EG31 MATHCAD NOTES

SESSION
1

WHAT CAN BE DONE WITH MATHCAD?

1. Combine the functionality of a spreadsheet with
the WYSIWYG text and features of a word processor to
generate a textbook-like document.

2. Update the entire document by making a change
on any page.

3. Read data from a file and do mathematical operations
such as adding a column of numbers , evaluating integrals
or derivatives and invert matrices.

GETTING AQUAINTED WITH THE MENUS AND TOOLS

1. Across the top of the Mathcad window are the
pulldown menus:

F
ile Edit
V
iew Insert Format
M
ath Symbolics W
indow H
elp

2. The second line contains the standard tool
icons found in a word processor along with some additional
ones for Mathcad. The function of each of these icons
can be identified by placing the cursor on it, without
clicking, and reading the prompt that appears

3. The Math tool palette consists of eight icons.
These functions can also be identified by placing the
cursor on the icon. Clicking once on any of these
icons makes the related function window appear.

4. The formatting toolbar, common to Microsoft
Word and Excel appears just above the main Mathcad
window.

To conserve screen space, any or all of the toolbars
can be hidden or shown. To do this, click on the
V
iew menu and click on the
T
oolbars selection with the cursor. This will expose
a menu with the choices: S
tandard,
Formatting and M
ath. Clicking on one of these causes a check mark to
either appear or disappear next to it and the corresponding
toolbar to do likewise.

UNITS & PHYSICAL CONSTANTS

US
CGS

feet (ft) centimeter (cm)

gram (gm) pound
(lb)

second (sec) second
(sec)

coulomb (coul) coulomb
(coul)

Kelvin (K) Kelvin
(K)

SI

meters (m) meters
(m)

kilogram (kg) kilogram
(kg)

second (sec) second
(sec)

amp (A) coulomb
(coul)

Kelvin (K) Kelvin
(K)

candela (cd)

mole (mole)

Mathcad starts in SI units, unless the user specify
some other system of units.

To change
the units:

1. Choose O
ptions
from the Math pull down menu at the top of the
Mathcad window.

2. Select the Units System tab at the top of the
Math Options window.

3. Click on whatever default system of units that
you would like to use.

4. Click on the OK button at the bottom of the
window to set the new units or click on the cancel
button to keep the same ones.If you set the units system
to None, you can define your own units in terms of
the special built-in constants 1L, 1M, 1T, 1Q, 1K,
1C and 1S which represent the dimensions length, mass, time
charge or current, absolute temperature, luminosity
and substance.

To display the units in the Mathcad worksheet:

1. Choose O
ptions
from the Math pull down menu at the top of the Mathcad window.

2. Click on the Dimensions tab at the top of the
Math Options window

3. Click on the box next to Display dimensions.

4. Click on the OK button at the bottom of the
window to confirm the Display dimensions mode or click
on the cancel button to keep the dimensions undisplayed
on the Mathcad worksheet.

A Simple
Calculation

1. Click anywhere in the worksheet. You see a
small crosshair. Anything you type appears
at the crosshair.

2. Type 15 - 8 / 104.4 =
. When you press the equals sign, Mathcad computes
and

shows the results.

As you type the equation, Mathcad shows a small
rectangular box called the placeholder
. This holds spaces open for numbers or expressions
not yet typed. As soon as you type a number, it replaces
the placeholder in the equation.

3. Once an equation is on the screen, you can edit
it by clicking on the appropriate spot and typing new
letter, digits or operators.

4. You can type many operators and Greek letters by
clicking on the Greek letters
in the
Math toolbar to make
the
Greek letter palette
appear. Then simply click on the Greek letter you
wish to appear at the placeholder in your equation.

Defining
Variables

To clear any previous equation and define
a variable t
, follow these steps:

1. Click in the equation you just typed and press
[Space] until the entire expression is held between
the two editing lines.

2. Chose Cut for the E
dit menu.

3. To begin defining
t
, type t: (the letter t followed by a colon). Mathcad
shows the colon as the definition symbol
:=

4.
Type 10
in the empty placeholder to complete the definition
for t
.

If you make a mistake, click on the equation
and press [Space] until the entire expression is between
the two editing limes, as you did earlier. Then delete
it by choosing Cu
t
from the Edit menu.

To enter another
definition:

1. Press the Enter key. This moves the crosshair
below the first equation.

2. To define acc
as -9.8, type
acc : -9.8
. Then press the Enter key again.

Calculating
Results

Now that the variables
t
and acc
are defined, they can be used in other expressions.

