Vector Calculus Review - University of Texas at El Pasoemlab.utep.edu/ee3321emf/Lecture 2 -- Vector Calculus Review.pdf · zx yz A Bz A By A Bx Step 3 –Make ... Vector Calculus

6/22/2018

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Vector CalculusReview

EE3321

Electromagnetic Field Theory

Course InstructorDr. Raymond C. RumpfOffice: A-337Phone: (915) 747-6958E-Mail: [email protected]

Outline

• Mathematical Preliminaries– Phasors, vectors, notation

• Math with Vectors

• Coordinate Systems– Notation, differentials

• Visualization of Fields & Operations

• Important Concepts

Vector Calculus Review

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Mathematical Preliminaries

Phasors (1 of 2)


A time-harmonic function can be written as

cosy t A t

Recall Euler’s Identity

cos sinje j

This let’s us write the function y(t) as

Re j ty t Ae

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Phasors (2 of 2)


In linear systems, frequency does not change.

We can express time-harmonic signals without explicitly using the frequency term .

This compact notation is a phasor.

jY Ae

Polar Vs. Rectangular Form


A phasor in polar form is written as

or jY Ae A

The same phasor written in rectangular form is

Y j

Rect Polar Polar Rect

2 2

1tan

A

cos

sin

A

A

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Phasor Arithmetic


Addition

1 2 1 2 1 2F F j

Subtraction

1 2 1 2 1 2F F j

Multiplication

1 2 1 2 1 2F F A A

Division

1 2 1 2 1 2F F A A

Scalars & Vectors


Scalar NumbersScalars contain only one piece of information, magnitude. Scalars can be real or complex. Phasors are scalar quantities.

Examples: 7, , -1.34, etc.

VectorsVectors have both a magnitude and a direction.

Examples: Velocity, force, electromagnetic fields

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Vector Notation


Parallel to paper Out of paperThink of seeing the point of an arrow

Into the paperThink of seeing the back of an arrow

direction

Note: Despite the arrow extending away from the point, a vector is describing something at that specific point and it does not actually extend outward.

What Can Vectors Convey?


Position Distance Disturbance

Position relative to the origin.

Vectors can indicate distance, but the origin is not given.

A vector can represent a directional disturbance. Thank of this as a push.

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Simple Vector Calculations


3D Vector

ˆ ˆ ˆx y zA A x A y A z

Vector Magnitude

2 2 2x y zA A A A

Unit Vector

ˆ AA

A

Math With Vectors

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Visualization of Vector Addition & Subtraction


Vector Addition Vector Subtraction

U

V

U V

x

y

UV

U V

x

y

Vector Addition & Subtraction


Cartesian Cylindrical Spherical

ˆ ˆ ˆ

ˆ ˆ ˆ

x x y y z z

x x y y z z

U U a U a U a

V V a V a V a

Starting Vectors

ˆ

ˆ

ˆ

x x x

y y y

z z z

U V U V a

U V a

U V a

Addition

ˆ

ˆ

ˆ

x x x

y y y

z z z

U V U V a

U V a

U V a

Subtraction

ˆ ˆ ˆ

ˆ ˆ ˆ

z z

z z

U U a U a U a

V V a V a V a

Starting Vectors

ˆ

ˆ

ˆz z z

U V U V a

U V a

U V a

Addition

ˆ

ˆ

ˆz z z

U V U V a

U V a

U V a

Subtraction

ˆ ˆ ˆ

ˆ ˆ ˆ

r r

r r

U U a U a U a

V V a V a V a

Starting Vectors

ˆ

ˆ

ˆ

r r rU V U V a

U V a

U V a

Addition

ˆ

ˆ

ˆ

r r rU V U V a

U V a

U V a

Subtraction

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The Dot Product,


A B

The dot product is all about projections. That is, calculating how much of one vector lies in the direction of another vector.

cos

x x y y z z

A B A B

A B A B A B

A

B

Projection of onto


A

B

2

DirectionMagnitudeˆB

B

BA BB

B B A BA A B

B B B

A

B

BA

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Projection of onto


B

A

A

B

AB

2

DirectionMagnitudeˆA

A

AB AA

A A B AB B A

A A A

The Dot Product Test


We can use the dot product to test of two vectors are perpendicular. If they are, the component of one along the other must be zero.

0 when A B A B

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The Cross Product,


A B

The cross product is all about area and calculating vectors that are perpendicular to and .

ˆsin nA B A B a

A

B

A

B

90°

90°

Area A B

ˆ is a unit vector perpendicular

to the plane defined by and .

na

A B

A B

Calculating Cross Products (1 of 2)


Suppose we wish to calculate the cross product .A B

ˆ ˆ ˆ ˆˆ

x y z x y

x y z x y

x y z x y

A A A A A

B B B B B

ˆ ˆ ˆx y zA A x A y A z

ˆ ˆ ˆx y zB B x B y B z

Step 1 – Construct an augmented matrix.

First two columns are repeated outside of the matrix.

