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Co-ordinate Systems
By,
S.T.Suganthi,
HOD/EEE,
S.Veerasamy Chettiar college of Engineering and
Technology,
Puliangudi-627855
Vector Calculus - Addition
1-2
Associative Law:
Distributive Law:
Scalar and Vector Fields
A scalar field is a function that gives us a single value of some variable for every point in space.
◦ Examples: voltage, current, energy, temperature
A vector is a quantity which has both a magnitude and a direction in space.
◦ Examples: velocity, momentum, acceleration and force
Examples of Vector Fields
Examples of Vector Fields
Examples of Vector Fields
Vector Field
1-7
We are accustomed to thinking of a specific vector:
A vector field is a function defined in space that has magnitude
and direction at all points:
where r = (x,y,z)
The Dot Product
1-8
Commutative Law:
Vector Projections Using the Dot Product
B • a gives the component of B
in the horizontal direction
(B • a) a gives the vector component
of B in the horizontal direction
Projection of a vector on another vector
Operational Use of the Dot Product
Given
Find
where we have used:
Note also:
Cross Product
1-12
Operational Definition of the Cross Product in Rectangular Coordinates
Therefore:
Or…
Begin with:
where
Orthogonal Vector Components
1-14
Orthogonal Unit Vectors
1-15
Vector Representation in Terms of Orthogonal Rectangular Components
1-16
Co-ordinate System
To describe a vector accurately such as lengths, angles, and projections etc…
Three types of Co-ordinate system Rectangular (or) Cartesian Co-ordinates
Cylindrical Co-ordinates
Spherical Co-ordinates
1-17
Rectangular Coordinate SystemCo-ordinates are (x,y,z)
1-18
Rectangular Coordinate System
A Point Locations in Rectangular Coordinates –Intersection of 3 orthogonal planes ( X-constant plane, Y- constant Plane, Z-constant plane)
1-19
Differential Volume Element
1-20
Vector Expressions in Rectangular Coordinates
1-21
General Vector, B:
Magnitude of B:
Unit Vector in the
Direction of B:
Example
1-22
Cylindrical Coordinate Systems
1-23
Cylindrical Coordinate Systems
1-24
Cylindrical Coordinate Systems
1-25
Cylindrical Coordinate Systems
1-26
Differential Volume in Cylindrical Coordinates
1-27
dV = dddz
Point Transformations in Cylindrical Coordinates
1-28
Dot Products of Unit Vectors in Cylindrical and Rectangular Coordinate Systems
1-29
Example
1-30
Transform the vector, into cylindrical coordinates:
Start with:
Then:
Finally:
Example: cont.
1-32
Spherical Coordinates
1-33
Spherical Coordinates
1-34
Spherical Coordinates
1-35
Spherical Coordinates
1-36
Spherical Coordinates
Spherical Coordinates
1-37
Point P has coordinates
Specified by P(r)
Differential Volume in Spherical Coordinates
1-38
dV = r2sindrdd
Dot Products of Unit Vectors in the Spherical and Rectangular Coordinate Systems
1-39
Example: Vector Component Transformation
1-40
Transform the field, , into spherical coordinates and components
Constant coordinate surfaces-Cartesian system
1-41
If we keep one of the coordinate variables
constant and allow the other two to vary,
constant coordinate surfaces are generated in
rectangular, cylindrical and spherical
coordinate systems.
We can have infinite planes:
X=constant,
Y=constant,
Z=constant
These surfaces are perpendicular to x, y and z axes respectively.
Constant coordinate surfaces-cylindrical system
1-42
Orthogonal surfaces in cylindrical
coordinate system can be generated as
ρ=constnt
Φ=constant
z=constant
ρ=constant is a circular cylinder,
Φ=constant is a semi infinite plane with its
edge along z axis
z=constant is an infinite plane as in the
rectangular system.
Constant coordinate surfaces-Spherical system
1-43
Orthogonal surfaces in spherical
coordinate system can be generated
as
r=constant
θ=constant
Φ=constant
θ =constant is a circular cone with z axis as its axis and origin at
the vertex,
Φ =constant is a semi infinite plane as in the cylindrical system.
r=constant is a sphere with its centre at the origin,
Differential elements in rectangularcoordinate systems
1-44
Differential elements in Cylindricalcoordinate systems
1-45
Differential elements in Sphericalcoordinate systems
1-46
1-47
Line integrals
Line integral is defined as any integral that is to be evaluated
along a line. A line indicates a path along a curve in space.
Surface integrals
1-48
Volume integrals
1-49
DEL Operator
1-50
DEL Operator in cylindrical coordinates:
DEL Operator in spherical coordinates:
Gradient of a scalar field
1-51
The gradient of a scalar field V is a vector that represents the
magnitude and direction of the maximum space rate of increase of V.
For Cartesian Coordinates
For Cylindrical Coordinates
For Spherical Coordinates
Divergence of a vector
1-52
In Cartesian Coordinates:
In Cylindrical Coordinates:
In Spherical Coordinates:
Gauss’s Divergence theorem
1-53
Curl of a vector
1-54
Curl of a vector
1-55
In Cartesian Coordinates:
In Cylindrical Coordinates:
In Spherical Coordinates:
Stoke’s theorem
1-57
Laplacian of a scalar
1-58
Laplacian of a scalar
1-59