1. Use the mouse to place the cursor a few line
below the two definitions and click.

2. Type acc / 2[Space] * t ^
2
. the caret symbol (
^ ) represents raising to a power. The
asterisk ( *
) is multiplication and the slash ( /
) represents division. Placing the space after the
2
lets Mathcad know that
the *t
does not belong in the denominator but is to multiple
the complete previous expression.

3. Press the equal
sign key ( =
).

This equation calculates the distance traveled
by a falling body in time t with acceleration acc.
We had previously defined t = 10 seconds and acc = -9.8 meters per second squared.

Entering
Text

To enter text, you click in an empty space
and do one of the following: choose Text Region from
the Insert menu or press the double-quote key ( " ).

Now do the following:

1. Click in the blank space to the right of the
equations you entered. You will see a small crosshair.

2. Press "
to tell Mathcad that you're about to enter some text.
The crosshair is changed into a vertical line called
the
insertion point
. the characters that you type appear behind this line.
A box surrounds the insertion point, indicating that
you are now in a text region. This box is called the
text
box. It grows as
you enter text.

3. Type the words:
Equations of motion.

Equations of motion

4. To enter a second line of text, press the enter
key and continue to type:

for falling body under
gravity.

Equations of motion

for falling body under gravity

5. To move out of the text box, click on a different
spot on the worksheet or press

[Shift][Enter
].

6. To edit the text after leaving the text box,
simply click on the text to reactive the text box.

Iterative
Calculations

Mathcad can do repeated or iterative calculations
as easily as individual ones. It uses a special variable
called a
range variable
to perform iterations.

Range variables take on a range of values,
such as the integers form 0 to 10. Whenever a range
variable appears in a Mathcad equation, Mathcad calculates
the equation once for each value of the range variable.

To compute equations for a range of values,
first create a
range variable
. In this case, we will compute the results for a range
of values of t
from 10 to 20 in
steps of 1.

1. change t into a range variable by editing its
definition. Click on the 10 in the equation
t : = 10. The insertion point should be next to the
10.

2. Type , 11. This tells Mathcad that the next number in the
range will be 11.

3. Type ;20
. This tells Mathcad that the last number in the range
will be 20
. Mathcad shows the semicolon as a pair
of dots. Clicking outside the equation for t
causes Mathcad to compute with t
defined as a range variable. Since t
takes on eleven different values, there must
be eleven different answers. These are displayed
in the table .

Mathcad displays on the first 16 rows of the table
of calculated

values. Single-clicking on the table displays horizontal
and

vertical scroll bars which can be used to view any group
of 16

rows of the calculated values.

Defining
a Function

Additional flexibility can be gained by defining
functions. Here is how to add a function to your worksheet:

1. First, delete the table. To do so, click anywhere
in the table and press the [Space] bar
until everything is enclosed between the two editing
lines.

2. Define the function
d(t)
by typing d( t ) :

and the expression : 1600 + acc / 2 [Space]
* t ^ 2

The definition just typed defines a function with
the name d
with the argument t
.

This function can be used to evaluate the expression
for different values of t.

To evaluate the function at the value 3.5, type
d
( 3.5 ) =.

Mathcad returns the correct value as shown below.

To evaluate the function once for each value of
t defined earlier, click below the equation

and type d ( t )=.

Mathcad shows a table of values. The first two
values are in exponential

(powers of 10)
notation.

Formatting
a Result

The display format can be set for any number
that Mathcad calculates and displays. The previous
table can be changed so that none of the numbers are
displayed in exponential notation.

1. Click on the table with the mouse and choose
Number
from the Format
menu. The Format Number dialog box is displayed.
The number of decimal places, the use of exponential
notation and whether the number is shown in decimal,
octal, or hexadecimal can be selected as settings.
The option button beside "Set for current region
only:" should be filled in. The Format Number
dialog box can also be displayed by double-clicking
on the table.

2. The default setting for Exponential Threshold
is 3. This means that only numbers greater or equal
to 103
are displayed in exponential notation. Click to the
right of the 3
, press [BkSp] and type 6
. Then click the "OK" button. The table
changes to reflect the new format and 1110 is no longer
shown in exponential notation.

Graphs

Mathcad can show both two-dimensional Cartesian
and polar graphs, contour plots, surface plots
and a variety of other two dimensional plots. These
are all done in a plot region.

To create a graph

1. Click in a blank space where you would like
to locate the graph and type the expression for the
dependent expression which is to appear on the y-axis
of your plot. Type d ( t ) and be sure that the editing
lines remained displayed on the expression.

2. Choose Graph=>X-Y Plot form the Insert
menu. This will produce an empty graph with placeholders
on the x-axis and y-axis for the expressions to be
graphed.

