SMath Studio Handbuch

Prof. Dr.-Ing. Martin Kraska

kraska@fh-brandenburg.de

29. March 2014

Fachhochschule Brandenburg

4 Mathematics

4.1 Definitions

Expression
Consists of numbers, variables, operators, functions and their arguments and can be evaluated to a result.
Identifier
Name for a variable or a function.
Function
Calculates a return value from its arguments. The arguments are enclosed in brackets, separated by the argument separator (semicolon in the German default setting)
Index, address
Subscript index for the specification of row and possibly column for vector or matrix-value identifiers and expressions. It is created by entering [ (square brackets open) and is separated by a space.
Index, descriptive
Subscriptive component of an identifier that is used for further explanation. The subscript is created by inserting a dot. There is no space between the identifier and the descriptive index.
Operator
Special symbol for a function, e.g. +, -, *, /, Σ
Variable
Contains an expression that can be addressed under a name (identifier).
Numbers
Whole numbers consist of a sequence of digits. Decimal fractions consist of a sequence of digits and the decimal separator (in the German default setting, the comma). Floating point numbers are represented in SMath as the product of a decimal fraction with a power of ten.

4.2 Arithmetic - calculating with numbers

SMath can perform the usual operations with integers.

8+7=

25-16=

34*49=

1000/8=

2^16=

If the results are too large or result in fractional values, they are displayed as a floating point number, i.e. as the product of a decimal fraction and a power of ten.

2^40=

10!=

1/23=


log(2 , 10 =

When a number is too large and how many decimal places are then displayed can be set in the context menu (right mouse button) under the items "Threshold exponential" and "Decimal places". This is explained in more detail in section 4.4.

However, a number can also become too large for the floating point display. The limit can be determined by trial and error:

Result is above max. allowed positive number.

This corresponds to an internal representation as a double-precision floating point number with 64-bit wide coding. There is 1 bit available for the sign, 11 bits for the exponent and 52 bits for the mantissa.
This means that SMath cannot distinguish between the numbers 2^52 and 2^52 + 1, i.e. the numbers can only be exact to 15 decimal places.

From an account balance of two-digit trillion amounts, cent values are therefore ignored.

The limitations of the number display also apply to symbolic evaluation, which can be switched to in the context menu or which can be activated with Ctrl -.
instead of the equals sign.

40 ! =

numeric answer

40 ! Ctrl+.

symbolic answer

If you want to calculate with exact integers of any length, you can do this with maxima:

Maxima(40 ! Ctrl+.

4.3 Symbolic calculation with variables

SMath can calculate symbolically with variables. This is always possible if no numerical value has been assigned to the variable. The following examples must therefore be reproduced in a new spreadsheet.

The following definition assigns the value a +b to the variable symb. However, since neither a nor b was previously assigned a number, symb, a and b are symbolic variables.

a and b are undefined, while symb contains the expression a + b just defined. The equals sign = is only suitable for displaying defined variables. For undefined variables, it generates an assignment operator :=, as if you had pressed :.

By default, the display attempts to determine a numerical value for the expression. However, as a and b are undefined in this case, a + b cannot be calculated either. The error message indicates the first problem with the calculation.

Instead of the value, the symbolic expression with which the variable was defined can also be displayed. This is done by changing the display in the context menu (right mouse button) to "Optimization symbolic":

The symbolic evaluation operator →, which you will find in the arithmetic panel of the sidebar, immediately displays the symbolic result (you can also use the keyboard shortcut Ctrl -.).

In earlier versions of SMath, this operator was represented with an arrow (→) so that a symbolic result could be distinguished from a numerically evaluated result. Currently, the equals sign is used in both cases. This is particularly annoying because the symbolic evaluation operator ignores the context menu setting "Numeric". Its use is therefore not recommended. Instead, the normal equal sign should always be selected with the appropriate setting in the context menu.

If a or b is now defined, this is taken into account the next time symb is displayed. First the definition

and now the display (symbolic, because of course you still don't get numerical values):

The expression stored for symb is still a +b. However, everything that SMath knows about a and b is used in the evaluation, even if this information was only added after the definition of symb. The decisive factor is what is known at evaluation time.

