o^h∫_o^2π ρ dφ dz = ρ (h - 0) (2π - 0) = 2 πρh

This answer describes the surface area of a cylinder. Hence the surface is a hollow cylinder.


- ds = dρ dz a_φ
 
This surface describes a simple ρ-z plane which is along the direction of φ.




Differential Volume (dv):

dv = ρdρ dφ dz ---(Scalar Quantity)

∫ dv = ∫_o^ρ ∫_o^2π ∫_o^h ρ dρ dφ dz = (ρ²/2) (2π -0) (h -0) = π ρ² h

This answer describes the volume of a cylinder. 

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- ... Back to Cylindrical Coordinate System.


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Field Theory - Circular Cylindrical Co-ordinate System...

A Vector in Cylindrical system is represented as
(A_ρ, A_φ, A_z)
or
A = A_ρa_ρ+ A_φa_φ+ A_za_z

Where a_ρ, a_φand a_z are the unit vectors in ρ, φ and z direction respectively.

The physical significance of each parameter of cylindrical coordinates:

- The value ρ indicates the distance of the point from the z-axis. It is the radius of the cylinder.

- The value φ, also called the azimuthal angle, indicates the rotation angle around the z-axis. It is basically measured from the x axis in the x-y plane. It is measured anti-clockwise.


- The value z indicates the distance of the point from z-axis. It is the same as in the Cartesian system. In short, it is the height of the cylinder.




Range of the variables:

It defines the minimum and the maximum value that ρ, φ and z can have in Cartesian system.

 0 ≤ ρ ≤ ∞
 0 ≤ φ ≤ 2π
-∞ ≤ z ≤ ∞

 Figure shows Point P and Unit vectors in Cylindrical Co-ordinate System.


Cylindrical System - Unit vectors:
Since the co-ordinate system is orthogonal, the unit vectors a_ρ, a_φ and a_z are mutually perpendicular to each other.
- a_ρ points in the direction of increasing ρ, i.e a_ρ points away from the z-axis.
- a_φ points in the direction of increasing φ (anticlockwise).
- a_z points in the direction of increasing z.


Relationship between Cylindrical and Cartesian Co-ordinate System Parameters:


Consider the parallelogram ABOC,

X = ρcosφ.
Y = ρsinφ.
Z = Z.

From the above equations we have,


Relationship between (a_x, a_y and a_z) and (a_ρ , a_φand a_z) :

Since a_z is common between the two coordinate system, our main focus is to find out the relation between a_x, a_y and a_ρ , a_φ

We know φ is the angle from the x-axis on the x-y plane.

From the above figures two equations can be deduced,



Transformation of vector A from (A_x, A_y, A_z) to (A_ρ, A_φ, A_z) i.e. transformation of Vector A from Cartesian to Cylindrical can be obtained as



Transformation of vector A from (A_ρ, A_φ, A_z) to (A_x, A_y, A_z) i.e. transformation of Vector A from Cylindrical to Cartesian can be obtained as





For more Information check out the following links:

- Cylindrical coordinate system - Wikepedia.

- Cylindrical coordinate system - Wolfram Mathworld.

NEXT TOPIC:

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- Differential Analysis (dl, ds, dv)- Cylindrical Coordinate System.

- Page 1 - Numericals on Coordinate system and Transformation.

- Page 2 - Numericals on Coordinate system and Transformation.

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Field Theory - Cartesian Co-ordinate System / Rectangular Co-ordinate System (X, Y, Z)

A Vector in Cartesian system is represented as
(A_x, A_y, A_z)Or
A = A_xa_x+ A_ya_y+ A_za_z
Where a_x, a_y and a_z are the unit vectors in x, y, z direction respectively.



Range of the variables:

It defines the minimum and the maximum value that x, y and z can have in Cartesian system.

-∞ ≤ x,y,z ≤ ∞

Differential Displacement / Differential Length (dl):

It is given as dl = dxa_x + dya_y + dza_z


Differential length for a surface is given as:

dl = dxa_x + dya_y --- ( For XY Plane or Z Constant Plane).
dl = dya_y + dza_z, ---( For YZ Plane or X Constant Plane).
dl = dxa_x + dza_z ---( For XZ Plane or Y Constant Plane).

