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  • . .. ()

    ..

    MathCAD 15

    2013

  • 2

    681.3.06

    , . . MathCAD

    [] / . . . : , 2013. 114 .

    .

    2 10.02.13

    MathCAD 15.

    ,

    .

    .

    , ,

    .

    .

    , ..

    .

    , .. -

    . -

    .

    .

    -

    .

  • 3

    .................................................................................................................. 7

    MathCAD ...................................................................... 9

    1 Mathcad ................................................................................................... 9

    1.1 Mathcad ............................................................................................. 9

    1.2 ........................................................................................................ 10

    1.3 ........................................................................................... 11

    2 , ............ 14

    2.1 Mathcad .................................................................................................. 14

    2.2 ................................................................... 15

    2.3 .......................................................................................... 16

    2.4 ....................................................................................... 16

    2.5 ....................................................................................... 17

    2.6 ............................................................................. 19

    2.7 .......................................................................................... 20

    2.8 . ....................................................... 22

    2.9 ......................................................................................... 22

    2.10 . ................................................. 23

    2.9 ........................................................................................ 24

    3 . ................................................................... 25

    3.1 ............................................................................ 25

    3.2 ............................................................................................................ 26

    4 ............................................................................................ 27

    4.1 ........................................................................................ 27

    4.2 .......................................................................................... 28

    4.3 ................................................................................... 28

    4.4 ....................................................................... 29

    4.5 ............................................................. 30

    4.6 ........................................................................................... 30

    4.7 ........................................................... 31

    4.8 ....................................................................... 33

  • 4

    . ................................................................................................ 34

    1 .................................................................................................. 34

    1.1 ....................................................................... 34

    2.2 ............................................................................. 35

    2.3 ........................................................................... 35

    2.4 ORIGIN ......................................................................... 36

    2.5 .................................................. 36

    2.6 .......................................... 38

    2.7 ....................................................................................... 39

    2.8 ........................................................................... 40

    2.9 ......................................................................................... 41

    2.9.1 ............................................................................. 41

    2.9.2 ................................................................ 41

    2.9.3 , ...................................................... 42

    2.9.4 , ,

    ....................................................................................................................... 42

    2.9.5 ..................................................................................... 45

    .................................................................................................. 46

    1 ............................................................................ 46

    2 ............................................................................ 46

    3 .............................................................................. 47

    4 ................................................................................... 49

    5 ....................................... 51

    5.1 ...................... 52

    5.2 ( ) 54

    6 ........................................................................... 55

    6.1 ................................................................ 55

    6.2 Given Find . 56

    7 ............................................................... 57

    ................................................. 58

    1 ......................................................... 58

    1.1 .............................................................. 60

  • 5

    1.2 ................................................................................ 64

    1.3 ................................................. 65

    3 ........................................................................ 68

    3.1 ............................................................. 69

    4 ..................................................................................... 70

    5 ................................................................................... 71

    6 ........................................................................................ 72

    7 .................................................................................. 73

    8 ......................................................................................... 73

    .............................................................. 75

    1 .......................................... 75

    1.1 ....................................................................................................... 76

    1.1.1 .................................................................................... 76

    1.1.2 - ....................................................................................... 77

    1.2 ............................................................................................................... 79

    1.2.1 ...................................................................................... 81

    1.2.2 ........ 85

    1.3 .......................................................................................... 87

    ............................................................................ 89

    1.1 .............................................................................................. 89

    1.1.1 .................................................................................... 89

    1.1.2. ..................................................................... 90

    1.2 if ........................................................................................... 91

    1.3 .................................................................................................. 94

    1.3.1 while ................................................................................................... 94

    1.3.2 for ....................................................................................................... 96

    1.3.3 break, continue, return ..................................................................... 97

    1.4 ......................................................... 98

    1.5 - ...................................................................................... 98

    1.6 ................................................................................................ 99

    1.7 ........................................................... 100

    2 ............................................................................................................... 103

  • 6

    2.1 ............................................................................................ 103

    ...................................................................................................... 106

    1. Mathcad ............................................................. 106

    1 .............................................................................................................. 106

    2 ................................................................................... 107

    3 ...................................................................................................... 107

    2. ................................................................. 108

    ................................................................................................... 114

  • 7

    Mathcad -

    , , . -

    Mathcad

    . Mathcad

    .

    Mathcad .

    , .

    , , -

    , Mathcad .

    Mathcad:

    Windows: -

    , , .

    , -

    .

    .

    .

    , .

    , ,

    .

    , , -

    , , -

    .

    , , -

    , , , ,

    .

    .

    , .

    : , -

  • 8

    , , -

    , , -

    , .

    .

    Mathcad 15. ,

    .. .

    :

    ,

    , ,

    [Ctrl]

    ( := )

    mean

  • 9

    MathCAD

    1 Mathcad

    Mathcad

    Windows- .

    1.1 Mathcad

    Mathcad.exe

    Mathcad : Standard (), Formatting

    (), Math ().

    1.1 Mathcad

    Untitled:1 (:1), -

    Normal () Mathcad, -

    Worksheet ( ). , Tip

  • 10

    of the day ( ) Mathcad Resource ( Mathcad). -

    . Tip of the day Show tip

    at startup ( ) Close (-

    ).

    1.2

    Mathcad . -

    , -

    .

    1 2 3 4 5 6 7 8 9 10

    1.2 Mathcad

    1. Mathcad.

    2. File () , , , -

    , -

    .

    3. Edit () , (,

    , . .).

    4. View () , -

    Mathcad, .

    5. Insert () .

    6. Format () , -

    .

    7. Tools () -

    .

    8. Symbolics ( ) -

    .

    9. Window () ,

    .

    10. Help () .

  • 11

    1.3

    . ()

    , -

    . ( )

    .

    , Hide ( )

    Customize ( ). Customize,

    .

    Standard () -

    , , . .

    1.3

    Normal ()

    ( )

    ( )

    Mathcad Help ( Mathcad)

  • 12

    Formatting ()

    .

    1.4

    Math ()

    .

    .

