MATLAB is a package plan that is widely available for digital computing machines at a big figure of universities and on a big assortment of machines. As will be noted in this text, we will do extended usage of it. The two and three dimensional plotting capablenesss will be exploited since a image or a graph can normally help in the physical reading of an equation. Herein, we will briefly present an debut of several germane characteristics of this plan that will be utile for electromagnetic theory. Assorted maps such as clean-cut maps appear in a MATLAB library that can be easy called and used. The user can custom-make and add to this list by composing a plan in a “ .m ” ( dot m ) file. Several illustrations and MATLAB plans will be included throughout this text. Extra plans are included in Appendix E. Matrix operations will non be examined since their application will have minimum attending in this text We assume that the reader is able to name MATLAB and have the familiar MATLAB prompt “ & gt ; & gt ; ” appear on the screen. Typing the words “ help subject ” after the prompt brings on-screen aid to the user.

As an illustration, we type

& gt ; & gt ; x = 3

ten =

3 ( 1.80 )

## & gt ; & gt ;

The computing machine has assigned a value for the variable ten that it will retrieve until it is changed or until we exit the plan. It is ready for the following input. Let us type Y = 4 and press the return key. MATLAB returns

& gt ; & gt ; y = 4

Y =

4 ( 1.81 )

## & gt ; & gt ;

Mathematical operations of these two Numberss follow and we write a mathematical operation at the prompt. In the tabular array given below, the undermentioned three lines will look after we push the return key.

Addition

Subtraction

Generation

Division

& gt ; & gt ; z = ten + Y

& gt ; & gt ; z = x – Y

& gt ; & gt ; z = ten * Y

& gt ; & gt ; z = ten / Y

omega =

omega =

omega =

omega =

7

– 1

12

0.7500

## & gt ; & gt ;

## & gt ; & gt ;

## & gt ; & gt ;

## & gt ; & gt ;

Note the four topographic point truth in the last column. This can be changed by the user.

As written, these operations appear to “ waste ” a considerable sum of computing machine screen since they instantly cause the computing machine to respond. We can conserve the screen by typing the semicolon “ ; ” after each operation. For illustration, typing the undermentioned sequence after the prompt

& gt ; & gt ; x = 3 ; y = 4 ; z = x + Y

omega =

7 ( 1.82 )

## & gt ; & gt ;

combines several lines. The semicolon will be really utile in a drawn-out computation if we do non wish to expose intermediate consequences. Another utile tool to retrieve is the symbol “ % ” since anything typed on the line after it will have no attending by the computing machine. It is a convenient manner to add remarks to a plan or to an operation.

A vector can be specified in MATLAB by saying its three constituents. We will utilize a capital missive to place a vector in utilizing MATLAB. Lower instance letters will be reserved for scalar variables or invariables. This is non required but it does add lucidity to the work. For illustration, a vector A = 1ux + 2uy + 3uz is written as

& gt ; & gt ; A = [ 1 2 3 ]

A =

1 2 3 ( 1.83 )

## & gt ; & gt ;

We must infix a infinite between the constituents of the vectors that are Numberss. A 2nd vector B = 2ux + 3uy + 4uz is written

& gt ; & gt ; B

= [ 2 3 4 ]

B =

2 3 4 ( 1.84 )

## & gt ; & gt ;

Having stored the two vectors A and B in the computing machine, we can execute the undermentioned mathematical operations. The vectors can be added C = A + B by typing

& gt ; & gt ; C = A + B

C =

3 5 7 ( 1.85 )

## & gt ; & gt ;

The vector is interpreted as C = 3 ux + 5 uy + 7 uz.