3. Type in t
at the placeholder on the x-axis of the graph. Click
anywhere outside the graph and
Mathcad calculates and graphs the points as shown
below.

Equation of motion for falling

body under gravity

Resizing a graph

1. Click the mouse just outside the graphics region
to anchor one corner of the selection rectangle.

2. Press and hold down the mouse button and drag
the mouse toward the plot region. A dashed selection
rectangle emerges from the anchor point.

3. With the selection rectangle just enclosing
the graphics region, let go of the mouse button. The
dashed rectangle will turn into a solid rectangle with
handles.

4. Move the mouse pointer to any of the handles.
When it is properly positioned, it will turn into
a double-headed arrow.

5. Press down and hold the mouse button and move
the mouse to stretch the graphics region in that direction.

6. Once the graphics region is the right size,
let go of the mouse button.

7. This operation can be repeated with any of
the handles.

8. When the graph has been stretched to the desired
size, click outside of the graphics region to deselect
it.

Zooming in on a portion of a graph

Sometimes it is desireable to be able to zoom in on
a portion of a graph in order to find out details such
as the value(s) at the intersection of two plotted
equations. These types of points are of interest since
they are the set(s) of values that are the solution
to both equations.

Zooming Method 1:

1. Click on the graph so that it appears in the
solid boarder with handles. It also has numerical
values at the extremes of each of its axes in addition
to the usual values along each of the axes.

2. Click on the leftmost value for the horizontal
axis. A cursor appears in that location.

3. Press the Delete key to remove the present value of
10 and replace it with a placekeeper.

4. Type in 15
and press the
Enter
key. The leftmost value on the horizontal scale is
changed from the default value of 10 to 15. The graph
has now zoomed in on the portion of the data between
the values of 15 and 20.

Note: The same steps are used to zoom out
on a graph. However, the graph will only plot the
values which are defined by the original range variable
and the resulting values computed for the dependent
variable. It will not extrapolate the plot to the
new range.

Zooming Method 2:

1. Click on the graph
to select it.

2. In the Format menu at the top of the window, choose G
raph and then Zoom
.

This causes the X-Y dialog box to appear.

3. If needed, the dialog box can be repositioned,
using the mouse, to allow the complete graph to
be seen.

4. Within the graph region, place the mouse at
a location which locates one corner of a box that
you want to draw to describe the region you want
to magnify.

5. Press and hold the mouse button down as you
define the box around the area to be magnified
by dragging the mouse through a diagonal path to describe
the region to be magnified. A dashed rectangle
appears as you drag the mouse to indicate the region.

6. If necessary, the region can be repositioned
by positioning the cursor in the dashed box,
pressing and hold the mouse button and move the mouse
to reposition the dashed rectangle.

Note that the coordinates of the corners of the
dashed box are displayed in the X-TY Zoom
dialog box and that they change as dashed box is moved.

7. Clicking on the Zoom button redraws the graph
within the dahed box as the full graph.

8. The limits of the axes are those of the dashed
box. However, they can be changed as described
in Zooming Method 1, above.

To unzoom a graph that has already been zoomed but which
the axis limits have not been changed.

1. Click on the graph to activate its graph region.

2. Choose G
raph => Zoom
from the F
ormat
menu to bring up the X-Y
Zoom box.

3. Click on the U
nzoom
button to get back to the previous level of zoom or
click on the Full View
button to see the original graph prior to any zooming.

Graph coordinates

To see a readout of the graph coordinates that make
up a trace:

1. Click on the graph region to select it.

2. Choose G
raph => Trace from the F
ormat
menu, or clic on the Trace button on the Graph
Palette, to show the X-Y Trace dialog box. Reposition
the box so that the entire graph can be
seen, if necessary. Note that the Track Data Points
box is checked.

3. In the graph region, click and drag the mouse
along the trace whose coordinates you want
to see. A dotted crosshair jumps from one point to
the next as you move along the trace.

4. If the mouse button is released, the left and
right arrrows can be used to move to the previous
or next data points. The up and down arrows move to
other traces on the same graph.

5. As the pointer reaches each point on the trace,
Mathcad displays the x and y values of that
point in the X-Value and Y-Value boxes.

6. The coordinate values of the last selected
point remain in the boxes. The crosshair remains
until you click outside the graph.

To copy a coordinate to
the clipboard:

1. Click "Copy X "or "Copy Y".
The value can then be pasted into a math region or
a text region on the Mathcad worksheet,
into a spreadsheet, or into any other application
that allows pasting from the clipboard.

2. Click outside the graph or on the "Close"
button to make the crosshairs disappear.

Formatting
a graph

A graph is produced with default characteristics:
numbered linear axes, no grid lines and points connected
with solid lines. These characteristics can be changed
for formatting the graph.