Differential length for a line parallel to x, y and z axis are respectively given as:

dl = dxa_x ---( For a line parallel to x-axis).
dl = dya_y ---( For a line Parallel to y-axis).
dl = dza_z, ---( For a line parallel to z-axis).


If there is a wire of length L in z-axis, then the differential length is given as dl = dz a_z. Similarly if the wire is in y-axis then the differential length is given as dl = dy a_y.


Differential Normal Surface (ds):

The differential surface (area element) is defined as ds = ds a_n, where a_n is the unit vector perpendicular to the surface.

For the 1st figure, ds = dydz a_x
2nd figure, ds = dxdz a_y
3rd figure, ds = dxdy a_z


Differential surface is basically a cross product between two parameters of the surface. For example, consider the first figure. The surface has two differential lengths, one is dy and dz. The differential surface (ds) is hence given as:
Where a_n is the unit vector normal to both dy and dz
i.e. a_n = a_y * a_z = a_x

In other words the differential surface element (ds) has an area equal to product dydz, and a normal vector that points in a_x direction.


Differential Volume element (dv)

The differential volume element (dv) can be expressed in terms of the triple product.

dv = dx . (dy * dz)

Consider a cubical surface having dimension x * y * z. The differential volume (dv) of the cubical surface is given as the triple product of the dimensions.

dv = dx . (dy * dz)
= dx a_x . (dy dz a _xsin θ _AB )
= dx a_x . (dy dz a _x)
= dx dy dz

Where dy and dz are mutually perpendicular to each other. Therefore the angle between them is 90^o.

One thing to remember isthat, the three parameters of Cartesian coordinate system i.e. X, Y, Z are all mutually perpendicular to each other.

Therefore a_x, a_y and a_z are all mutually perpendicular to each other.




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For more Information, check out the following links:

- Cartesian co-ordinate system - Wikepedia.

- Understanding Cartesian co-ordinate system with diagrams.

NEXT TOPIC:

- Cylindrical Coordinate System.

- Differential Analysis (dl, ds, dv)- Cylindrical Coordinate System.

- Spherical Coordinate System.

- Differential Analysis (dl, ds, dv)- Spherical Coordinate System.

Your suggestions and comments are welcome in this section. If you want to share something or if you have some stuff of your own, please do post them in the comments section.

Field Theory - Introduction to Co-ordinate System....

In order to describe the spatial variations of the quantities, appropriate coordinate system is required.

- A point or vector can be represented in a curvilinear coordinate system that may be orthogonal or non-orthogonal.

- An orthogonal system is one in which the coordinates are mutually perpendicular to each other.

- The different co-ordinate system available are:

- Cartesian or Rectangular.
- Circular cylindrical.
- Spherical.
- Elliptical Cylindrical.
- Hyperbolic Cylindrical.
- Parabolic Cylindrical .

The choice depends on the geometry of the application.


The frequently used and hence discussed herein are

- Rectangular Co-ordinate system.(Example: Cube, Cuboid)
- Cylindrical Co-ordinate system.(Example : Cylinder)
- Spherical Co-ordinate system.(Example : Sphere)


- A set of 3 scalar values that define position and a set of unit vectors that define direction form a co-ordinate system.


- The 3 scalar values used to define position are called co-ordinates. All coordinates are defined with respect to an arbitrary point called the origin.



- The 3 unit vectors used to define direction are also called base vectors.



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For more Information, check out the following links:

- Co-ordinate System - Wikepedia.

- Co-ordinate - Wolfram MathWorld.


NEXT TOPIC:

- ... Back To Index.

- Cartesian Coordinate System.

- Cylindrical Coordinate System.

- Spherical Coordinate System.


Your suggestions and comments are welcome in this section. If you want to share something or if you have some stuff of your own, please do post them in the comments section.