    1.5

    ,

  • 13

    :

    Calculator () -

    , , .

    Graph () .

    Matrix ( )

    .

    Evaluation ()

    .

    Calculus ( ) -

    , , .

    Boolean ( ) () .

    Programming () ,

    .

    Greek ( ) .

    Symbolic ( ) -

    Recources ( )

    , Mathcad.

    Controls ()

    Mathcad-.

    Debug () Mathcad 13,

    .

    Resources (. 1.6)

    , Mathcad:

    1.6

  • 14

    2 , -

    .

    , -

    . - -

    .

    , , ,

    [ _ ], [ ], [ % ], .

    , -

    , , , . -

    .

    Mathcad -

    , , ,

    .

    Mathcad . -

    f(), f, -

    f(x) - f.

    Mathcad 12 . Mathcad 13-

    15 ,

    . Preference (), Warnings

    (), .

    Mathcad ,

    . , -

    , -

    .

    2.1 Mathcad

    ( ) -

    , .

    .

    .

    ([],[],[],[]). -

  • 15

    [PageUp]

    [PageDn].

    [Ctrl]+[PagegUp] [Ctrl]+[PageDn].

    , . -

    . -

    [],[], [], []

    . [] []

    .

    , , . -

    .

    . -

    [Ins].

    2.2

    , , -

    , . -

    :

    1. .

    2. [ : ],

    ( := ) .

    3. . Mathcad

    -

    .

    ,

    . ,

    ,

    .

  • 16

    :

    ,

    [ ; ]

    .

    1. , -

    ;

    [ ^ ];

    - [ . ];

    [ [ ];

    , .. , -

    .

    2.3

    , -

    . ,

    , :

    1. [ ( ].

    2. , -

    [ ) ].

    3. [ : ],

    ( := ) .

    4. .

    2.4

    , , :

    1. , .

    2. [ = ].

    x 1 3 20

    x 0 5

    x

    y1

    M

    2 32

    2 32

    11

  • 17

    2.5

    :

    :

    1. . ,

    [], .

    [] [].

    2. .

    , [Backspace].

    :

    1. . ,

    [],

    .

    2. .

    , [Backspace].

    3. -

    .

    :

    1. , -

    . -

    [ ].

    2. (, , ,

    ), .

    4.

    [Ins] , .

    :

    1. / .

    2. [Backspace]/[Del], .

    3. .

  • 18

    :

    1. .

    2. [ ' ] .

    3. [Del], .

    :

    1. , -

    . [ ], -

    .

    2. [Ins].

    3. .

    4. ,

    , . [Back-

    space].

    :

    1. , ,

    .

    2.

    , -

    .

    3. -

    , .

    4. , -

    .

  • 19

    1. .

    2. [ ' ].

    3. [] .

    4. [Ins].

    5. .

    2.6

    :

    1. .

    2. . -

    .

    3. .

    , .

    4. , .

    [Enter], .

    ( " ).

    , .

    :

    1. . -

    [] [].

    2. . -

    , [Backspace].

    :

    1. .

    2. ,

    , .

  • 20

    :

    1. , .

    2. .

    :

    2. .

    3. .

    4. , . -

    .

    :

    :

    1. Style Normal Modify Font

    ( ).

    2. (Font),

    (Style Font) (Size).

    Mathcad . , -

    Word Times New Roman .

    System Ms Sans Serif.

    2.7

    :

    1. ,

    . .

    2. ,

    .

    .

    3.

    , . -

    -

    .

  • 21

    :

    1. , -

    . -

    .

    2. , . -

    .

    3. , -

    .

    :

    1. .

    2. ,

    .

    3. , . -

    , .

    4. .

    -

    : [Ctrl]+[C] ( ) [Ctrl]+[V]

    ( ).

    .

    :

    1. .

    2. .

    :

    1. .

    2. .

    3. -

    .

    .

    Mathcad

    . ,

    , , ,

    Cut (), (), Paste ().

  • 22

    2.8 .

    -

    , =2,718.

    ,

    ( := ), : :=2.

    -

    .

    -

    Calculator (), ,

    .

    . -

    , ,

    . ,

    [Ctrl]+[G].

    , -

    .

    a b c d e h g l m n w f p y r s t q x z

    - -

    , ,

    .

    2.9

    Mathcad -

    . -

    Mathcad, .

    -

    ( ), ,

    ( := ), .

    , , -

    . , -

  • 23

    ,

    . Mathcad , -

    .

    This variable is not definited above

    ( ).

    2.10 .

    :

    ,

    . -

    , .

    .

    := 0..5, -

    = 0, 1, 2, 3, 4, 5.

    , -

    , , :

    := 0..5 0 5;

    := 1,1.1..5 , 1 , 1.1

    , 5 . 1.1-1=0.1;

    : min, min ..maxx , min , (-

    ), max (min, max ).

    ,

    .

    ,

    . [ ; ] ( )

    Matrix ( ) m..n.

    , , -

    , Mathcad -

    , .

    . ( = ).

  • 24

    2.9

    Mathcad ,

    . Mathcad 20 ,

    .

    ,

    . -

    Result Format ( ), Number

    Format ( ). :

    General ()

    .

    . -

    Expo-

    nential threshold ( -

    ).

    Scientific ()

    : 1,22105.

    Decimal (-

    ) : 12,2564.

    Engineering () ,

    3: 1,22106.

    Fraction () :

    / 1

    .

    (Level of accuracy)

    (Use fixed number).

    ( ) -

    ( Set as default ( )).

    1.3

  • 25

    3 .

    1. :

    2. :

    3. :

    4. :

    5. n- :

    n 3

    6. :

    3.1

    1. .

    1.1 :

    1.2 :

    a 23.4 b 78

    =a b 101.4 =a b 54.6

    =.a b 1.825103 =

    a

    b0.3

    =a 4.837

    =b 8.832

    =a

    1

    n2.86

    =b

    1

    n4.273

    =ab6.29310

    106

    x 3.34

    =sin( )x 0.197 =cos( )x 0.98

    =sec( )x 1.02 =csc( )x 5.073

    =tan( )x 0.201 =cot( )x 4.974

  • 26

    2. .

