Boolean Algebra. ECE 152A Winter 2012

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Boolean Algebra. ECE 152A Winter 2012"

Transcription

1 Boolen Algebr ECE 52A Wnter 22

2 Redng Assgnent Brown nd Vrnesc 2 Introducton to Logc Crcuts 2.5 Boolen Algebr 2.5. The Venn Dgr Notton nd Ternology Precedence of Opertons 2.6 Synthess Usng AND, OR nd NOT Gtes 2.6. Su-of-Products nd Product of Sus Fors Jnury, 22 ECE 52A - Dgtl Desgn Prncples 2

3 Redng Assgnent Brown nd Vrnesc (cont) 2 Introducton to Logc Crcuts (cont) 2.7 NAND nd NOR Logc Networks 2.8 Desgn Exples 2.8. Three-Wy Lght Control Multplexer Crcut Jnury, 22 ECE 52A - Dgtl Desgn Prncples 3

4 Redng Assgnent Roth 2 Boolen Algebr 2.3 Boolen Expressons nd Truth Tbles 2.4 Bsc Theores 2.5 Couttve, Assoctve, nd Dstrbutve Lws 2.6 Splfcton Theores 2.7 Multplyng Out nd Fctorng 2.8 DeMorgn s Lws Jnury, 22 ECE 52A - Dgtl Desgn Prncples 4

5 Redng Assgnent Roth (cont) 3 Boolen Algebr (Contnued) 3. Multplyng Out nd Fctorng Expressons 3.2 Exclusve-OR nd Equvlence Operton 3.3 The Consensus Theore 3.4 Algebrc Splfcton of Swtchng Expressons Jnury, 22 ECE 52A - Dgtl Desgn Prncples 5

6 Redng Assgnent Roth (cont) 4 Applctons of Boolen Algebr Mnter nd Mxter Expressons 4.3 Mnter nd Mxter Expnsons 7 Mult-Level Gte Crcuts NAND nd NOR Gtes 7.2 NAND nd NOR Gtes 7.3 Desgn of Two-Level Crcuts Usng NAND nd NOR Gtes 7.5 Crcut Converson Usng Alterntve Gte Sybols Jnury, 22 ECE 52A - Dgtl Desgn Prncples 6

7 Boolen Algebr Axos of Boolen Algebr Axos generlly presented wthout proof = + = = + = = = + = + = f X =, then X = f X =, then X = Jnury, 22 ECE 52A - Dgtl Desgn Prncples 7

8 Boolen Algebr The Prncple of Dulty fro Zv Kohv, Swtchng nd Fnte Autot Theory We observe tht ll the precedng propertes re grouped n prs. Wthn ech pr one stteent cn be obtned fro the other by nterchngng the OR nd AND opertons nd replcng the constnts nd by nd respectvely. Any two stteents or theores whch hve ths property re clled dul, nd ths qulty of dulty whch chrcterzes swtchng lgebr s known s the prncple of dulty. It stes fro the syetry of the postultes nd defntons of swtchng lgebr wth respect to the two opertons nd the two constnts. The plcton of the concept of dulty s tht t s necessry to prove only one of ech pr of stteents, nd ts dul s henceforth proved. Jnury, 22 ECE 52A - Dgtl Desgn Prncples 8

9 Boolen Algebr Sngle-Vrble Theores Theores cn be proven wth truth tbles Truth tble proof.k.., Perfect Inducton X = X + = X = X X + = X X X = X X + X = X X X = X + X = (X ) = X Jnury, 22 ECE 52A - Dgtl Desgn Prncples 9

10 Boolen Algebr Two- nd Three-Vrble Propertes Couttve X Y = Y X X + Y = Y + X Assoctve X (Y Z) = (X Y) Z X + (Y + Z) = (X + Y) + Z Dstrbutve X (Y + Z) = X Y + X Z X + (Y Z) = (X + Y) (X + Z) Jnury, 22 ECE 52A - Dgtl Desgn Prncples

11 Boolen Algebr Absorpton (Splfcton) X + X Y = X X ( X + Y ) = X X Y X Y X X Y X+Y X Jnury, 22 ECE 52A - Dgtl Desgn Prncples

12 Boolen Algebr Cobnng (Splfcton) X Y + X Y = X (X + Y) (X + Y ) = X X Y X Y X Y X X Y X+Y X X+Y Jnury, 22 ECE 52A - Dgtl Desgn Prncples 2