If a and b are changed, the symbolic evaluation result of symb also changes:

The variables xx and yy are now irrelevant for the expression symb = a + b, as they are no longer related to a and b. Therefore, changes to xx and yy no longer have any effect:

If numerical values are known, they are displayed as a quotient for symbolic evaluation and as a floating point number for numerical evaluation:

Symbolic

Numerical

Symbolic optimization for definitions

The optimization setting ("Optimization" context menu with the values "Symbolic", "Numeric" or "Disabled") also has an effect on definitions.

Symbolic
(default setting for definitions) The expression on the right-hand side is simplified symbolically, but without using existing variable assignments outside the definition. There are cases in which symbolic optimization introduces errors into the definition. For example, simplification rules are applied for the multiplication of numbers that are not valid if the variables are vector or matrix-valued. This is usually not known at definition time.

Numerical
The result of the expression on the right-hand side is calculated. If this is not successful because one of the values involved is not defined, the expression is saved unchanged as a definition.

None
The expression on the right-hand side is saved unchanged. In view of the uncertain symbolic simplification, this should be the default setting should be the default setting.

4.3.1 Data types

There are scalar variables that can have a single expression as a value or fields that can hold several expressions that can be addressed by address indices.

There are three types of fields:
• Matrices, which are two-dimensional fields that can be addressed with one or two indices and are displayed in round brackets. Internal representation: mat()
• Lists, which are one-dimensional fields that are displayed with a curly bracket from the left. Internal representation: sys()
• Statement blocks, these are one-dimensional fields whose expressions are executed line by line when called. They are delimited on the left by a vertical line. Internal representation: line()

Expressions are combinations of identifiers, functions, operators, number constants, character strings and units of measurement.

• Identifiers are character strings that do not begin with a number and do not contain reserved operation characters. They are used as names for variables and functions.
• Functions are identifiers with subsequent arguments in brackets. The arguments are separated by the argument separator.
• Operators are special symbols that mathematically link associated expressions (the operands). They can be understood as functions with a special display format.
• Number constants are sequences of digits that can contain a decimal separator. In SMath, the sign is not part of the number constant, but a unary operator (character that stands for a function with exactly one argument).
• Character strings are any character strings enclosed in quotation marks.
They are displayed with the quotation marks in red-brown color. Functions for character strings are explained in chapter "Programming", section 7.4.
• Units of measurement are special identifiers that begin with an apostrophe. The apostrophe is suppressed in the display and the identifier is displayed in blue italics. Multiplication signs before units of measurement are suppressed.

Argument separators in functions and decimal separators in number constants depend on the individual program settings (Main menu> Tools> Options> Interface).

In Germany, the comma is commonly used as a decimal separator. Therefore, it cannot be used as an argument separator, as is common in many programming languages.
The semicolon is therefore used instead.
The English-language default setting is the point as the decimal separator (decimal point) and the comma as the argument separator.

Logical (Boolean) values are represented by numerical values:
• Zero for false
• Not equal to zero (typically one) for true
Powers of ten are not represented and entered as part of the number constant, but as the corresponding arithmetic operation.
Complex numbers are represented as the sum of the real part (number constant) and the imaginary part (product of imaginary unit i and a number constant).

4.3.2 Delete variables

By default, SMath has no way of deleting a variable once it has been defined, e.g. to be able to use it again further down in the spreadsheet. The Custom Functions plugin (page 215) contains the Clear() function (page 244), which can be used to delete variables:

Note that the function must be evaluated for it to actually be executed. The deletion is not quite complete, because the variable contains a statement block with itself as the return value (you can see this if you place the mouse over the argument A in Clear(A):

It can therefore not be defined element by element as a matrix like virgin variables.

The variable must first be explicitly declared as a matrix by assigning a value to it. You can then expand this matrix dymanically as usual by assigning elements.

4.3.3 Automatic multiplication

If a letter is entered after a number, SMath automatically understands this as a multiplication and inserts a point. No point is inserted between numbers and units.

2x

3sin ⇆ 3p Ctrl -G

10N ⇆

4.3.4 Caution with symbolic evaluation!

Examples of errors in the symbolic evaluation are given here. As a rule, these problems can be avoided by numerical evaluation. They illustrate the survival rule
"Extreme caution with symbolic evaluation! If in doubt, do not use this feature."
The incorrect results are marked in red.