    2.1 :

    2.2 :

    3.2

    1. 10.

    10 -

    .

    1.1 :

    1.2 :

    1.3 10:

    2.

    -

    .

    2.1 :

    2.2 :

    2.3 :

    x 300 x .xdeg

    =sin( )x 0.866 =cos( )x 0.5

    =sec( )x 2 =csc( )x 1.155

    =tan( )x 1.732 =cot( )x 0.577

    x 5.67

    =ln( )x 1.735

    =log( )x 0.754

    x 12.78

    b 2

    logb( ),b xln( )x

    ln( )b

    =logb( ),b x 3.676

  • 27

    4

    4.1

    Simplify ()

    . -

    , ,

    , .

    , . -

    ,

    .

    :

    :

    23 3 21 1 3

    9 1 3

    x x x

    x x

    :

    1. , -

    , ..

    .

    2. , -

    .

    3. .

    4. .

    13 x 3 x

    2

    9x 1

    1 32x

    3x

    13

    3

    3 x 3 x2

    9 x 1

  • 28

    4.2

    . ,

    -

    . , -

    .

    :

    :

    5( 5)x

    :

    1. .

    2. . -

    .

    4.3

    Factor ( ) -

    ,

    .

    .

    :

    :

    3

    3 2

    2 3 1

    2 9 18

    x x

    x x x

    :

    1. .

    x5

    25 x4

    250 x3

    1250x2

    3125x 3125

  • 29

    2. .

    .

    :

    124678488

    :

    1. .

    2. .

    .

    4.4

    Collect ( ) , -

    .

    :

    :

    2 3 2 2 32 3 8 6 15 50 10x x x x x x x

    :

    1. .

    2. .

    .

    x 1( ) 2 x2

    2 x 1 x 3( ) x 3( ) x 2( )

    23

    3 11 347 1361

    13 x3

    24 x2

    8 x 50

  • 30

    4.5

    -

    .

    . -

    .

    :

    b:

    3 2 3( )a b c

    :

    1. .

    2. -

    . .

    4.6

    Mathcad

    :

    1. , -

    Calculus (), -

    . .

    2. lim.

    3. lim ,

    , .

    4. .

    a9

    3 a6

    c2

    3 a3

    c4

    c6

    6 a3

    c2

    3 a6

    3 c4

    3 a3

    3 c2

    1

  • 31

    5. Mathcad Symbolics Evaluate Symbolical-

    ly ( ). Mathcad

    , .

    .

    :

    :

    2 2lim

    3 6

    x

    xx

    :

    3. .

    4. -

    . .

    4.7

    Symbolics -

    :

    1. , -

    , ,

    . Symbolics Variable Differen-

    tiate ( ) Symbol-

    ics Variable Integrate ( -

    ).

    2. Calculus () -

    .

    Symbolics Evaluate Symbolically ( -

    ).

    1

    3

  • 32

    :

    2

    (5 3 7)x x dx :

    1. .

    2. .

    .

    :

    10

    3

    1

    ( 3 5)x x dx

    :

    1. .

    2. -

    . .

    (=).

    :

    '

    cos( )sin

    xx

    x

    x5x2

    3x 7

    d5 x

    3

    3

    3 x2

    2 7 x

    1

    10

    xx3

    3x 5

    d10773

    4 2.693 10

    3

  • 33

    :

    1. .

    2. -

    . .

    4.8

    :

    1. .

    2. Symbolics Variable Convert to

    Partial Fraction ( -

    - ).

    Mathcad ,

    .

    :

    2

    3 2

    2 3 1

    2 9 18

    x x

    x x x

    :

    1. .

    5.

    - .

    .

    xsin x( )

    cos x( )

    x

    d

    dcos x( )

    cos x( )

    x2

    sin x( )

    x

    1

    3 x 3( )

    3

    x 2

    14

    3 x 3( )

  • 34

    .

    1

    :

    1. , -

    , ( 100 ).

    2. . ,

    .

    3. .

    1.1

    1. , .

    2. Vector and Matrix

    Toolbar ( ),

    Matrix or Vector ( ).

    Mathcad Insert Matrix ( )

    [Ctrl]+[m].

    3. (Rows)

    (Columns), .

    .

    2.1

    4. ,

    .

    . ,

    , .

  • 35

    2.2

    , :

    1. . -

    .

    2. -

    [Ctrl]+[m]. ,

    .

    3. Insert () Delete (),

    . , -

    1, 0 Insert Delete.

    Mathcad -

    , ,

    .

    , -

    [Delete] ( ) Cut -

    Mathcad ( ). -

    Cut ,

    .

    Mathcad

    , , . -

    , .

    2.3

    ,

    .

    Xn Subscript ( ), [ [ ]

    ( ),

    .

    ( ), -

    . :

  • 36

    1. .

    2. Matrix and Vectors Tool-

    bar ( )

    Matrix Column

    ( ).

    3. .

    2.2

    2.4 ORIGIN

    -

    ORIGIN. ORIGIN=0,

    , .

    , -

    ( ) ORIGIN:=l.

    ORIGIN . -

    Tools Worksheet Options ( -

    ), Built-in Variables

    ( ) Array Origin (ORIGIN)

    . ORIGIN ,

    .

    Matrix. ORIGIN

    , .

    Mathcad ,

    , , mean (

    ) fft ( ). ORIGIN:=l -

    , .

    2.5

    M2

    4

    3

    2

    1

    0V1

    < >M

    0 =V12

    4

    V2< >M

    1

    =V23

    2

  • 37

    Mathcad -

    :

    , :

    rows(M) ;

    m() min() -

    ;

    , :

    cols(M) ;

    tr(M) , -

    .

    , :

    last(M) ;

    length(M) ;

    V

    Matrix;

    2.3

    2.4

    2.5

    M

    2

    4

    5

    1

    3

    2

    n rows( )M =n 3

    m cols( )M =m 2

    V

    4

    1

    7

    SV V =SV 2

    ORIGIN 1 V

    4

    3

    1

    5

    K last( )V

    =K 4

  • 38

    2.6

    2.7

    2.8

    2.6

    -

    :

    augment(A, ) .