13 Boolen Algebr Redundnt Coverge (splfcton) X + X Y = X + Y X (X + Y) = X Y X Y X Y X X+Y X Y X +Y X Y X Jnury, 22 ECE 52A - Dgtl Desgn Prncples 3

14 Boolen Algebr The Consensus Theore XY + X Z + YZ = XY + X Z X YZ XY X Z YZ Jnury, 22 ECE 52A - Dgtl Desgn Prncples 4

15 Boolen Algebr DeMorgn s Theore (X Y) = X + Y (X + Y) = X Y X Y X Y (X Y) X +Y (X+Y) X Y Jnury, 22 ECE 52A - Dgtl Desgn Prncples 5

16 Boolen Expressons Precedence of Opertons Order of evluton. NOT 2. AND 3. OR Or forced by prentheses Exple: F = b c + b + bc + b c =, b= nd c= NOT: AND: OR: Jnury, 22 ECE 52A - Dgtl Desgn Prncples 6

17 Boolen Expressons, Logc Networks, Krnugh Mps, Truth Tbles & Tng Dgrs Derve Logc Network, Krnugh Mp, Truth Tble nd Tng Dgr fro: F = b c + b + bc + b c 3 vrbles, lterls, 4 product ters Expresson s n Stndrd Su-of-Products for.e., the functon s the su (or logcl OR) or the four product (or logcl AND) ters The lterntve stndrd for s Product-of-Sus The expresson ples structure Drect relzton wth AND, OR nd NOT functons Jnury, 22 ECE 52A - Dgtl Desgn Prncples 7

18 Boolen Expressons, Logc Networks, Krnugh Mps, Truth Tbles & Tng Dgrs Logc Network F = b c + b + bc + b c Jnury, 22 ECE 52A - Dgtl Desgn Prncples 8

19 Boolen Expressons, Logc Networks, Krnugh Mps, Truth Tbles & Tng Dgrs Krnugh Mp F = b c + b + bc + b c bc bc b c b c b Jnury, 22 ECE 52A - Dgtl Desgn Prncples 9

20 Boolen Expressons, Logc Networks, Krnugh Mps, Truth Tbles & Tng Dgrs Note possble splfcton Redundnt coverge (elntes lterl) nd bsorpton (elntes product ter) bc b c b b Jnury, 22 ECE 52A - Dgtl Desgn Prncples 2

21 Jnury, 22 ECE 52A - Dgtl Desgn Prncples 2 Boolen Expressons, Logc Networks, Krnugh Mps, Truth Tbles & Tng Dgrs Truth Tble F = b c + b + bc + b c F c b

22 Boolen Expressons, Logc Networks, Krnugh Mps, Truth Tbles & Tng Dgrs Tng Dgr (Functonl Sulton) F = b c + b + bc + b c Input Output Jnury, 22 ECE 52A - Dgtl Desgn Prncples 22

23 Mnters nd Mxters Mnter A product ter whch contns ech of the n vrbles s fctors n ether copleented or uncopleented for s clled nter Exple for 3 vrbles: b c s nter; b s not Mxter A su ter whch contns ech of the n vrbles s fctors n ether copleented or uncopleented for s clled xter For 3 vrbles: +b+c s xter; +b s not Jnury, 22 ECE 52A - Dgtl Desgn Prncples 23

24 Mnters nd Mxters Mnter nd Mxter Expnson Three vrble exple: ( )' M nd ( M )' Jnury, 22 ECE 52A - Dgtl Desgn Prncples 24

25 Su-of-Products For Cnoncl Su-of-Products (or Dsjunctve Norl) For The su of ll nters derved fro those rows for whch the vlue of the functon s tkes on the vlue or ccordng to the vlue ssued by f. Therefore ths su s n fct n lgebrc representton of f. An expresson of ths type s clled cnoncl su of products, or dsjunctve norl expresson. Kohv Jnury, 22 ECE 52A - Dgtl Desgn Prncples 25

26 Mnters nd Mxters Truth Tble fro erler exple F = b c + b + bc + b c b c F M = M = 2 M 2 = 2 3 M 3 = 3 4 M 4 = 4 5 M 5 = 5 6 M 6 = 6 7 M 7 = 7 Jnury, 22 ECE 52A - Dgtl Desgn Prncples 26

27 Jnury, 22 ECE 52A - Dgtl Desgn Prncples 27 Su-of-Products Cnoncl Su-of-Products F = b c + b + bc + b c (,2,3,4,5) ' ' ' ' ' ' ' ' ' F c b c b bc bc c b F F F F

