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2.8: Untitled Page 20

  • Page ID
    18153
  • Chapter 2

    and an array can be raised to an array power that leads to the relation b

    b

    b

    b

    b

    b

    b

    b

    1

    a

    a 2 a 3 a 4  1 2

    3

    4 

     1 2 3 4

    1

    a

    2

    a

    a 3

    a 4 

    (2‐28)

    One must remember that these are defined operations and they are defined because they are so common and so useful in engineering analysis. It is important that these array operations not be confused with matrix operations that are specified by the classic rules of matrix algebra.

    In the previous paragraph, several algebraic operations were clearly defined for one‐dimensional arrays. Under certain circumstances, we would like to represent these operations in compact notation and this requires the introduction of special nomenclature. We have already done this in Sec. 2.2 where we made use of lower case, boldface font to identify the force and velocity vectors, i.e., f and v.

    To express one‐dimensional arrays in compact notation, we use Goudy Handtooled BT font for arrays in order to obtain

    a   1

    a

    a 2 a 3 a 4

    (2‐29)

    This allows us to express Eq. 2‐28 in the following compact form b

    a

     c

    (2‐30)

    This type of nomenclature is only useful if one visualizes Eq. 2‐28 when one sees Eq. 2‐30. This means that one must be able to interpret the array c according to b

    b

    b

    b

    c   1

    2

    3

    4

    1

    a

    a 2

    3

    a

    a 4 

    (2‐31)

    In using Eq. 2‐30 as a compact version of Eq. 2‐28, we are confronting two classic problems associated with nomenclature. First, compact nomenclature is only useful if one can visualize the details and second, there are not enough letters and simple fonts to take care of all our needs.

    Two‐dimensional arrays are also useful in engineering calculations, and we identify them using uppercase letters according to

    11

    a

    12

    a

    13

    a

    A   a 21 a 22 a 23 

    (2‐32)

    a 31 a 32 a 33 

    If an array has the same number of rows and columns, it is called a square array.

    The following two square arrays have the special characteristic that the rows of

    Units

    27

    array A are equal to the columns of array B and that the columns of array A are equal to the rows of array B .

     3

    4

    2 

     3

    8

    5 

    A   8 2 7  ,

    B   4 2

    3

     

    (2‐33)

    5

    3

    4

    ‐2 

    7

    4 

    that the columns of array A are equal to the rows of array B . We call the array B the transpose of array A and we give it the special symbol T

    A so that Eqs. 2‐33

    take the form

     3 4

    2 

     3

    8

    5 

    T

    A   8 2 7  ,

    A

      4 2 3 

    (2‐34)

    5 3

    4

     2 7

    4 

    Given that T

    A is the transpose of A , one can see that A is also the transpose of T

    A , and one is obtained from the other by interchanging rows and columns.

    2.5.1 Units

    The addition and subtraction of arrays, in particular 1 4 arrays, can be used to determine the units of various derived quantities. This is accomplished by the use of the 1 4 array of exponents in which the four entries are the exponents of the four fundamental standards given by

    mass, length, time, electric charge

    For example, in terms of the four fundamental units, we represent the units of force as

     units of 

    M L

    1 1

    2

    M L T

    (2‐35a)

    2

     force 

    T

    in which we have used M, L, and T to represent the units of mass, length, and time. The 1 4 array of exponents associated with force is given by f   1 1

    2

    0 

    (2‐35b)

    The units of area are given by

     units of 

    2

      L

    (2‐36a)

     area 

    and the array of exponents in this case takes the form

    28