    .

    2.9

    stack(A, ) . -

    .

    2.10

    V

    4

    3

    1

    5

    n length( )V =n 4

    V

    1

    4

    2

    1

    MV max( )V

    MV min( )V

    =MV 4

    =MV 2

    M

    2

    3

    8

    4

    2

    2

    1

    7

    3

    =tr( )M 3

    M11

    1

    0

    4

    5

    2M augment( ),M1 M2 =M

    1

    1

    0

    4

    5

    2

    2

    4

    3

    3

    1

    1

    M1 1 2 3( )

    M2 4 5 6( )M stack M1 M2( ) M

    1

    4

    2

    5

    3

    6

  • 39

    submatrix(A, irows, jrows, icols, jcols) ,

    . A,

    irows jrows, icols jcols.

    2.11 ,

    2.7

    Mathcad . -

    , 100

    . , -

    augment stack.

    -

    - .

    2.12

    M

    1

    6

    11

    16

    21

    2

    7

    12

    17

    22

    3

    8

    13

    18

    23

    4

    9

    14

    19

    24

    5

    10

    15

    20

    24

    M1 submatrixM 1 3 1 3( ) M1

    7

    12

    17

    8

    13

    18

    9

    14

    19

    ORIGIN 1

    n 5 m 5

    i 1 n j 1 m

    xi

    0.1i yj

    0.2j

    fi j

    sin xi

    yj

    f

    0.02

    0.04

    0.06

    0.08

    0.1

    0.04

    0.08

    0.12

    0.159

    0.199

    0.06

    0.12

    0.179

    0.238

    0.296

    0.08

    0.159

    0.238

    0.315

    0.389

    0.1

    0.199

    0.296

    0.389

    0.479

  • 40

    2.8

    Mathcad -

    :

    sort(V) ;

    2.13

    reverse(V) ;

    2.14

    csort(M, i)

    i- ;

    2.15

    rsort(, i) -

    i- .

    2.16

    V

    5

    2

    4

    1

    3

    V sort V( ) V

    1

    2

    3

    4

    5

    V

    1

    2

    3

    4

    5

    V reverse V( ) V

    5

    4

    3

    2

    1

    M

    3

    4

    6

    4

    5

    8

    5

    1

    2

    6

    2

    1

    M csort M 2( ) M

    4

    6

    3

    5

    8

    4

    1

    2

    5

    2

    1

    6

    M

    3

    4

    6

    4

    5

    8

    5

    1

    2

    6

    2

    1

    M rsort M 2( ) M

    6

    2

    1

    5

    1

    2

    3

    4

    6

    4

    5

    8

  • 41

    2.9

    2.9.1

    ,

    , .

    :

    1. .

    2. .

    3. Matrix () -

    , .

    .

    ( -

    ), ,

    , -

    .

    2.17

    2.9.2

    Matrix -

    ||, :

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    VTTV =VT 2 3 1( )

    M

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    7

    11

    4

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    M MT

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    4

    5

    6

    7

    8

    9

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    M2

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    2NM M =NM 1

  • 42

    2.9.3 ,

    , , -

    X-1

    Matrix.

    .

    .

    Mathcad identity(n),

    n .

    2.19

    2.9.4 , , -

    .

    A, -

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    M M1

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    M1 identity 4( ) M1

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  • 43

    ( ). Mathcad-

    .

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    17

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    18

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    22

    28

    18

    21

    26

    26

  • 44

    , -

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    2.23

    ,

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    n

    i ii

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    , X Y ,

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    Matrix X Y ,.

    2.25

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    V2 ( )3 2 1 M .V1V2 =M

    6

    9

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    2

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    5

    V1

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    V2

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    P V1 V2 =P

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    6

    3

  • 45

    2.9.5

    Mathcad , -

    .

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    1. .

    2. ( )f M . -

    ;

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    2.26

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  • 46

    1

    :

    2( )f x ax bx c

    :

    1. .

    2. , ,

    . .

    3. polyroots

    :

    : 2

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    1. .

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    3. polyroots.

    2

    :

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    .

    3. Symbolics Variable Solve

    ( ).

    a 1 b 2 c 8

    vT

    ( )c b a

    r polyroots( )v =r2

    4

  • 47

    :

    h:

    32 0h h b

    1. . -

    ( [Ctrl]+[=]).

    2. , .

    3. Symbolics Variable

    Solve ( ).

    Symbolics , -

    .

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    root -

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    , .

    .

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    8 b 1

    4

    1

    4

  • 48

    :

    -4 3:

    36 2 0x x

    1. , .

    2. .

    3. , .

    3 (-3-2, 01, 2..3),

    3 .

    4. .

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    f x( )

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    -2

    11

    TOL 105

    x3 2

    2 x 2.5

    X1 root f x( ) x( ) X1 2.602

    f X1( ) 3.553 1015

  • 49

    8. .

    9.

    4

    root ,

    . .

    :

    1. ,

    . , -

    .

    2. root, ,

    , ;

    3. .

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    x2 3

    2 x 2.5

    X3 root f x( ) x( ) X3 2.262

    f X3( ) 3.052 1010

    x ..4 3

    f( )x x3 .6 x 2

  • 50

    2 (-2-1, 12),

    2 .

    4. .

    5. .

    ,

    .

    6.

    7. .

    8. .

    x

    -4

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    d

    42

    21

    6

    -3

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    6

    21

    TOL 105

    x2 1

    2 x 1.5

    X1 rootxf x( )

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    dx

    X1 1.414

    f X1( ) 7.657

    x1 2

    2 x 1.5

    X2 rootxf x( )

    d

    dx

    X2 1.414

    f X2( ) 3.657

  • 51

    5

    , AX=B.

    X=A-1B.

    MathCAD -

    lsolve(A,B), X

    .

    -

    .

    .

    , ,

    :

    ;

    ;

    ,

    .

    :

    :

    2 3 1

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    x y z

    x y z

    2. :

    3.