28 Product-of-Sus For Cnoncl Product-of-Sus (or Conjunctve Norl) For An expresson fored of the product of ll xters for whch the functon tkes on the vlue s clled cnoncl product of sus, or conjunctve norl expresson. Jnury, 22 ECE 52A - Dgtl Desgn Prncples 28

29 Product-of-Sus Cnoncl Product-of-Sus F = b c + b + bc + b c F ( F M )( ( M M )( )( M 2 M 2 )( M )( 2 3 M )( M F ( M 3 )( )( M )( M M )( )( M )( M M F ( b c')( ' b' c)( ' b' c') F M (,6,7) ) 5 5 )( 6 )( M M 6 6 )( 7 )( M 7 M ) 7 ) Jnury, 22 ECE 52A - Dgtl Desgn Prncples 29

30 Jnury, 22 ECE 52A - Dgtl Desgn Prncples 3 Generl Su-of-Product (SOP) nd Product-of-Sus (POS) Fors s the Boolen vlue of the functon n the th row of n n-vrble Truth Tble ) ' ( ' ' ) ( ) ( ) )( ( n n n n M F M M M M F F

31 Jnury, 22 ECE 52A - Dgtl Desgn Prncples 3 Equvlence of SOP nd POS Fors Mnter / Mxter Lsts (,2,3,4,5) (,6,7) ' ) ' ( ' ' (,6,7) (,2,3,4,5) ) ( M F M F nd M F M F exple exple n n n n

32 Functonlly Coplete Opertons A set of opertons s sd to be functonlly coplete (or unversl) f nd only f every swtchng functon cn be expressed entrely by ens of opertons fro ths set [Snce] every swtchng functon cn be expressed n cnoncl su-of-products [nd product-of-sus] for, where ech expresson conssts of fnte nuber of swtchng vrbles, constnts nd the opertons AND, OR nd NOT [ths set of opertons s functonlly coplete] Jnury, 22 ECE 52A - Dgtl Desgn Prncples 32

33 SOP Relzton wth NAND/NAND The NAND operton s functonlly coplete Jnury, 22 ECE 52A - Dgtl Desgn Prncples 33

34 POS Relzton wth NOR/NOR The NOR operton s functonlly coplete Jnury, 22 ECE 52A - Dgtl Desgn Prncples 34

Square & Square Roots

Square & Square Roots Squre & Squre Roots Squre : If nuber is ultiplied by itself then the product is the squre of the nuber. Thus the squre of is x = eg. x x Squre root: The squre root of nuber is one of two equl fctors which

More information

Implementation of Boolean Functions through Multiplexers with the Help of Shannon Expansion Theorem

Implementation of Boolean Functions through Multiplexers with the Help of Shannon Expansion Theorem Internatonal Journal o Computer pplcatons (975 8887) Volume 62 No.6, January 23 Implementaton o Boolean Functons through Multplexers wth the Help o Shannon Expanson Theorem Saurabh Rawat Graphc Era Unversty.

More information

Newton-Raphson Method of Solving a Nonlinear Equation Autar Kaw

Newton-Raphson Method of Solving a Nonlinear Equation Autar Kaw Newton-Rphson Method o Solvng Nonlner Equton Autr Kw Ater redng ths chpter, you should be ble to:. derve the Newton-Rphson method ormul,. develop the lgorthm o the Newton-Rphson method,. use the Newton-Rphson

More information

Resistive Network Analysis. The Node Voltage Method - 1

Resistive Network Analysis. The Node Voltage Method - 1 esste Network Anlyss he nlyss of n electrcl network conssts of determnng ech of the unknown rnch currents nd node oltges. A numer of methods for network nlyss he een deeloped, sed on Ohm s Lw nd Krchoff

More information

Chapter Solution of Cubic Equations

Chapter Solution of Cubic Equations Chpter. Soluton of Cuc Equtons After redng ths chpter, ou should e le to:. fnd the ect soluton of generl cuc equton. Ho to Fnd the Ect Soluton of Generl Cuc Equton In ths chpter, e re gong to fnd the ect

More information

Basics of Counting. A note on combinations. Recap. 22C:19, Chapter 6.5, 6.7 Hantao Zhang

Basics of Counting. A note on combinations. Recap. 22C:19, Chapter 6.5, 6.7 Hantao Zhang Bscs of Countng 22C:9, Chpter 6.5, 6.7 Hnto Zhng A note on comntons An lterntve (nd more common) wy to denote n r-comnton: n n C ( n, r) r I ll use C(n,r) whenever possle, s t s eser to wrte n PowerPont

More information

5.6 POSITIVE INTEGRAL EXPONENTS

5.6 POSITIVE INTEGRAL EXPONENTS 54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL PRODUCTS AND FACTORIZATION MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

More information

Level Annuities with Payments Less Frequent than Each Interest Period

Level Annuities with Payments Less Frequent than Each Interest Period Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Symoblc approach

More information

Boolean Algebra Part 1

Boolean Algebra Part 1 Boolean Algebra Part 1 Page 1 Boolean Algebra Objectives Understand Basic Boolean Algebra Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand and Use First Basic Theorems

More information

Algebra Review. How well do you remember your algebra?

