90 / Chapter 8 Calculating in Mathcad

Recursive Function Definitions

Mathcad supports recursive function definitions — you may define the value of a function in terms of a previous value of the function. As shown in Figure 8-7, recursive functions are useful for defining arbitrary periodic functions, as well as implementing numerical functions like the factorial function.

Note that a recursive function definition should always have at least two parts:

•An initial condition that prevents the recursion from going forever.

•A definition of the function in terms of some previous value(s) of the function.

A simple recursive function definition.

Using recursion to define a periodic sawtooth wave.

Figure 8-7: Mathcad allows recursive function definitions.

Note If you do not specify an initial condition that stops the recursion, Mathcad generates a “stack overflow” error message when you try to evaluate the function.

Online Help The programming operators in Mathcad also support recursion. See Recursion in Help.

Units and Dimensions

One of Mathcad’s strengths is units and unit conversion. Units are used just like builtin variables. To assign units to a number or expression, just multiply it by the name of the unit.

Mathcad recognizes most units by their common abbreviations. By default Mathcad uses the SI unit system (the International System of Units) in the results of any calculation, but you may use any supported units you wish in creating your expressions. You can change the default system of units to MKS, CGS, U.S. or None under the Unit

Units and Dimensions / 91

Systems tab of Worksheet Options in the Tools menu. See “Displaying Units of Results” on page 95 for how to set a unit system for results.

For example, type expressions like the following:

mass:75*kg

acc:100*m/s^2 acc_g:9.8*m/s^2 F:mass*(acc + acc_g)

Figure 8-8 shows how these equations appear in a worksheet.

Figure 8-8: Equations using units. Mathcad treats the multiplication as implied when you type an expression like mass:75kg.

Tip If you define a variable that consists of a number followed immediately by a unit name, you can omit the multiplication symbol; Mathcad inserts a very small space and treats the multiplication as implied. See the definition of mass at the bottom of Figure 8-8.

To insert a unit:

1. Click in the empty placeholder and choose Unit from the Insert menu or click on the Standard toolbar. Mathcad opens the Insert Unit dialog box.

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2.The bottom list shows built-in units, along with their Mathcad names, corresponding to the physical quantity selected in the top list. To see all available built-in units select “Dimensionless” in the top list.

3.Double-click the unit you want to insert or click the unit you want and then click “Insert.” Mathcad inserts the unit into the empty placeholder.

Note Mathcad performs some dimensional analysis by trying to match the dimensions of your selected result with one of the common physical quantities in the top list. If it finds a match, all the builtin units corresponding to the highlighted physical quantity appear in the bottom list. If nothing matches, Mathcad simply lists all available built-in units on the bottom.

Dimensional Checking

Whenever you enter an expression involving units, Mathcad checks it for dimensional consistency. If you add or subtract values with incompatible units or violate other principles of dimensional analysis, Mathcad displays an appropriate error message.

For example, suppose you had defined acc as 100 m ⁄ s instead

of 100 m ⁄ s2 as shown at right. Since acc is in units of velocity and g is in units of acceleration, it is inappropriate to add them together. When you attempt to do so, Mathcad displays an error message that specifies how the dimensions mismatch.

Other unit errors are usually caused by one of the following:

•An incorrect unit conversion.

•A variable with the wrong units.

• Units in exponents or subscripts (for example v3 acre or 23 ft ).

•Units as arguments to inappropriate functions (for example, sin(2 s)).

Defining Your Own Units

You may want to define your own units or use your own abbreviation for a unit.

Define your own units in terms of existing units in exactly the same way you define a variable in terms of an existing variable. Figure 8-9 shows how to define new units as well as how to redefine existing units.

Note Since units behave just like variables, you may run into unexpected conflicts. For example, if you define the variable m in your worksheet, you cannot use the built-in unit m for meters anywhere below that definition. However, Mathcad automatically displays the unit m in any results involving meters. Mathcad will warn you if you redefine a built-in unit by underlining the redefinition with a wavy, green line.