    A

    1

    2

    1

    2

    3

    2

    3

    1

    5

  • 52

    4.

    5.

    6.

    7. Isolve

    5.1

    1. ORIGIN .

    2. .

    3. ( ).

    4. . -

    , .

    5. , -

    .

    6. .

    b

    1

    2

    3

    X .A1b

    =X

    8

    5

    1

    =.A X b

    0

    0

    0

    X lsolve( ),A b

    =X

    8

    5

    1

  • 53

    :

    :

    2 3 1

    2 3 2

    2 5 3

    x y z

    x y z

    x y z

    1. :

    1 2 3 1

    2 3 1 2

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    x y z

    x y z

    x y z

    2. :

    3.

    4. ORIGIN 1.

    5. .

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    7.

    A

    1

    2

    1

    2

    3

    2

    3

    1

    5

    b

    1

    2

    3

    ORIGIN 1

    A = 2

    1

    1

    2

    3

    2

    3

    2

    3

    1

    5

    2

    1

    2

    1

    1

    2

    3

    3

    1

    5

    3

    1

    2

    1

    2

    3

    2

    1

    2

    3

    =1 16 =2 10 =3 2

    x1

    y

    2

    z

    3

    =x 8 =y 5 =z 1

  • 54

    5.2 ( -

    )

    :

    1. ORIGIN .

    2. - .

    3.

    augment(A,b).

    4.

    rref(Ar).

    5. sub-

    matrix(Ag,1,3,4,4).

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    :

    2 3 1

    2 3 2

    2 5 3

    x y z

    x y z

    x y z

    1. :

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    x y z

    x y z

    x y z

    2. :

    3.

    4. ORIGIN 1.

    A

    1

    2

    1

    2

    3

    2

    3

    1

    5

    b

    1

    2

    3

    ORIGIN 1

  • 55

    5.

    6.

    7.

    6

    6.1

    , ,

    . -

    . F(x)=0 -

    root. -

    , 0.

    :

    3- :

    3 26 21 52 0x x x

    1. .

    2. .

    3. root.

    Ar augment( ),A b =Ar

    1

    2

    1

    2

    3

    2

    3

    1

    5

    1

    2

    3

    Ag rref( )Ar =Ag

    1

    0

    0

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    1

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    8

    5

    1

    x submatrix( ),,,,Ag 1 3 4 4 =x

    8

    5

    1

    a3 1 a2 6 a1 21 a0 52

    f( )x .a3 x3 .a2 x

    2 .a1 x a0

    x 0 x1 root( ),f( )x x =x1 4

  • 56

    4. .

    6.2 Giv-

    en Find

    , Given

    :

    1.

    2. Given

    3.

    4. find minerr

    :

    :

    2

    3

    2 5

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    z

    x y z

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    xy z e

    1. .

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    Given

    i 1 x 1 .1 i

    x2 root ,f( )x

    x x1x =x2 1 +3.464i

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    x 1 y 1 z 0

    .2 x y 5 .2 z2

    y3 .4 z 4

    .xy z ez

    vec find( ),,x y z

  • 57

    3. .

    7

    , ,

    :

    1. Given.

    2. Given.

    3. Find -.

    4. , -

    , .

    5. [Ctrl]+[+]. Mathcad

    .

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    Find( ),x y

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    ( )1 .8 ( )..2 b a

    ( ).4 a b

    ( )1 .8

  • 58

    1

    :

    1.

    2. , .

    3. Graph Toolbar (-

    ),

    .

    4.

    , .

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    .

    ,

    -10 10.

    :

    ( ) sin( )f x x

    1.

    2. ,

    .

    1 2 3

    4.1 , 1 ,

    2 , 3

    f x( ) sin x( )

  • 59

    -

    , -

    , -

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    ).

    ( )

    . .

    ,

    , f1(x) f2(y), ( )

    , ( ) -

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    4.3

    ORIGIN 1

    Data

    2

    5

    8

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    20

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    i 1 last Data( )

    0 2 4 6 80

    5

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    15

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    Datai

    i

    4.2

    2 0 21

    0.5

    0

    0.5

    1

    sin x( )

    cos x( )

    x

  • 60

    1.1

    (. -

    4.4). ( ,

    ) , -

    .

    4.4 X,Y

    (. 4.4)

    :

    1. X-Y Axes ( X-Y) -

    : , ;

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    (Boxed ), .

    ( ).

    Log Scale ( ) -

    .

    ;

    Grid Lines ( ) ;

    Numbered () -

    ;

  • 61

    Auto Scale () -

    , -

    . , -

    ;

    Show Markers ( ) . -

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    ( -

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    , .

    .

    1 2 3

    4.5 , 1 , 2

    Y, 3

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    ( ).

    -

    ,

    . , -

    Enable Secondary

    Y-axis ( Y).

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    holder ( ),

    , .

    4.6 -

  • 62

    2. Trace () -

    .

    4.7

    :

    Symbol Frequency ( )

    , , 2 . .;

    Symbol () (

    , , , );

    Symbol Weight ( ) , -

    ;

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    , Dadot -;

    Line Weight ( ) ;

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    bar , Step .;

    Y-Axis ( Y) Mathcad Y Y2

    ,

    .

  • 63

    3. Number Format ( ) , -

    .

    4.8

    4. Labels () . Title

    () ,

    . , , (Axis Label).

    4.9

    5. Defaults ( ) ,

    (Change to default), -

    (Use for default) .

  • 64

    4.10

    1.2

    :

    1. .

    Zoom ().

    2. -

    , .

    3. X-Y Zoom ( X-Y)

    Zoom ().

    .

    1 2 3

    4.11 , 1 , 2

    , 3

  • 65

    ,

    Unzoom ( ) Full View ( ).

    , .

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    , Trace ().

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    . , -

    . -

    X-Y Trace , -

    . ,

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    .

    1.3

    , :

    1. .

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    3.

    , , .

    1 2 3

    4.12 , 1 -

    , 2 , 3

  • 66

    , :

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    , .

    4.13

    : ,

    , .