Algebra Review. How well do you remember your algebra? Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

Factoring Polynomials

Factoring Polynomials Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

More information

Inequalities for the internal angle-bisectors of a triangle

Inequalities for the internal angle-bisectors of a triangle Mtheticl Counictions 2(997), 4-45 4 Inequlities for the internl ngle-bisectors of tringle Wlther Jnous nd Šefket Arslngić Abstrct. Severl ne inequlities of type α ± for ngle-bisectors re proved. Certin

More information

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C; B-26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndom-numer genertor supplied s stndrd with ll computer systems Stn Kelly-Bootle,

More information

+ + + - - This circuit than can be reduced to a planar circuit

+ + + - - This circuit than can be reduced to a planar circuit MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

More information

Reporting Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (including SME Corporate), Sovereign and Bank Instruction Guide

Reporting Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (including SME Corporate), Sovereign and Bank Instruction Guide Reportng Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (ncludng SME Corporate), Soveregn and Bank Instructon Gude Ths nstructon gude s desgned to assst n the completon of the FIRB

More information

MULTIPLYING OUT & FACTORING

MULTIPLYING OUT & FACTORING igitl ircuit Engineering MULTIPLYING OUT & FTORING I IGITL SIGN Except for #$&@ fctoring st istributive X + X = X( + ) 2nd istributive (X + )(X + ) = X + (X + )(X + )(X + ) = X + Swp (X + )(X + ) = X +

More information

DETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc.

DETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc. Chpter 4 DETERMINANTS 4 Overview To every squre mtrix A = [ ij ] of order n, we cn ssocite number (rel or complex) clled determinnt of the mtrix A, written s det A, where ij is the (i, j)th element of

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

Incorporating Negative Values in AHP Using Rule- Based Scoring Methodology for Ranking of Sustainable Chemical Process Design Options

Incorporating Negative Values in AHP Using Rule- Based Scoring Methodology for Ranking of Sustainable Chemical Process Design Options 20 th Europen ymposum on Computer Aded Process Engneerng ECAPE20. Perucc nd G. Buzz Ferrrs (Edtors) 2010 Elsever B.V. All rghts reserved. Incorportng Negtve Vlues n AHP Usng Rule- Bsed corng Methodology

More information

Irregular Repeat Accumulate Codes 1

Irregular Repeat Accumulate Codes 1 Irregulr epet Accumulte Codes 1 Hu Jn, Amod Khndekr, nd obert McElece Deprtment of Electrcl Engneerng, Clforn Insttute of Technology Psden, CA 9115 USA E-ml: {hu, mod, rjm}@systems.cltech.edu Abstrct:

More information

Section A-4 Rational Expressions: Basic Operations

Section A-4 Rational Expressions: Basic Operations A- Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr open-topped bo is to be constructed out of 9- by 6-inch sheets of thin crdbord by cutting -inch squres out of ech corner nd bending the

More information

and thus, they are similar. If k = 3 then the Jordan form of both matrices is

and thus, they are similar. If k = 3 then the Jordan form of both matrices is Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

More information

Extending Probabilistic Dynamic Epistemic Logic

Extending Probabilistic Dynamic Epistemic Logic Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set

More information

VRT012 User s guide V0.1. Address: Žirmūnų g. 27, Vilnius LT-09105, Phone: (370-5) 2127472, Fax: (370-5) 276 1380, Email: info@teltonika.