The two vectors can besides be subtracted D = A – Bacillus

& gt ; & gt ; D = A – Bacillus

D =

-1 -1 -1 ( 1.86 )

## & gt ; & gt ;

The vector is written as D = -1 ux + -1 uy + -1 uz

It is besides possible to execute the scalar and the vector merchandises of the two vectors. MATLAB denotes these merchandises with the footings point and cross severally. The scalar merchandise degree Celsius = A aˆ? B is given by

& gt ; & gt ; c = point ( A, B )

degree Celsiuss =

20 ( 1.87 )

## & gt ; & gt ;

We, of class, will obtain the same consequence if we had typed

& gt ; & gt ; c = point ( B, A )

degree Celsiuss =

20 ( 1.88 )

## & gt ; & gt ;

The vector merchandise C = A ten B is given by

& gt ; & gt ; C = cross ( A, B )

C =

-1 2 -1 ( 1.89 )

## & gt ; & gt ;

We write the vector C = – 1ux + 2 uy – 1 uz. For these two vectors, we find that

D = B x A = – C as will be observed after typing

& gt ; & gt ; D = cross ( B, A )

D =

1 -2 1 ( 1.90 )

## & gt ; & gt ;

which we write as D = + 1ux – 2 uy + 1 uz. In add-on to these mathematical operations, MATLAB provides extended two and three dimensional graphical plotting modus operandis. The information to be plotted can be generated internally in a plan or it can be imported from an external plan. The bid “ fplot ” specifies and plots a known map. Labels and rubrics utilizing different founts and font sizes and manners can be placed on the graphs and the secret plans can be distinguished with different symbols. We will show several illustrations here in order to exemplify the assortment of two dimensional secret plans that are available. Extra graphs can be placed on one secret plan with the “ keep ” bid. Either axis can hold a logarithmic graduated table. In add-on, the bid “ subplot ” permits us to put more than one graph on a page, either vertically or horizontally displaced. The bid “ subplot ( 1, 2, 1 ) ” states that there are to be two graphs following to each other and this bid will be used to choose the left 1. Other bids that follow detail the features of that peculiar graph. The bid “ subplot ( 2, 1, 1 ) ” states that there are besides to be two graphs, one above the other, and this bid will be used to choose the top 1.

Let us build a series of graphs.

a ) Plot five Numberss – saloon graph

& gt ; & gt ; subplot ( 2,2,1 ) ;

& gt ; & gt ; a = [ 2 4 6 8 10 ]

& gt ; & gt ; saloon ( a )

& gt ; & gt ; xlabel ( ‘number ‘ ) ( 1.91 )

& gt ; & gt ; ylabel ( ‘a ‘ )

& gt ; & gt ; rubric ( ‘ ( a ) Bar graph ‘ )

## & gt ; & gt ;

B ) Plot one set of Numberss versus another set of Numberss – symbol

& gt ; & gt ; subplot ( 2,2,2 ) ;

& gt ; & gt ; a = [ 2 4 6 8 10 ] ;

& gt ; & gt ; b = [ 5 4 3 2 1 ] ;

& gt ; & gt ; secret plan ( a, B, ‘* ‘ ) ;

& gt ; & gt ; xlabel ( ‘a ‘ ) ; ( 1.92 )

& gt ; & gt ; ylabel ( ‘b ‘ ) ;

& gt ; & gt ; rubric ( ‘ ( B ) Plot of two Numberss ‘ ) ;

& gt ; & gt ; grid

## & gt ; & gt ;

degree Celsius ) Plot a known map between prescribed bounds

& gt ; & gt ; subplot ( 2,2,3 ) ;

& gt ; & gt ; fplot ( ‘sin ‘ , [ 0 4*pi, ] ) ;

& gt ; & gt ; xlabel ( ‘theta ‘ ) ;

& gt ; & gt ; ylabel ( ‘Y ‘ ) ; ( 1.93 )

& gt ; & gt ; rubric ( ‘ ( degree Celsius ) A sine moving ridge ‘ )

## & gt ; & gt ;

vitamin D ) Plot a known map between prescribed bounds – increases

& gt ; & gt ; subplot ( 2,2,4 ) ;

& gt ; & gt ; x= [ 0:0.5:3 ] ;

& gt ; & gt ; y=exp ( x ) ;

& gt ; & gt ; secret plan ( x, y, ‘o ‘ ) ;

& gt ; & gt ; xlabel ( ‘X ‘ ) ;

& gt ; & gt ; ylabel ( ‘Y ‘ ) ; ( 1.94 )

& gt ; & gt ; rubric ( ‘ ( vitamin D ) Exponential ‘ )

& gt ; & gt ; grid

## & gt ; & gt ;

The graphs, shown in Figure 1-24 have labels on their axes and a rubric.