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    Bring to Front ( )

    Send to Back ( ).

    ,

    .

    , -

    , . -

    . Mathcad -

    Format Separate Region ( ).

    .

    Separate Region,

    , Mathcad

    .

    . Mathcad

    .

  • 67

    2

    :

    1. Graph Toolbar , -

    .

    2. : ,

    . -

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    .

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    .

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    Log Scale (

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    30 2

  • 68

    3

    :

    1. , -

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    2. , .

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    ). .

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    1.

    2. ,

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    1 2 3

    4.14 , 1 -

    , 2 , 3

    -

    . :

    f x y( ) x2

    y2

    f

  • 69

    1. -

    .

    2. , ,

    .

    3. . 3-5 -

    4.15 -

    3.1

    -

    .

    :

    Appearance Fill

    surface Colormap (

    )

    .

    -

    Lighting Enable Lighting On ( -

    ). (Light-

    ing schema). 6 .

    i 0 7 j 0 7

    Xi

    0.5i

    Zi j

    sin Xi

    Yj

    Y

    j0.2j

    4.16

  • 70

    4

    , ,

    .

    :

    1.

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    3.

    4.

    4.17 -

    :

    ,

    2

    ( ) sin ( )f x x x

    1.

    n 40 i 1 n j 1 n

    i

    i

    n

    jj 2

    n 5

    Xi j

    sin i cos j Yi j sin i sin j Zi j cos i

    X Y Z( )

    f x( ) x sin x( )2

  • 71

    2.

    3.

    4.18 2

    ( ) sin ( )f x x x

    4.

    5.

    4.19 , -

    2

    ( ) sin ( )f x x x

    5

    CreateS-

    pace. CreateSpace

    x 0.01 2

    6 4 2 0 2 4 6

    2

    0

    2

    4

    6

    x

    x

    f x( ) f x( )

    S u v( )

    u

    f u( ) cos v( )

    f u( ) sin v( )

    S

  • 72

    -

    .

    :

    CreateSpace(F, t0, t1, tgrid, fmap),

    F ; t0 t1

    , -5 5; tgrid ,

    20; fmap . fmap -

    . -

    .

    4.20 , CreateSpace

    6

    4.21

    H t( )

    t cos t( )

    t sin t( )

    t

    t0 0 t1 16

    C CreateSpace H t0 t1 200( )

    n 40 i 1 n j 1 ni 0 N

    ii 2

    N j 0 N

    jj 2

    N

    Xi j

    cos2 i

    n

    Yi j

    sin2 i

    n

    Zi j

    j 20

    5

    xi j

    5 2 cos i cos j

    yi j

    5 2 cos i sin j

    zi j

    2 sin i

  • 73

    7

    Mathcad

    . 80 . -

    , Mathcad , -

    : Resource Center QuickSheets and Reference Tables Reference

    Tables Geometry Formulas Polyhedra (

    ).

    , -

    D- Polyhedron ("# 1 80").

    .

    4.22

    8

    . -

    ( ). -

    , . -

    . -

    : , , . .

    . 4.22

    G, X Y H1 H2 -

    . ,

  • 74

    ,

    .

    4.23

    F1 x y( ) sin xy( ) F2 x y( ) cos x y( ) n 10 m 10

    i 0 n j 0 m xi

    2 ni y

    j

    2 mj

    H1i j

    F1 xi

    yj

    H2i j F2 xi yj Gi j H1i j i H2i j

  • 75

    1

    ,

    ,

    . , -

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    . -

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    ).

  • 76

    1.1

    ,

    , .

    1.1.1

    Mathcad

    . linterp.

    linterp(X, Y, x) X Y, -

    y, -

    x. X Y . X

    , -

    . ,

    . x y

    .

    :

    :

    F, 4,7 5,0 5,2 5,4 5,9

    S, 3,0 3,0 4,0 7,0 10,0

    S F=5,6 . -

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    1.

    2.

    3. F

    4. S .

    5. .

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    F FT

    S ST

    Fz 5.6

    Sz linterp F S Fz( ) Sz 8.2

    i ORIGIN last F( )

  • 77

    1.1.2 -

    -. -

    ,

    . ,

    . , -

    -, ,

    ( : spline ).

    MathCAD -

    .

    - :

    cspline(X, Y) -

    ;

    pspline(X, Y) -

    ;

    lspline(X, Y) -

    ;

    interp(S, X, Y, ) y(x)

    S, X, Y .

    , - .

    cspline, pspline lspline

    y(x), X Y (

    ). , -

    y(x) c interp.

    4.5 5 5.5 60

    2.5

    5

    7.5

    10

    Si

    Sz

    Fi Fz

  • 78

    :

    -

    :

    X, 4,7 5,0 5,2 5,4 5,9

    Y, 3,0 3,0 4,0 7,0 10,0

    Y X=5,75. -

    .

    1.

    2.

    3.

    4. .

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    X 4.7 5 5.2 5.4 5.9( ) Y 3 3 4 7 10( )

    F FT

    S ST

    xz 5.75

    yc interp D2C X Y xz( )

    yp interp D2P X Y xz( )

    yl interp D2L X Y xz( )

    yc 11.263

    yp 10.072

    yl 9.676

    h 0.05

    x minX( ) minX( ) h maxX( )

    gc x( ) interp D2C X Y x( )

    gp x( ) interp D2P X Y x( )

    gl x( ) interp D2L X Y x( )

    i ORIGIN last X( )

    5 5.333 5.667 60

    5

    10

    15

    Yi

    gc x( )

    yc

    Xi x xz

    4.5 5 5.5 60

    5

    10

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    yp

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    4.5 5 5.5 60

    5

    10

    15

    Yi

    gl x( )

    yl

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  • 79

    1.2

    y(x). -

    , -

    ( V VY)

    . -

    .

    Mathcad

    interp(s, , , t); regress(x, , n)

    ,

    ; ; s -

    ,

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    ; n .

    .

    , 5- -

    .

    :

    s:=regress(X,,n), YY(t):=interp(s,,,t) YY(t):=interp(regress(X,,n),X,,t)

    interp(s, , , t)

    loess(, , span), s:=loess(X, , span)

    2- , interp;

    span>0 , .

    span .