VRT012 User s guide V0.1. Address: Žirmūnų g. 27, Vilnius LT-09105, Phone: (370-5) 2127472, Fax: (370-5) 276 1380, Email: info@teltonika. VRT012 User s gude V0.1 Thank you for purchasng our product. We hope ths user-frendly devce wll be helpful n realsng your deas and brngng comfort to your lfe. Please take few mnutes to read ths manual

More information

Traffic-light a stress test for life insurance provisions

Traffic-light a stress test for life insurance provisions MEMORANDUM Date 006-09-7 Authors Bengt von Bahr, Göran Ronge Traffc-lght a stress test for lfe nsurance provsons Fnansnspetonen P.O. Box 6750 SE-113 85 Stocholm [Sveavägen 167] Tel +46 8 787 80 00 Fax

More information

greatest common divisor

greatest common divisor 4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no

More information

3 The Utility Maximization Problem

3 The Utility Maximization Problem 3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best

More information

Understanding Logic Design

Understanding Logic Design Understanding Logic Design ppendix of your Textbook does not have the needed background information. This document supplements it. When you write add DD R0, R1, R2, you imagine something like this: R1

More information

AVR32723: Sensor Field Oriented Control for Brushless DC motors with AT32UC3B0256. 32-bit Microcontrollers. Application Note. Features.

AVR32723: Sensor Field Oriented Control for Brushless DC motors with AT32UC3B0256. 32-bit Microcontrollers. Application Note. Features. AVR7: Sensor Feld Orented Control for Brushless DC motors wth ATUCB056 Fetures Stndlone Spce Vector Modulton lbrry for AVR UC mcrocontroller. Prk nd Clrke mthemtcl trnsformton lbrry for AVR UC mcrocontroller.

More information

Positive Integral Operators With Analytic Kernels

Positive Integral Operators With Analytic Kernels Çnky Ünverte Fen-Edeyt Fkülte, Journl of Art nd Scence Sy : 6 / Arl k 006 Potve ntegrl Opertor Wth Anlytc Kernel Cn Murt D KMEN Atrct n th pper we contruct exmple of potve defnte ntegrl kernel whch re

More information

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6)

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6) Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called

More information

Multiplication Algorithms for Radix-2 RN-Codings and Two s Complement Numbers

Multiplication Algorithms for Radix-2 RN-Codings and Two s Complement Numbers Multplcaton Algorthms for Radx- RN-Codngs and Two s Complement Numbers Jean-Luc Beuchat Projet Arénare, LIP, ENS Lyon 46, Allée d Itale F 69364 Lyon Cedex 07 jean-luc.beuchat@ens-lyon.fr Jean-Mchel Muller

More information

Faraday's Law of Induction

Faraday's Law of Induction Introducton Faraday's Law o Inducton In ths lab, you wll study Faraday's Law o nducton usng a wand wth col whch swngs through a magnetc eld. You wll also examne converson o mechanc energy nto electrc energy

More information

Generalized Inverses: How to Invert a Non-Invertible Matrix

Generalized Inverses: How to Invert a Non-Invertible Matrix Generlized Inverses: How to Invert Non-Invertible Mtrix S. Swyer September 7, 2006 rev August 6, 2008. Introduction nd Definition. Let A be generl m n mtrix. Then nturl question is when we cn solve Ax

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00

Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00 COMP20212 Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE Digitl Design Techniques Dte: Fridy 16 th My 2008 Time: 14:00 16:00 Plese nswer ny THREE Questions from the FOUR questions provided

More information

Matrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA

Matrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA CHAPTER 1 Mtrix Algebr PREAMBLE Tody, the importnce of mtrix lgebr is of utmost importnce in the field of physics nd engineering in more thn one wy, wheres before 1925, the mtrices were rrely used by the

More information

Learning Outcomes. Computer Systems - Architecture Lecture 4 - Boolean Logic. What is Logic? Boolean Logic 10/28/2010

Learning Outcomes. Computer Systems - Architecture Lecture 4 - Boolean Logic. What is Logic? Boolean Logic 10/28/2010 /28/2 Lerning Outcomes At the end of this lecture you should: Computer Systems - Architecture Lecture 4 - Boolen Logic Eddie Edwrds eedwrds@doc.ic.c.uk http://www.doc.ic.c.uk/~eedwrds/compsys (Hevily sed

More information

ORIGIN DESTINATION DISAGGREGATION USING FRATAR BIPROPORTIONAL LEAST SQUARES ESTIMATION FOR TRUCK FORECASTING

ORIGIN DESTINATION DISAGGREGATION USING FRATAR BIPROPORTIONAL LEAST SQUARES ESTIMATION FOR TRUCK FORECASTING ORIGIN DESTINATION DISAGGREGATION USING FRATAR BIPROPORTIONAL LEAST SQUARES ESTIMATION FOR TRUCK FORECASTING Unversty of Wsconsn Mlwukee Pper No. 09-1 Ntonl Center for Freght & Infrstructure Reserch &

More information

Derive the material derivative

Derive the material derivative Fld Mechncs: Dertons & Proofs Dere the mterl derte No, f: Then: (,,, t) ρ (.) δρ ρ ρ δt δ (.) t Dde (.) bδ t, nd te lmt: δρ lm δt 0 δt ρ δ lm t δt 0 δt (.) No: lm δ δt 0 δt t (.4) We defne the LH of (.)