Example 1

When forming mixed second derivatives, the result depends on the order and can therefore be incorrect (forum post):

This problem is avoided by using the Maxima plug-in.

Example 2

Invalid simplification of expressions if the variables are vector-valued. The following simplifications are correct for scalar numbers:

However, if the variables are vector-valued (which SMath is aware of here), the same simplifications are still made in symbolic evaluation, although they are no longer permitted:

This example is also discussed again in section 4.10.3 in the context of matrix operations.

Example 3

Incorrect evaluation of user-defined functions that contain vector operations.
The following function can take either scalars or 2×2 matrices as arguments:

For matrices, the 1 is subtracted element by element. Correct results are, for example

The result in SMath depends on the optimization setting for both the function definition and the function call (evaluation). Incorrect results are marked in red:

Here, too, the cause lies in symbolic transformations of the definition, which are only permitted for scalar arguments. This can be seen in the dynamic selection (hover the mouse pointer over the definition).

The symbolically transformed expression is then processed correctly, but the transformation was not valid for matrix arguments:

The line() function (actually intended for forming statement blocks) suppresses the symbolic simplification in the definition.

4.4 Format of the results display

The result display can be modified with the following settings.
• Number of decimal places (with or without trailing zeros at the end)
• Exponential threshold (from which number of digits before the decimal point the display switches to exponential representation)
• Separation of a specified factor (unit of measurement, reference value)

4.4.1 Representation of decimal numbers

The exponential threshold can also be set in the context menu, i.e. the power of ten from which numbers are displayed in exponential form. In the context menu, the current setting for the formula range is marked with a √ and the default setting with a *. The default setting can be set under Main menu> Tools > Options > Calculation.

The number of decimal places is set using the "Decimal places" option in the context menu.

3

2

1

0

Optionally, you can specify that zeros should be displayed at the end of the decimal fraction ("trailing zeros"):

Unfortunately, there is no built-in function for rounding to a certain total number of digits. You can, of course, write a corresponding function:

4.4.2 Separating a factor (power of ten, constant, unit of measurement)

You can subtract an arbitrary factor from each result. As soon as you click on a displayed result, a placeholder ■ appears in which you can enter the factor to be displayed separately. This mechanism is intended for units of measurement, but it also works with numbers and variables.

In the following example, the value of arcsin(1) is displayed once unmodified, then as a multiple of π and then in the unit of measurement °:

asi ⇆ 1= ↵
asi ⇆ 1= → → p Ctrl -G ↵
asi ⇆ 1= → → ° ⇆ ↵

4.4.2 Separating a factor (power of ten, constant, unit of measurement)

There is a unit of measurement %1 for percentage calculations. This can only be entered via the dynamic help, as the actual percentage sign cannot be entered in formula areas.
For example, to enter 10%, first type
10'

The dynamic help appears with a list of all units of measurement, starting with the percent sign.

Select the percentage sign with the ↓ arrow key

and confirm with ⇆ :

The percentage sign stands for a factor of 0.01 and can therefore be used largely intuitively.

Alternatively, the percentage sign can also be inserted via the "Insert unit" dialog in the "All" category. The dialog is opened with Main menu> Insert> Unit or Ctrl-W.

4.5 Operators

In SMath, operators are those functions that are also represented internally by special characters without brackets around the arguments. When entering operators, the required operands are represented by placeholders (■), which must be filled in with expressions.
The "Insert operator" dialog can be found under Main menu> Insert>Operators:

The Boolean operators are explained below in section 4.6.2.
The following overview lists the remaining operators:

Operator

Keyboard input

Meaning

+

Positive sign (pure optics) or addition

-

Negative sign or subtraction

Plus / Minus

Minus / Plus

*

Multiplication

/

Division

^

Potentiation

Ctrl+8

Cross product (vectors)

!

Factorial

:

Assignment, definition

=

Evaluation (can be switched from numerical to symbolic in the context menu)

Table 4.1: Operation in SMath

The operator formatting of some arithmetic operations, which are written internally as functions, i.e. as function name(argument1; argument2...), contributes to the readability of SMath documents.
These functions with special formatting are listed in the following table.