    0,2< span

  • 80

    :

    3- 2-

    :

    1 2 3 4 5 6 7 8

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    0,000 0,349 0,698 1,047 1,396 1,745 2,094 2,443

    9 10 11 12 13 14 15 16

    Y 144 156 164 166 152 137 117 93

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    2.

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    4. 0,4

    5.

    Y 38 15 5.5 23 51 79 105 126 144 156 164 166 152 137 117 93( )

    Y YT

    T

    Yreg x( ) interp regress Y 3( ) Y x( )

    span 0.4 Yloess x( ) interp loess Y span( ) Y x( )

    i 0 15

    0 2 4 60

    50

    100

    150

    200

    Yi

    Yreg x( )

    Yloessx( )

    i x

  • 81

    1.2.1

    . ,

    ,

    . , , -

    .

    linfit , . -

    , -

    :

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    linfit (X, Y, F) , ,

    , F,

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    , ,

    .

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    2

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    :

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    x2

    x

    1

    1 x

  • 82

    2.

    3.

    4.

    , linfit

    , ,

    . , -

    0 1( ) sin(2 ) tanh(3 )f x a x a x

    a0 a1,

    linfit.

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    Y YT

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    S

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    a2 0.515

    x minX( ) minX( ) 0.05 max X( )

    f x( ) F x( ) S

    i 0 length X( )

    0 0.2 0.4 0.6 0.8 10

    0.5

    1

    1.5

    2

    Yi

    f x( )

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  • 83

    1 2( ) 2 sin( ) 3 tanh( )f x a x a x

    a1 a2,

    genfit.

    genfit(X, Y, G, F) , n

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    . G n- n -

    .

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    :

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    1.

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    3.

    X 0.3 0.4 1 1.4 2 4( )

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    2

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    2

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    2

    z2

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    2

    X XT

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  • 84

    4.

    5.

    linfit genfit -

    .

    . -

    . ,

    genfit , linfit

    .

    G

    1

    0

    1

    P genfit X Y G F( )

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    u2 0.036

    x minX( ) minX( ) 0.05 max X( )

    f x( ) F x P( )0

    i 0 length X( )

    0 1 2 3 40

    5

    10

    15

    Yi

    f x( )

    Xi x

  • 85

    1.2.2

    Mathcad -

    .

    ,

    , . -

    , -

    , .

    :

    expfit(X, , g) ( )b t

    f t a e c

    ;

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    f t a t c ;

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    b e

    ;

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    ( )f t a bt ;

    medfit(X, ) , -

    ( )f t a bt . line medfit

    , ;

    lnfit(X, ) ( ) ln( )f t a t b .

  • 86

    -

    rr.

    ( 1), -

    .

    :

    -

    -

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    Y, 3 3 4 7 10

    1.

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    g 1 1 1( )

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    X XT

    Y YT

    g gT

    C expfit X Y g( )

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    Yexp x( ) C0

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    4.5 5 5.5 62

    4

    6

    8

    10

    12

    Y

    Yexp x( )

    X x corr Yexp X( ) Y( ) 0.967

  • 87

    1.3

    y ( -

    x) y, ,

    . , -

    y, , -

    .

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    , ,

    . -

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    . b -

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    .

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    -

    k- , k

    .Y X n- . X

    .

    medsmooth ,

    .

    , ,

    .

  • 88

    medsmooth(Y, n) m- , -

    Y . Y m-

    . n , . n

    , , Y.

    :

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    , -

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    i 0 120 xi

    10i

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    0 10 20 30 400.2

    0.1

    0

    0.1

    0.2

    vi

    zi

    xi

  • 89

    1

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    cad . Mathcad

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    , ,

    . Mathcad . -

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  • 90

    Add Line (

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    Add Line (. 6.1).

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    , ,

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    Cut ();

    Add Line;

    ;

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  • 91

    Mathcad Mathcad-

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  • 92

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  • 93

    3. if

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  • 94

    1.3

    , -

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    , .

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  • 95

    3. while .

    .

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    , .

    , -

    Add Line ,

    . ,

    . .

    while, Mathcad .

    , Mathcad .

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    1.

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    3.

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    ORIGIN 1 j ORIGIN length V( )

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  • 96

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    ).

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  • 97

    1.3.3 break, continue, return

    -

    :

    continue ;

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    ;

    return .

    :

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    . ,

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    " (). -

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    , .

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    S 5( ) 9

    S 6( ) "Error"

  • 98

    1.4

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  • 99

    1.6

    Mathcad . -

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    . , Mathcad -

    -

    . , .

    .

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    Debug, . Mathcad 13 -

    Mathcad, Trace Window (

    ), Debug ()

    trace pause. trace -

    ; pause -

    , -

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    :

    trace("format string", , , z...);

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    format string , ( ), -

    ; , , z... , -

    .

    View

    Toolbars Debug ( )

    , Touggle Debugging Trace Window (-

    )

    6.1

  • 100

    , , ( = ) -

    , ,

    Trace Window ( ).

    :

    , 1,5.

    X, 1,25 1,28 1,34 1,45 1,72 1,81 1,98

    trace

    [ = ] -

    trace , [ = ].

    1.7

    :

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    .

    V 1.25 1.28 1.34 1.45 1.72 1.81 1.98( ) K 1.5

    V VT

    ORIGIN 1 j ORIGIN length V( )

    t j 1

    trace "K={0}, j={1}, Vj={2}"K j Vj

    j j 1

    Vj

    Kwhile

    j

    Vj

  • 101

    3. .

    .

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    ,

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    Value of subscript or superscript is too big

    (or too small) for this array (

  • 102

    ). , -

    , .

    , , ,

    . ,

    , ORIGIN.

    ORIGIN = 1, 0.

    ,

    :

    1.

    , Cut ().

    2.

    , Paste ().

    3.

    ( := ).

    4. , (

    ) , , [ = ].

    . .

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    , ( := ) , Add line,

    .

    6. .