More information

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange

More information

Small-Signal Analysis of BJT Differential Pairs

Small-Signal Analysis of BJT Differential Pairs 5/11/011 Dfferental Moe Sall Sgnal Analyss of BJT Dff Par 1/1 SallSgnal Analyss of BJT Dfferental Pars Now lets conser the case where each nput of the fferental par conssts of an entcal D bas ter B, an

More information

2.016 Hydrodynamics Prof. A.H. Techet

2.016 Hydrodynamics Prof. A.H. Techet .01 Hydrodynics Reding #.01 Hydrodynics Prof. A.H. Techet Added Mss For the cse of unstedy otion of bodies underwter or unstedy flow round objects, we ust consider the dditionl effect (force) resulting

More information

Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Luby s Alg. for Maximal Independent Sets using Pairwise Independence Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

More information

Lesson 28 Psychrometric Processes

Lesson 28 Psychrometric Processes 1 Lesson 28 Psychrometrc Processes Verson 1 ME, IIT Khrgpur 1 2 The specfc objectves of ths lecture re to: 1. Introducton to psychrometrc processes nd ther representton (Secton 28.1) 2. Importnt psychrometrc

More information

Exponents base exponent power exponentiation

Exponents base exponent power exponentiation Exonents We hve seen counting s reeted successors ddition s reeted counting multiliction s reeted ddition so it is nturl to sk wht we would get by reeting multiliction. For exmle, suose we reetedly multily

More information

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001 CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic

More information

Laws of Electromagnetism

Laws of Electromagnetism There are four laws of electromagnetsm: Laws of Electromagnetsm The law of Bot-Savart Ampere's law Force law Faraday's law magnetc feld generated by currents n wres the effect of a current on a loop of

More information

Mathematics Higher Level

Mathematics Higher Level Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:

More information

Fuzzy Clustering for TV Program Classification

Fuzzy Clustering for TV Program Classification Fuzzy Clusterng for TV rogrm Clssfcton Yu Zhwen Northwestern olytechncl Unversty X n,.r.chn, 7007 yuzhwen77@yhoo.com.cn Gu Jnhu Northwestern olytechncl Unversty X n,.r.chn, 7007 guh@nwpu.edu.cn Zhou Xngshe

More information

Basic Queueing Theory M/M/* Queues. Introduction

Basic Queueing Theory M/M/* Queues. Introduction Basc Queueng Theory M/M/* Queues These sldes are created by Dr. Yh Huang of George Mason Unversty. Students regstered n Dr. Huang's courses at GMU can ake a sngle achne-readable copy and prnt a sngle copy

More information

How Much to Bet on Video Poker

How Much to Bet on Video Poker How Much to Bet on Vdeo Poker Trstan Barnett A queston that arses whenever a gae s favorable to the player s how uch to wager on each event? Whle conservatve play (or nu bet nzes large fluctuatons, t lacks

More information

Section 7-4 Translation of Axes

Section 7-4 Translation of Axes 62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

More information

Maitra Cascade Minimization

Maitra Cascade Minimization Matra Cascade Mnmzaton Voudours Dmtros Dr. apakonstantnou George Natonal Techncal Unversty o Athens Basc Dentons Comple Matra Term [] Constant 0 Boolean uncton Lteral M Matra Term a lteral G Arbtrary varable

More information

ALABAMA ASSOCIATION of EMERGENCY MANAGERS

ALABAMA ASSOCIATION of EMERGENCY MANAGERS LBM SSOCTON of EMERGENCY MNGERS ON O PCE C BELLO MER E T R O CD NCY M N G L R PROFESSONL CERTFCTON PROGRM .. E. M. CERTFCTON PROGRM 2014 RULES ND REGULTONS 1. THERE WLL BE FOUR LEVELS OF CERTFCTON. BSC,

More information

Calculus of variations with fractional derivatives and fractional integrals

Calculus of variations with fractional derivatives and fractional integrals Anis do CNMAC v.2 ISSN 1984-820X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro 3810-193 Aveiro, Portugl