Figure 1-24. Two dimensional graphs. ( a ) Bar graph. ( B ) Plot of a versus B. ( degree Celsius ) Functional secret plan of a sine moving ridge. ( vitamin D ) Plot of an exponential at distinct points.

We can custom-make a graph by altering the features of the line. This is done with an operation called the “ grip. ” For illustration, allow us ( a ) foremost secret plan two rhythms of a sine moving ridge utilizing 100 points. We have broken the line between 0 and 4Iˆ into 100 points. These values, as computed in ( 1.95 ) , are stored in the computing machine. We will so modify the graph: ( B ) by altering the line thickness, and ( degree Celsius ) by altering the line manner.

In the above treatment, we have been forced to separately compose a short plan utilizing the MATLAB linguistic communication. Each plan was separately written and debugged to our satisfaction. MATLAB contains an extended library of plans that are built into it. For illustration, in ( 1.93 ) , we wrote

“ Y = wickedness ( T ) ; ”

severally. The “ wickedness ” map is incorporated into this library.

The inquiry so arises, “ Can a plan written by us which we expect to utilize once more besides be included in the library? ” The reply is “ Yes ” and the procedure is given the name of making a “ .m ” ( dot m ) file with its alone name, say “ custom.m ” The creative activity of the “ custom.m ” file involves a text editor, the signifier of which depends upon the local computing machine or work station to which the user has entree. Once the file is created, it becomes a portion of our library. In order to utilize this file, all we have to make is type the word “ usage ” after the prompt “ & gt ; & gt ; ” and the file is activated at that point. These files are often shared over the cyberspace.

Let us besides create a “ dot m ” file and name it “ si.m ” . Hence, whenever we type “ Si ” after the MATLAB prompt & gt ; & gt ; , these three graphs will look. Rather than infix a rubric in the graphs, we insert the letters ( a ) , ( B ) and ( degree Celsius ) . This is accomplished with the bid “ s= ‘ ( a ) ‘ . ” The bid “ text ” locates the missive on the graph. the bid “ s ( 2 ) ” sequences the missive. Variations of this labeling are included in Appendix E. The MATLAB plan “ si.m ” is

clear

clg

s= ‘ ( a ) ‘ ; ( 1.95 )

t=linspace ( 0,4*pi,100 ) ;

y=sin ( T ) ;

subplot ( 1,3,1 )

hL=line ( T, Y ) ;

text ( 15,1, s ) ( 1.96 )

s ( 2 ) =setstr ( s ( 2 ) +1 ) ;

subplot ( 1,3,2 )

hL=line ( T, Y ) ;

set ( hectoliter, ‘linewidth’,3 ) ; ( 1.97 )

text ( 15,1, s )

s ( 2 ) =setstr ( s ( 2 ) +1 ) ;

subplot ( 1,3,3 )

hL=line ( T, Y ) ;

set ( hectoliter, ‘linestyle ‘ , ‘ — ‘ ) ( 1.98 )

text ( 15,1, s )

s ( 2 ) =setstr ( s ( 2 ) +1 ) ;

The consequences are shown in Figure 1-25.

Figure 1-25. Two dimensional graphs. ( a ) Sine moving ridge. ( B ) Change line thickness. ( hundred ) Change linestyle. The axes can be labeled as in Figure 1-24.

