(2) vector analysis

8/3/2019 (2) Vector Analysis

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Electromagnetics (I)

Chapter 2Vector Analysis

Vector algebra

Coordinate systems

Vector calculus


Maxwells EquationsThe governing equations for electromagnetic energy involves vectorand scalar calculation.


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Vector Algebra


Vector Addition and Subtraction


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Product of Vector

Multiplication by a scalar

Dot (scalar) product

Cross (vector) product


Product of Vector Multiplication by a scalar

Dot (scalar) product


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Product of Vector


Product of Vector Cross (vector) productA new product perpendicular to the plane containing the twoVectors.


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Product of Three Vectors


Vector Triple Product

2


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Orthogonal Coordinate Systems


Metric Coefficients

A conversion factor , h i , is defined and used to convert a differential change of the coordinates, du i , into a change in length, dl , where u i may not be a length and thus a conversion factor is needed.


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Differential Length change, Area, and Volume


Orthogonal Coordinate Systems

(r, , z) (r, , )


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Cartesian Coordinates


Cartesian Coordinates


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Cylindrical Coordinates


Cylindrical Coordinates


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Spherical Coordinates


Spherical Coordinates


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Coordinate TransformationCylindrical coordinateCartesian coordinate

where


Coordinate TransformationChange of variables

Cylindrical coordinateCartesian coordinate


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Coordinate TransformationChange of variables

Spherical coordinateCartesian coordinate


Coordinate TransformationSpherical coordinateCartesian coordinate


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Coordinate TransformationSpherical coordinateCartesian coordinate

)where

A sin

similarly,


Metric Coefficients


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Differential Length change, Area, and Volume

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Basic Orthogonal Coordinate Systems


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Electromagnetics (I) Integrals Containing Vector Functions

Line IntegralsIntegral of scalar functions

Integral of vector function (scalar line integral)

Cartesian coordinate

Cartesian coordinate


Scalar Line Integral

(i) In Cartesian coordinates.


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Scalar Line Integral

(ii) In cylindrical coordinates.


Scalar Surface Integral


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Scalar Surface Integral


Vector Calculus


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Gradient


Gradient

For Cartesian coordinate


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Example 2-16

The electrostatic field intensity E is derivable as the negative gradient of a scalar electric potential V; that is, E = V.

(a)

(b)


Vector FieldsField : directed field lines flux lines or streamlines

(i) Direction of the vector field: directed line or curve.

(ii) Magnitude of the field: density of the lines.


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Divergence


Expression for Divergence

Similarly,

front surface

back surface

(expanded as Taylor series)


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Expression for Divergence

Recall

A = a x A x + a y A y + a z A z

Divergence of a vector field


Divergence of a Vector Field General orthogonal coordinates

Cartesian coordinates

Spherical coordinates

Cylindrical coordinates


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Divergence Theorem

Recall


Example 3-20


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Vector Calculus


For Cartesian coordinate

Gradient


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Divergence

Divergence of a vector field


Divergence of a Vector Field General orthogonal coordinates





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Divergence Theorem

Recall


Curl

(Vortex source can cause a circulation of a vector field around the field)


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Expression for Curl


Expression for Curl

Similarly,


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Expression for Curl

Curl of a vector field in Cartesian coordinates

Curl of a vector field in general orthogonal curvilinear coordinates


General orthogonal coordinates




Expression for Curl


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Basic Orthogonal Coordinate Systems


Example 2-21


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Stokes Theorem

where


Divergence Theorem

Recall


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Example 2-22

For surface integral


Example 2-22


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Two Null Identities Identity I


Two Null Identities Identity II


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Identity II


Vector Field Classification


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Helmholtzs Theorem


Helmholtzs Theorem


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Helmholtzs Theorem


Helmholtzs Theorem


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Example

(2) vector analysis

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