More information

CS 316: Gates and Logic

CS 316: Gates and Logic CS 36: Gtes nd Logi Kvit Bl Fll 27 Computer Siene Cornell University Announements Clss newsgroup reted Posted on we-pge Use it for prtner finding First ssignment is to find prtners P nd N Trnsistors PNP

More information

A Hadoop Job Scheduling Model Based on Uncategorized Slot

A Hadoop Job Scheduling Model Based on Uncategorized Slot Journl of Communctons Vol. 10, No. 10, October 2015 A Hdoop Job Schedulng Model Bsed on Unctegored Slot To Xue nd Tng-tng L Deprtment of Computer Scence, X n Polytechnc Unversty, X n 710048, Chn Eml: xt73@163.com;

More information

Implementation of Deutsch's Algorithm Using Mathcad

Implementation of Deutsch's Algorithm Using Mathcad Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages - n "Machnes, Logc and Quantum Physcs"

More information

Communication Networks II Contents

Communication Networks II Contents 8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

More information

Compilers. 3 rd year Spring term. Mick O Donnell: Alfonso Ortega: Topic 5: Semantic analysis

Compilers. 3 rd year Spring term. Mick O Donnell: Alfonso Ortega: Topic 5: Semantic analysis Complers 3 rd year Sprng term Mck O Donnell: mchael.odonnell@uam.es Alfonso Ortega: alfonso.ortega@uam.es Topc 5: Semantc analyss 5.0 Introducton 1 Semantc analyss What s the Semantc Analyser? Set of routnes

More information

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1. Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose

More information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn 33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

More information

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.

More information

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100 hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

More information

Section 5.3 Annuities, Future Value, and Sinking Funds

Section 5.3 Annuities, Future Value, and Sinking Funds Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme

More information

Operating Network Load Balancing with the Media Independent Information Service for Vehicular Based Systems

Operating Network Load Balancing with the Media Independent Information Service for Vehicular Based Systems CHI MA et l: OPERATING NETWORK LOAD BALANCING WITH THE MEDIA INDEPENDENT... Opertng Network Lod Blncng wth the Med Independent Inforton Servce for Vehculr Bsed Systes Ch M, End Fllon, Yunsong Qo, Brn Lee

More information

Time Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University

Time Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University Tme Seres Analyss n Studes of AGN Varablty Bradley M. Peterson The Oho State Unversty 1 Lnear Correlaton Degree to whch two parameters are lnearly correlated can be expressed n terms of the lnear correlaton

More information

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a. Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles

More information

MENT STATUS. cd subject to resolution of indicated comments, :ptance or approval of design detiuls, calculations, the supplier and does not relieve

MENT STATUS. cd subject to resolution of indicated comments, :ptance or approval of design detiuls, calculations, the supplier and does not relieve . COGMA-A-79, Rev. STRUCTURAL NTrGRlT ASSSSMNT OF TH LOW ACTVT WAST FACLT (LAW) SCONDAR OFFGASNSSL VNT PROCSS SSTM (LVP) MSCLLANOUS TRATMNT UNT (MTU) SUBSSTMS ANCLLAR QUPMNT JobNo. 259 MNT STATUS cd subject

More information

Research on performance evaluation in logistics service supply chain based unascertained measure

Research on performance evaluation in logistics service supply chain based unascertained measure Suo Junun, L Yncng, Dong Humn Reserch on performnce evluton n logstcs servce suppl chn bsed unscertned mesure Abstrct Junun Suo *, Yncng L, Humn Dong Hebe Unverst of Engneerng, Hndn056038, Chn Receved

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

WiMAX DBA Algorithm Using a 2-Tier Max-Min Fair Sharing Policy

WiMAX DBA Algorithm Using a 2-Tier Max-Min Fair Sharing Policy WMAX DBA Algorthm Usng 2-Ter Mx-Mn Fr Shrng Polcy Pe-Chen Tseng 1, J-Yn Ts 2, nd Wen-Shyng Hwng 2,* 1 Deprtment of Informton Engneerng nd Informtcs, Tzu Ch College of Technology, Hulen, Twn pechen@tccn.edu.tw

More information

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

Algebra (Expansion and Factorisation)

Algebra (Expansion and Factorisation) Chapter10 Algebra (Expansion and Factorisation) Contents: A B C D E F The distributive law Siplifying algebraic expressions Brackets with negative coefficients The product (a + b)(c + d) Geoetric applications

More information

Comparison of Control Strategies for Shunt Active Power Filter under Different Load Conditions