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Calculating Cross Products (2 of 2)


Step 2 – Multiply elements along the diagonals.

ˆ

ˆ

ˆ

y x

z y

x z

A B z

A B x

A B y

ˆ

ˆ

ˆ

x y

z x

y z

A B z

A B y

A B x

Step 3 – Make left-hand side products negative.

Step 4 – Add up all of the products.

ˆ ˆ ˆy z z y z x x z x y y xA B A B A B x A B A B y A B A B z

The Cross Product Test


We can use the cross product to test of two vectors are parallel. If they are, the cross product will be zero because the angle between the vectors is zero.

0 when ||A B A B

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Vector Algebra Rules


Commutative Laws

A B B A

Distributive Laws

A B C A B A C

Self-Product2

A A A

A B B A

A B C A B A C

0A A

Associative Laws

A B C A B C

Vector Triple Products


Scalar Triple ProductThe scalar triple product is the volume of a parallelpiped.

B C A C A B A B C

Vector Triple ProductThe vector triple product arises when deriving the wave equation.

A B C B A C C A B

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Coordinate Systems

Types of Integrations


L

d

L

d

S

ds

S

ds

V

dv

v

dV

Ordinary Line Integral

Closed-Contour Line Integral

Ordinary Surface Integral

Closed-Contour Surface Integral

Ordinary Volume Integral

Closed-Contour Volume Integral

L

L

S

S

V

?

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Cartesian Coordinates


Image courtesy of Wikipedia.

Cartesian Differentials (1 of 2)


x

y

z

dy dx

dz

dv dxdydz

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Cartesian Differentials (2 of 2)


x

y

z

dy dx

dz

dv dxdydz

ˆz zds dxdya

ˆy yds dxdza

Cylindrical Coordinates



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Cylindrical Differentials (1 of 2)


x

y

z

d

d

dz

dv d d dz

d d

Cylindrical Differentials (2 of 2)


x

y

z

d

d

dz

dv d d dz

ˆz zds d d a

ˆds d dza

ˆds d dza

d d

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Spherical Coordinates



Spherical Differentials


0

0

0 2

r

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Summary of Differentials


Cartesian Coordinates

Cylindrical Coordinates

Spherical Coordinates

Visualization of Fields &

Operations

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Scalar Field Vs. Vector Field


magnitude , ,x y z

magnitude , ,

direction , ,

x y z

x y z

Scalar Field, ,f x y Vector Field, ,v x y

Alternate Visualization of a Vector Field


Arrows convey magnitude and direction. Background color also conveys magnitude.

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Isocontour Lines


Isocontour lines trace the paths of equal value. Closely space isocontours conveys that the function is varying rapidly.

Gradient of a Scalar Field (1 of 3)


We start with a scalar field…

,f x y

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…then plot the gradient on top of it. Color in background is the original scalar field.

,f x y



The gradient will always be perpendicular to the isocontour lines.

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Divergence of a Vector Field (1 of 2)


Suppose we start with the following vector field…

,A x y

Divergence of a Vector Field (2 of 2)


We then plot the divergence as the color in the background. The arrows are the original vector function.

,A x y

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Curl of a Vector Field (1 of 2)


Suppose we start with the following vector field…

,B x yy

x

z

y

xz

Curl of a Vector Field (2 of 2)


y

x

z

y

xz

The color in the background is the magnitude of the curl. The direction is either into, or out of, the screen. Red indicates +z direction while blue indicates –z direction.

,B x y

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Summary of Vector Operations


Operation Input Output

Vector Addition & Subtraction Vectors Vectors

Dot Product Vectors Scalar

Cross Product Vectors Vector

Gradient Scalar Function Vector Function

Divergence Vector Function Scalar Function

Curl Vector Function Vector Function

U V

U V

U V

f

U

U

Important Concepts

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Flux


Flux is the total amount of a vector field that passes straight through a surface.

S

dsA Vector Field A

Normal component counted as flux

Surface S

Tangential component ignored

Stoke’s Theorem


L S

F d F ds

Stoke’s theorem allows us to write a closed-contour line integral as a surface integral.

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Divergence Theorem


S V

F ds F dv

The divergence theorem allows us to write a closed-contour surface integral as a volume integral.

Curl of Gradient is Zero


0f

The curl of the gradient of any scalar function is always zero.

Why?

1. The gradient f always points in the direction that f increases.

2. If a vector field as curl, then that vector field forms closed loops.

3. How can a function always be increasing around a closed loop?

M. C. Escher (1898 – 1972)

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Product Rule for Divergence


fA f A A f

ˆ ˆ ˆx x y y z zfA fA a fA a fA a

yx zx y z

AA Af f ff A f A f Ax x y y z z

x y zfA fA fAx y z

yx zx y z

AA A f f ff A A A

x y z x y z

f A

A f

Documents

Vector Calculus Review - University of Texas at El Pasoemlab.utep.edu/ee3321emf/Lecture 2 -- Vector Calculus Review.pdf · zx yz A Bz A By A Bx Step 3 –Make ... Vector Calculus