Chapter 3: Vector Analysis

The divergence measures the outflow per unit volume of a vector field at a point. 20[r]

Luong Vinh Quoc Danh

Bài giảng: TRƯỜNG ĐIỆN TỪ (CT361)

(ELECTROMAGNETICS)

Chapter 3: Vector Analysis (Giải tích Vector)

Giảng viên: GVC.TS Lương Vinh Quốc Danh

Bộ môn Điện tử Viễn thông, Khoa Công Nghệ

• Basic Laws of Vector Algebra

• Orthogonal Coordinate Systems

• Transformations between Coordinate Systems

• Gradient of a Scalar Field

• Divergence of a Vector Field

• Curl of a Vector Field

• Laplacian Operator

Content

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Basic Laws of Vector Algebra

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Luong Vinh Quoc Danh

Basic Laws of Vector Algebra (cont.)

 Equality of two vectors:

 Vector Addition and Subtraction:

C = B + A = A + B

) ( ) ( ) ( ) ( )

(x^ Ax  y^ Ay  z^ Az  x^ Bx  y^ By  z^ Bz  x^ Ax  Bx  y^ Ay  Bx  z^ Az  Bz



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Position and Distance Vectors

Position Vector:

From origin to point P

Distance Vector:

Between two points

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Vector Multiplication

Dot Product -> Scalar

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Vector Multiplication (cont.)

Cross Product -> Vector

(3.28)

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Scalar Triple Product

Vector Triple Product

Vector Multiplication (cont.)

(3.29)

(3.33)

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Coordinate Systems • Cartesian

• Cylindrical

• Spherical

Differential Length, Area, and Volume

Example 3-5

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Transformation between Coordinate Systems • To solve a problem, we select the coordinate system that best

fits its geometry

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Example 3-8

Leads to: Using the relations:

Solution

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Each GPS satellite sends radio signals down to Earth all the time The GPS receivers on the ground pick up these signals A computer in each receiver compares the arrival times of signals from different satellites It can then calculate the position from this information

GPS comprises a constellation of 24 satellites Each is in a 12 hour orbit at a height of 20,000 km above the ground

Source: http://www.suntrek.org/solar-spacecraft/satellites-rockets/what-satellites-do/gps-satellites.shtml

Global Positioning System

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Gradient of a Scalar Field

Example: Temperature Distribution T(x,y,z)

- The direction of T is along the maximum increase of T - The magnitude of T is equal

to the maximum rate of change of T

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Directional Derivative

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Divergence of a Vector Field

Total flux crossing a closed surface S:

Scalar -> Vector

Net outward flux per unit volume:

(3.95)

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Divergence of a Vector Field (cont.)

Physical meaning of Divergence

Imagine the two vectors below give the velocity of fluid flow

An explosion outwards The origin is a source of fluid

Flow inward The origin is a sink

The divergence measures the outflow per unit volume of a vector field at a point.

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Divergence Theorem

Useful tool for converting integration over a volume to one over the surface enclosing that volume, and vice versa.

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Uniform field

, ld B n

Circulatio    

, ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 0 0          dy y B x dx x B x dy y B x dx x B x n Circulatio a d d c c b b a

 Circulation of a uniform field is zero.

Curl of a Vector Field

(3.100)

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, , ˆ ˆ 0 I d r I l d B n Circulatio               , ˆ r I B      (3.102)

Curl of a Vector Field (cont.)

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Curl of a Vector Field (cont.)

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Curl of a Vector Field (cont.)

Physical meaning of Curl

- The curl measures the rotation of a vector field.

Imagine placing a ball in a flow, holding its position fixed, allowing it to spin freely

If the ball spins, it indicates rotation

We indicate the direction of the rotation by the axis of rotation using the right-hand rule (Fingers with rotation, thumb indicates direction)

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Stokes’s Theorem

Laplacian Operator

Laplacian of a Scalar Field

Laplacian of a Vector Field

Useful Relation

(3.112)

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Quick Review

• Coordinate Systems

• Unit vector

• Differential length, surface, volume

• Transformation between coordinate systems • Gradient

• Divergence

• Divergence theorem

• Curl

• Stokes theorem

• Laplacian

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