Comparison of Control Strategies for Shunt Active Power Filter under Different Load Conditions Comparson of Control Strateges for Shunt Actve Power Flter under Dfferent Load Condtons Sanjay C. Patel 1, Tushar A. Patel 2 Lecturer, Electrcal Department, Government Polytechnc, alsad, Gujarat, Inda

More information

Boolean Algebra (cont d) UNIT 3 BOOLEAN ALGEBRA (CONT D) Guidelines for Multiplying Out and Factoring. Objectives. Iris Hui-Ru Jiang Spring 2010

Boolean Algebra (cont d) UNIT 3 BOOLEAN ALGEBRA (CONT D) Guidelines for Multiplying Out and Factoring. Objectives. Iris Hui-Ru Jiang Spring 2010 Boolean Algebra (cont d) 2 Contents Multiplying out and factoring expressions Exclusive-OR and Exclusive-NOR operations The consensus theorem Summary of algebraic simplification Proving validity of an

More information

ErrorPropagation.nb 1. Error Propagation

ErrorPropagation.nb 1. Error Propagation ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then

More information

Multiple stage amplifiers

Multiple stage amplifiers Multple stage amplfers Ams: Examne a few common 2-transstor amplfers: -- Dfferental amplfers -- Cascode amplfers -- Darlngton pars -- current mrrors Introduce formal methods for exactly analysng multple

More information

AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

More information

MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!

MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent! MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

Optimal Pricing Scheme for Information Services

Optimal Pricing Scheme for Information Services Optml rcng Scheme for Informton Servces Shn-y Wu Opertons nd Informton Mngement The Whrton School Unversty of ennsylvn E-ml: shnwu@whrton.upenn.edu e-yu (Shron) Chen Grdute School of Industrl Admnstrton

More information

Description of the Force Method Procedure. Indeterminate Analysis Force Method 1. Force Method con t. Force Method con t

Description of the Force Method Procedure. Indeterminate Analysis Force Method 1. Force Method con t. Force Method con t Indeternate Analyss Force Method The force (flexblty) ethod expresses the relatonshps between dsplaceents and forces that exst n a structure. Prary objectve of the force ethod s to deterne the chosen set

More information

A Secure Password-Authenticated Key Agreement Using Smart Cards

A Secure Password-Authenticated Key Agreement Using Smart Cards A Secure Password-Authentcated Key Agreement Usng Smart Cards Ka Chan 1, Wen-Chung Kuo 2 and Jn-Chou Cheng 3 1 Department of Computer and Informaton Scence, R.O.C. Mltary Academy, Kaohsung 83059, Tawan,

More information

Vectors 2. 1. Recap of vectors

Vectors 2. 1. Recap of vectors Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

More information

MAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date

MAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPP-ATBD-ClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller

More information

PROGRAMOWANIE STRUKTUR CYFROWYCH

PROGRAMOWANIE STRUKTUR CYFROWYCH PROGRAMOWANIE STRUKTUR CYFROWYCH FPGA r inż. Igncy Pryk, UJK Kielce Mteriły źrółowe:. Slies to ccompny the textbook Digitl Design, First Eition, by Frnk Vhi, John Wiley n Sons Publishers, 7, http://www.vhi.com.

More information

Unit 6: Exponents and Radicals

Unit 6: Exponents and Radicals Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -

More information

Elastic Systems for Static Balancing of Robot Arms

Elastic Systems for Static Balancing of Robot Arms . th World ongress n Mechans and Machne Scence, Guanajuato, Méco, 9- June, 0 _ lastc Sstes for Statc alancng of Robot rs I.Sonescu L. uptu Lucana Ionta I.Ion M. ne Poltehnca Unverst Poltehnca Unverst Poltehnca

More information

BERNSTEIN POLYNOMIALS

BERNSTEIN POLYNOMIALS On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

More information

A new algorithm for generating Pythagorean triples

A new algorithm for generating Pythagorean triples A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf

More information

Multi-Market Trading and Liquidity: Theory and Evidence

Multi-Market Trading and Liquidity: Theory and Evidence Mult-Mrket Trdng nd Lqudty: Theory nd Evdence Shmuel Bruch, G. Andrew Kroly, b* Mchel L. Lemmon Eccles School of Busness, Unversty of Uth, Slt Lke Cty, UT 84, USA b Fsher College of Busness, Oho Stte Unversty,

More information

Scalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra

Scalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra Sclr nd Vector Quntities : VECTO NLYSIS Vector lgebr sclr is quntit hving onl mgnitude (nd possibl phse). Emples: voltge, current, chrge, energ, temperture vector is quntit hving direction in ddition to

More information