    7. .

    8. .

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    .

    10. -

    .

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    .

    12. ,

    . .

  • 103

    2

    Mathcad -

    AVI- Mathcad-.

    AVI- Windows .

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    . FRAME

    0 n ,

    FRAME .

    FRAME .

    FRAME -

    Mathcad-. FRAME

    Tools Animation Record ( -

    ). ,

    Mathcad Tools .

    2.1

    :

    1. FRAME , -

    .

    2. Tools Animation Record (

    ).

    3. -

    .

    4. -

    . 10

    . 2-3 ,

    .

    5. Animate () -

    . .

    .

  • 104

    Playback ( ,

    1:1, ) .

    - 10 ,

    30 %. 3 -

    , , -

    .

    , . -

    .

    6. Save As (

    ). Mathcad-.

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    cad-, Mathcad

    Insert Object ( ). .

    . -

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    ,

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    .

    ,

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    AVI- . -

    .

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    ( ) .

    :

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  • 105

    . ,

    , , .

    (AVI-) .

    Mathcad .

    , .

    Animate Options (),

    Compressor Options ( ) -

    ( ).

    -

    ,

    . -

    .

    6.2

  • 106

    1. Mathcad

    1

    ,

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    .

    ()

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    .

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    & .

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    \

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    | .

    Ctrl+1 .

    Ctrl+3 .

    Ctrl+4 .

    Ctrl+8 .

    Ctrl+9 .

    Ctrl+0 .

    Ctrl+Shift+4 .

    Ctrl+Shift+3 .

    Ctrl+Shift+? n- .

    Ctrl+\ n- .

    Ctrl+Enter .

  • 107

    2

    .

    ()

    Alt+Esc Windows.

    Alt+Tab Windows

    Ctrl+F4 .

    Ctrl+F6 .

    Alt+F4 Mathcad.

    Ctrl+R .

    F1 .

    F5 .

    F6 .

    F7 .

    F9 .

    Shift+F1 .

    3

    -

    Mathcad.

    ()

    Ctrl+F9 .

    Ctrl+F10 .

    Ctrl+F5 .

    Shift+F5

    Alt+BkSp .

    Ctrl+C .

    Ctrl+V .

    Ctrl+X .

    Ctrl+U .

    Ins

  • 108

    2.

    -

    . -

    , , Mathcad -

    .

    Mathcad , -

    , ,

    . , ,

    .

    . . . .

    Array size

    mismatch

    ,

    .

    ,

    , linterp

    .

    Cannot be

    denned

    (:=) -

    .

    Cannot take

    subscript

    , -

    .

    Cannot take

    subscript

    ,

    - .

    Definition stack

    over- How

    .

    Did not find

    solution

    Mathcad .

    -

    , Minerr -

    Find.

  • 109

    Domain error

    , ,

    Duplicate

    . ,

    .

    Equation too

    large

    Mathcad -

    . -

    .

    Error in constant -

    Mathcad -

    . Mathcad -

    .

    Error in list -

    -

    .

    : f(, , z ...).

    , -

    .

    Error in solve

    block

    ,

    ,

    .

    File error -

    READ READPRN.

    File not found ,

    READ READPRN,

    .

    Illegal array

    operation

    , . ,

    -

    .

    Illegal context -

    , -

    Mathcad.

  • 110

    Illegal factor

    ,

    , . -

    .

    Illegal function

    name

    , Mathcad

    , . -

    , ,

    : 6(). ,

    *, Mathcad

    , -

    .

    Illegal ORIGIN

    ORIGIN

    ORIGIN

    , 16 000 000.

    -

    ORIGIN.

    Illegal range

    .

    Illegal tolerance

    , TOL -

    , Root, Find Minerr,

    TOL>1 TOL 0)

    , ,

    .

    Invalid order -

    , 0 5

    .

  • 111

    List too long

    ,

    .

    , Mathcad,

    .

    Missing operand

    . ,

    .

    Mathcad ( )

    ,

    Misplaced

    comma

    , .

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    . ,

    .

    Mathcad ( )

    ,

    Missing operator

    -

    .

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    ,

    . -

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    ,

    , .

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    sionless

    ,

    , .

    (-

    , cos In) .

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    creasing

    , -

    ,

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    .

  • 112

    Must be integer

    , ,

    identity ,

    .

    Must be nonzero

    ,

    .

    Must be positive

    ,

    , ,

    . Mathcad

    .

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    , -, -

    .

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    , Mathcad .

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    ,

    ,

    identity.

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    , -

    .

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    Given

    -

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    Find Minerr -

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    , ,

    , .

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    ,

    Mathcad ,

    root.

    Only one array

    allowed

    . Mathcad

    ,

    .

    Overflow , -

    , Mathcad

    ( 10).

  • 113

    Significance lost -

    -

    , ,

    .

    Too many points

    , Mathcad -

    .

    Too many sub-

    scripts

    .

    Undefindet

    . ,

    (:)

    , . ,

    (=) .

    Unmatched

    parenthesis

    ,

    . , -

    .

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    vector

    -

    ,

    , -

    , fft -

    2n, n - , 1. ifft

    1+2n , n ,

    0. ORIGIN , Mathcad -

    -.

  • 114

    1. Mathcad (+ CDROM): , ,

    2009 . 384 .

    2. Mathcad 11/12/13 . (+ CDROM): . . -

    , , 2007 . 960 .

    3. Mathcad 12. : ,

    , 2005 . 464 .

    4. Mathcad 12. (+ CDROM):

    , , 2005 . 566 .

    5. Mathcad 12: . . , . . ,

    , 2005 . 352 .

    6. Mathcad 14 . : -

    , , 2009 . 512 .

    7. Mathcad: ..

    , 2006 . 264 .

    8. . 1. MathCAD: . . -

    , . . , , 2009 . 132 .

    9. Mathcad 12,

    MATLAB 7, Maple 9: . . , . . , , 2006

    . 496 .

    10. Mathcad (+ CDROM): . . -

    , , 2004 . 512 .

    11. Mathcad. : . .

    , , 2005 . 752 .