In add-on to plotting the graphs in Cartesian co-ordinates, we can besides plot them in polar co-ordinates. Since we have the Numberss from the old figure already stored in the computing machine, we merely will hold to make the secret plan. A word of cautiousness must, nevertheless, be invoked. If we were to compose the MATLAB bid “ polar ( t, Y ) ” , merely one half of the figure would look. This is because Y = wickedness ( x ) is negative in the scope Iˆ & lt ; x & lt ; 2Iˆ . This quandary can be removed by happening the absolute value of Y. The manner of the line can besides be modified. The consequences are shown in Figureb1-26. In Chapter 7, an extra technique to make a polar graph will be suggested.

Polar secret plan

& gt ; & gt ; x = acrylonitrile-butadiene-styrenes ( Y ) ;

& gt ; & gt ; subplot ( 1,2,1 )

& gt ; & gt ; polar ( t, x ) ( 1.99 )

& gt ; & gt ; text ( -.5, -1.4, ‘ ( a ) polar secret plan ‘ )

## & gt ; & gt ;

Change linestyle

& gt ; & gt ; subplot ( 1,2,2 )

& gt ; & gt ; polar ( t, x, ‘ — ‘ ) ( 1.100 )

& gt ; & gt ; tex ( -.9, -1.4, ‘ ( B ) alteration linestyle ‘ )

## & gt ; & gt ;

Figure 1-26. Examples of a polar secret plan. ( a ) Normal linestyle. ( B ) Dashed line.

We can besides custom-make a graph by adding text to it, altering founts, or altering the size of the type. This is besides done with an operation called the “ grip. ” For illustration, allow us ( a ) foremost plot a Gaussian signal Y = exp [ -q2 ] . In MATLAB, we write this as Y = exp ( -t.^2 ) . The period that separates the T and the ^ is important. It causes each value of T to be individually squared. We will obtain a sequence of values. If we divide one map by another, a period must follow the term in the numerator. ( B ) We will add text to the graph and alter the size of the fount in the labels on the axes. One of the labels will utilize the symbol fount. The MATLAB plan is given below.

& gt ; & gt ; t = linspace ( -2,2,100 ) ;

& gt ; & gt ; y = exp ( -t.^2 ) ;

& gt ; & gt ; secret plan ( T, Y )

& gt ; & gt ; t = -1.8 ; y = 0.8 ; ( 1.101 )

& gt ; & gt ; hT = text ( T, Y, ‘Gaussian ‘ )

& gt ; & gt ; xlabel ( ‘ Q ‘ , ‘fontsize ‘ , 18, ‘fontname ‘ , ‘symbol ‘ )

& gt ; & gt ; ylabel ( ‘y ‘ , ‘fontsize ‘ , 18 )

## & gt ; & gt ;

The ensuing graph with the customized text and axis labels is shown in Figure

1-27. Note that Q is somewhat displaced from the centre of the graph for lucidity.

Figure 1-27. Graph with added text and customized axis labels.

Three dimensional graphs can besides be created with MATLAB. The secret plans can be customized and the resulting figures can be viewed from different locations. The default position is ( -37.5, 30 ) . Each graph can go a subplot in a big figure as was done antecedently. We illustrate the plan by happening the possible V caused by a dipole charge which we will analyze in the following chapter. This possible is relative to

( 1.102 )

where the charges Q1 = – Q2 = Q and the distances R1 and R2 are the distances between the charge location and the point of observation. The distance R1 is computed from. In MATLAB, this is written as

R1 = ( ( x – x1 ) .^2 + ( y – y1 ) .^2 ) .^1/2 ( 1.103 )

The point “ . ” that appears before the circumflex symbol “ ^ ” indicates that an action will be taken on each single point. We will calculate the potency at equal intervals in the xy plane. In order to avoid the uniqueness of calculating the potency at the location of the charges, we place the charges at

( x1 = 0, y1 = + 1/2 ) and ( x2 = 0, y2 = – 1/2 ) .