To do this:

1. Double-click on the graph to bring up the appropriate
dialog box. This box contains settings for
all available plot format options.

2. Click on the Traces tab in the dialog box to see the correct
page.

3. Click on "trace 1" in the scrolling
list under "Legend Label". Mathcad places
the current settings for trace 1 in the boxes
under the corresponding columns of the scrolling list.

4, Click on the arrow under the "Type"
column to see a drop-down list of trace types.

5. Choose "bar" from this drop-down
list by clicking on it.

6. Click on the "OK" button to show
the result of changing the setting. Mathcad shows
the graph as a bar chart instead of connecting
the points with lines.

7. Click outside the graph to deselect it.

Building
Expressions

Mathematical expressions can be created in Mathcad
by simply typing in a stream of characters. Characters
like
* and +
represent operators.

Mathcad behaves a lot like a standard word processor.
As you type, characters appear behind a vertical editing
line. There are, two important differences:

1. As the editing line moves to the right, it
leaves behind a horizontal editing line.

2. Unless you have clicked on an equation that
already has an operator in it, pressing the
[Space] bar will turn the the math region into a text
region. You cannot turn a text region back
into a math region.

Typing in operators

Operators like the
symbols +
and -
link variables and numbers to form expressions. These
expressions are called operands
. The key to working with operators is learning to
specify which variable or expression is to become the
operand. This can be done in two ways:

1. You can type the operator first and fill in
the placeholders with operands.

2. You can learn how to use the editing lines
to specify what variables or expression you
want to turn into an operand.

Try creating the expression
a x+y
using the first method:

1. Press ^ to create the exponent operator.
It appears with two placeholders.

2. Click in the lower placeholder and type the
letter
a.

3. Click in the upper placeholder and type +
.

4. Click in the remaining placeholders and type
x
and y
.

Now try creating the same expression using the editing
lines.

1. Type a
. The line beneath the a
indicates the a
will be the first operand of whatever
operator you type.

2. Press ^ to create the exponent operator.
The editing lines now surround the new placeholder.

3. Type x + y in this placeholder to complete
the expression.

Now type the expression
a
x
+ y instead of
a
x+y
.

1. Type a^x
.

2. Press the [Space] bar to cause the editing
line to hold the entire expression.

3. Now type +
. Whatever was held between the editing lines has
now become the first operand of the
plus sign.

4. In the remaining
placeholder, type y .

EG 31 MATHCAD EXERCISES

SESSIONS
1

Exercise
1 : Units
Conversion

1. Go to options in the Math pull down menu.
Select the Dimensions tab and click on the
box next to "Display dimensions".

2. In the box next
to "Mass" type
kg
. In the box next to "
L
ength" type m
. In the box next to "Time" type sec
.

3. Type in the following:

4.
Now type F
= to get the answer
with units:

To change the units:

Click on the placeholder to the right of the units
and type in dyne
and press the Enter
key.

Exercise
2: X-Y plots

1. Define the range
of the variable
x to start at
-8 and end at +8
and to increment by 0.1

2. Define the equation for the dependent variable
y(x)
as 3x3 + 7x +3

3. Plot the graph.

4. Redefine the range
of the variable
x to start at
-5 and end at +5
.

5. Add the title
X-Y
PLOT
(Hint: double click on the graph)

6. Deselect Autoscale

7. Change the plotted line to a dashed line.

8. Change the number of grid line on the x
axis to 10 and the y(x) axis to 4.

9. Generate a table of x and y values by typing
: x
= and y(x) =

Exercise 3: Finding the zero of a function using a
graph

1. Plot the function
y(x)
= 3x3 + 7x + 3
over the range of -8 to 8
in increments of
0.1

2. Adjust the x
axis to the range with grid lines at an interval of
2
units.

3. Adjust the y(x) axis to the range with gridlines at an interval
400 units.

4. From this plot, estimate the value of x
where the value of y(x)
equals zero.

5. Create a new plot which has a narrower range
for the values of x and y so that you can
zoom-in and add gridlines

6. Continue narrowing the range until you can
view the value of x
for which the value of y(x)
is zero.

Exercise
4: Polar plots

1. Create a polar plot for the functions R(q) = sin(q)
over the range of 0 to 2p
in increments
of 0.1

2. Repeat the plot
using the function r(
q)
= q sin(
q)

3.
Repeat the plot using the function r(q) = cos(q)
sin(q)

4.
Repeat the plot using the function r(q1) = cos(4q1)
sin(4q1)
between 0 and 2
p
in increments
of .05