The MATLAB plan is ( Note the periods in ( 1.104 ) )

& gt ; & gt ; [ x, y ] = meshgrid ( -2: .2:2, -2: .2:2. ) ;

& gt ; & gt ; R1= ( x.^2 + ( y – .5 ) .^2 ) .^.5 ;

& gt ; & gt ; R2= ( x.^2 + ( y + .5 ) .^2 ) .^.5 ; ( 1.104 )

& gt ; & gt ; V = ( 1./R1 ) – ( 1./R2 ) ;

Plot possible – default mesh position

& gt ; & gt ; subplot ( 2, 2, 1 )

& gt ; & gt ; mesh ( x, y, V )

& gt ; & gt ; xlabel ( ‘X ‘ ) ( 1.105 )

& gt ; & gt ; ylabel ( ‘Y ‘ )

& gt ; & gt ; zlabel ( ‘V ‘ )

& gt ; & gt ; rubric ( ‘ ( a ) ‘ )

## & gt ; & gt ;

Plot possible – alteration position

& gt ; & gt ; subplot ( 2, 2, 2 )

& gt ; & gt ; mesh ( x, y, V )

& gt ; & gt ; xlabel ( ‘X ‘ ) ( 1.106 )

& gt ; & gt ; ylabel ( ‘Y ‘ )

& gt ; & gt ; zlabel ( ‘V ‘ )

& gt ; & gt ; rubric ( ‘ ( B ) ‘ )

## & gt ; & gt ;

Plot possible – changed position – equipotential contours

& gt ; & gt ; subplot ( 2, 2, 3 )

& gt ; & gt ; meshc ( x, y, V )

& gt ; & gt ; xlabel ( ‘X ‘ ) ( 1.107 )

& gt ; & gt ; ylabel ( ‘Y ‘ )

& gt ; & gt ; zlabel ( ‘V ‘ )

& gt ; & gt ; rubric ( ‘ ( degree Celsius ) ‘ )

## & gt ; & gt ;

Plot possible – changed position – default surface

& gt ; & gt ; subplot ( 2, 2, 4 )

& gt ; & gt ; breaker ( x, y, V )

& gt ; & gt ; xlabel ( ‘X ‘ ) ( 1.108 )

& gt ; & gt ; ylabel ( ‘Y ‘ )

& gt ; & gt ; zlabel ( ‘V ‘ )

& gt ; & gt ; rubric ( ‘ ( vitamin D ) ‘ )

## & gt ; & gt ;

The graphs, shown in Figure 1-28 have labels on their axes and a rubric.

Figure 1-28. Three dimensional secret plans. ( a ) Default mesh secret plan. ( B ) Sing location changed. ( degree Celsius ) Contour secret plan beneath default secret plan. ( vitamin D ) Default surface secret plan.

We will do extended usage of MATLAB in this text in order to bring forth the figures. The plans that were used to make the figures are included with the figures in several instances. Additional drawn-out plans are given in Appendix E. All plans can be customized and modified by the reader and can go “ .m ” files. We will meet the numerical solution of differential equations in Chapter 3.

Decision

The electromagnetic Fieldss that will be described in the remainder of this book will do usage of vectors and the assorted differential operations that have been given in this chapter. The usage of generalised extraneous co-ordinates has faciliAtated the stating of vector operations in different co-ordinate systems. In add-on, the two theorems that allowed us to change over a surface built-in into a closed line built-in ( Stokes ‘ theorem ) or a volume inteAgral into a closed surface built-in ( divergence theorem ) will be really of import in deriving an apApreciation of these Fieldss. They will besides be employed in ulterior derivations to really develop the basic Torahs of electroAmagnetic theory from the equaAtions that arise from experiAmental obAservations. We have ab initio interpreted several applications of vecAtors utilizing fluids. This was done since most of us have gone to the beach at one clip or have seen mechanical systems. At this phase, electric and magnetic Fieldss may look instead opaque.

Problems

1. Find the vector that connects the two opposite corners of a regular hexahedron whose volume is a3. One corner of the regular hexahedron is located at the centre of a Cartesian co-ordinate system. Write this vector besides in footings of the magnitude times a unit vector.

2. Find the vector B from the beginning to the opposite corner that lies in the xy plane.

3. Given two vectors A = 3ux + 4uy + 5uz and B = -5ux + 4uy – 3uz, find

A + B and A – Bacillus.

4. Sketch a vector field defined by A = y2ux – xuy. The length of the vectors in the field should be relative to the field at that point. Find the magniAtude of this vector at the point ( 5,7 ) .

5. Find the scalar merchandise of the two vectors defined by A = 3ux + 4uy + 5uz and B = -5ux + 4uy – 3uz. Determine the angle between these two vectors.

6. Find the projection of a vector from the beginning to a point defined at ( 1,2,3 ) on the vector from the beginning to a point defined at ( 2,1,6 ) . Find the angle between these two vectors.

7. Express the vector field A = 3ux + 4uy + 5uz in cylindrical coordiAnates.

8 Express the vector A = 3ur + 4uj + 5uz that is in cylindrical co-ordinates into Cartesian co-ordinates.

9. Express the vector field A = 3ux + 4uy + 5uz in spherical coordiAnates.

10 Express the vector A = 3ur + 4uj + 5uq that is in spherical co-ordinates into Cartesian co-ordinates.

11. Find the vector merchandise ( A x B ) of the two vectors defined by A = 3ux + 4uy + 5uz and B = -5ux + 4uy – 3uz. Show that the vector found from

( B x A ) is in the opposite way.

12. For the vectors A = 3ux + 4uy + 5uz, B = -5ux + 4uy – 3uz, and C = 2ux +3uy + 4uz ; show that A x ( B x C ) = B ( A aˆ? C ) – C ( A aˆ? B ) .

13. For the vectors A = 3ur + 4uj and B = 4ur + 3uj, and C = 5ur + 5uj show that A x ( B x C ) = B ( A aˆ? C ) – C ( A aˆ? B ) .

14. Find the country of the parallelogram utilizing vector notation. Compare your consequence with that found diagrammatically.

15. Show that we can utilize the vector definitions A aˆ? B = 0 and A x B = 0 to exApress that two vectors are perpendicular and parallel to each other.

16. Let A = -2ux + 3uy + 4uz ; B = 7ux + 1uy + 2uz ; and C = -1ux + 2uy + 4uz. Find ( a ) A ten B. ( B ) ( A ten B ) aˆ? C. ( degree Celsius ) A aˆ? ( B x C ) .

17. Calculate the work required to travel a mass m against a force field

F = 5ux + 7uy

along the indicated diArect way from point a to point B.

18. Calculate the work required to travel a mass m against a force field

F = yux + xuy along the way rudiment and along the way adc. Is this field conAservative?

19 Calculate the work required to travel a mass m against a force field

F = rur + juj along the way rudiment.

20 Calculate the work required to travel a mass m against a force field F = juj if the radius of the circle is a and 0 a‰¤ J a‰¤ 2Iˆ .

21. Calculate the closed surface inteAgral A aˆ? Ds if A = xux + yuy and the surface is the surface of a regular hexahedron.

22. Measure the closed surface integral of the vector A = xyz ux + xyz uy + xyz uz over the cubelike surface shown in Problem 21.

23 Evaluate the closed surface integral of the vector A = 3 Ur over the spherical surface that has a radius a.

24 Find the surface country of a cylindriAcal surface by puting up and evaluatAing the integrals.

25. A hill can be modeled with the equation H = 10 – x2 – 3y2 where H is the lift of the hill. Find the way that a frictionless ball would take in order that it experiAenced the greatAest alteration of lift in the shortest alteration of horizontal place. Assume that the gesture of the ball is unconstrained.

26. For the hill described in job 25, find the angle Q with reAspect to the x axis that the ball makes as it rolls down the hill at the point ( 2,1 ) .

27. Find the gradient of the map H = x2yz and besides the direcAtional derived function of H specified by the unit vector u = ( ux + uy + uz ) / ( 3 ) 1/3 at the point ( 1, 2, 3 ) .

28. By direct distinction show that where

and ‘ denotes differentiation with regard to the variables x ‘ , Y ‘ , and omega ‘ .

29. Calculate the divergency of the vector A = x3y wickedness ( Iˆz ) ux + xy wickedness ( Iˆz ) uy + x2y2z2uz at the point ( 1,1,1 ) .

30 Show that the divergency theorem is valid for a regular hexahedron located at the centre of a Cartesian co-ordinate system for a vector A = xux + 2uy.

31. Show that the divergency theorem is valid for a domain of radius a located at the centre of a co-ordinate system for a vector A = R Ur

32. The H2O that flows in a channel with sides at x = 0 and x = a has a veAlocity distribution. The botAtom of the river is at omega = 0. A little paddle wheel with its axis analogue to the omega axis is inAserted into the channel and is free to roAtate. Find the comparative rates of rotary motion at the points

## .

Will the paddle wheel rotate if its axis is parallel to the x axis or the Y axis?

33. Measure the line integral of the vector map A = xux + x2yuy + xyzuz around the square contour C. Integrate x A over the surAface bounded by C. Show that this examAple satisfies Stokes ‘ theorem.

34. Show that x A = 0 if A = uj ( 1/r ) in cylindrical co-ordinates.

35 Show that x A = 0 if A = ur r2 in spherical co-ordinates.

36. In rectangular co-ordinates, verify that aˆ? x A = 0 where

A = x2y2z2 [ ux + uy + uz ]

by transporting out the elaborate distinctions.

37. In rectangular co-ordinates, verify that x o = 0 where

o = 3x2y + 4z2x

by transporting out the elaborate distinctions.

38. In rectangular co-ordinates, verify that x oA = ( o ) x A + o x A where A = x2y2z2 [ ux + uy + uz ] and o = 3x2y + 4z2x

by transporting out the elaborate distinctions.

39. In rectangular co-ordinates, verify that aˆ? ( oA ) = A aˆ? o + o aˆ? A where A = x2y2z2 [ ux + uy + uz ] and o = 3x2y + 4z2x

by transporting out the elaborate distinctions.

40. By direct distinction, show that at all points where

R a‰ 0 where.

41. Express the signal V = 100 cos ( 120 Iˆ t – 45o ) in phasor notation.

42. Given a phasor V = 10 + J 5. Find the sinusoidal signal this represents if the frequence = 60 Hz.

43. Find the phasor notation of V = cos [ 120 Iˆt – 60o ] – wickedness [ 120 Iˆt ] .

44. Find the current in the circuit if 5 = 10 cos ( 120 Iˆt )

45. Repeat job 44 with V = 10 cos ( 120 Iˆt + 45o ) .

46. Using MATLAB, compose a plan to change over grades C to degrees F. Plot the consequences.

47. Using MATLAB, compose a plan to change over a pace stick to a metre stick. Plot the consequences.

48. Using MATLAB, add the two vectors A = 3ux + 4uy + 5uz and B = 5ux + 4uy + 3uz.

49. Using MATLAB, deduct the two vectors A = 3ux + 4uy + 5uz and B = 5ux + 4uy + 3uz.

50. Using MATLAB, find A aˆ? B where A = 3ux + 4uy + 5uz and B = 5ux + 4uy + 3uz

51. Using MATLAB, find A ten B where A = 3ux + 4uy + 5uz and B = 5ux + 4uy + 3uz

52. Using MATLAB, secret plan y= exp ( -x ) on a additive and a semilog graph.

53. Using MATLAB, secret plan two rhythms of y= cos ( x ) on a additive and a polar graph.

54. Using MATLAB, secret plan y= ( x ) exp ( -x ) on a additive graph for -3 a‰¤ x a‰¤ 3..

55. Using MATLAB, secret plan y= ( x ) exp ( -x ) on a additive graph for -3 a‰¤ x a‰¤ 3. Add the secret plan of Y = x2 with a different linestyle to this graph..