Operator

Keyboard input

Meaning

Function

Substitution

at (#,#)

Substitution

at (#,#,#)

Case differentiation

cases (#,#,#)

\

Square root

sqrt (#)

Ctrl + \

n-th root

nthroot (#,#)

Absolute value

abs (#)

Determinant of a matrix

det (#)

next largest integer

Ceil (#)

next smaller integer

Floor (#)

[

Element addressing in fields

el (#,#)

[ → ,

with two indices

el (#,#,#)

diff (

Differentiation (with maxima)

diff (#)

Differentiation

diff (#,#)

Differentiation

diff (#,#,#)

Integration

int (#,#,#)

Sum

sum (#)

sum (#,#,#,#)

Product

Product (#,#,#,#)

range (#,#)

Value range (vector)

range (#,#,#)

Table 4.2: Functions with operator representation in SMath

It is not always clear from the operators how far their effect extends. When editing, this can be recognized by the insertion point (length of the underline). You may need to use brackets to ensure clarity in the printed document.

4.6 Function

This section describes the functions available in SMath. At present, however, the description is incomplete and is limited to the angle functions and their inversion, as well as logical and comparison operations.
The definition of your own functions is explained in the "Programming" chapter in section 7.1.

4.6.1 Angle functions

SMath knows the elementary trigonometric functions. Their arguments can be specified in radians (left-hand image) and in degrees (right-hand image). Values in degrees are generated by multiplication with the unit of measurement °:

x*° ⇆

Argument in radians

Argument in degrees

Some values are known symbolically (display with "Optimization symbolic")

The secant and cosecant functions are not provided directly, but are fed back internally to the sine and cosine.

sec(x)

csc(x)

However, the symbolic computing power is not sufficient to actually use the known values.

Inverse functions of the angle functions:

Result in radians

Result in degrees

Symbolic optimization

Symbolic optimization

Symbolic optimization

Numerical optimization

4.6.2 Logical and comparison operations

SMath has the usual logical and comparison operations. These return the numerical value 1 if they are true and 0 if they are false. The operations can be accessed via the "Boolean" panel in the sidebar, and there are also some keyboard shortcuts.

Function

Key

Priority over :=

equal to

Ctrl + =

lower

Less than

<

higher

greater than

>

higher

Less than or equal to

Ctrl + 9

higher

Greater than or equal to

Ctrl +0

higher

Not equal

Ctrl +3

lower

Logical not

Logical and

&

lower

Logical OR

|

lower

Exclusive OR (XOR)

lower

Table 4.4: Binary logical operators

The behavior of the operators can be checked using a truth table:

Operator

Function

lele (#,#,#)

lelt (#,#,#)

ltle (#,#,#)

ltlt (#,#,#)

gege (#,#,#)

gegt (#,#,#)

gtge (#,#,#)

gtgt (#,#,#)

Table 4.6: Ternary logical operators from the Functions Extensions plugin

Due to the low priority of some operators, the right-hand side of assignments must be placed in brackets in such cases.

a:1& → (first assignment, then AND operation)

a:(1& → (first AND operation, then assignment)

Logical expressions can also be used to define piecewise continuous functions:

4.7 Complex numbers

Notes:
• Avoid redefining the imaginary unit i (e.g. by using it as a loop counter). You are on the safe side if you always write √-1 instead of i.

• Some important functions such as Re(), Im(), arg() do not work with units of measurement.

• Please note that some functions for complex numbers only work properly if complex solution sets are permitted. If you encounter problems, please always check the effect of the setting under Main menu> Tools> Options> Calculation: Result (quantities).

4.8 Derivatives

Maxima is very good at calculating the derivatives of expressions symbolically. The diff() function, which is available in three operator forms depending on the number of arguments, is responsible for this.

4.9 Integrals

SMath can only calculate certain integrals (with integration limits) and only numerically.

• Units of measurement at the integration limits are ignored. The integration variable is therefore regarded as dimensionless.
• Functions must not contain an if() statement. If piecewise continuous functions are to be integrated, they can be described as described in the previous section using
arithmetic expressions with logical operators as described in the previous section.
• The integration accuracy is set globally under Main menu> Tools> Options> Calculation> Integrals: Accuracy. The value specifies the number of intervals between the integration limits used in the numerical calculation. If in doubt, increase the value and check the extent to which the result depends on the setting. For sufficiently smooth functions, the default setting (100) is well suited; for functions with discontinuities, higher values may quickly be required.
• In the interest of accuracy, it is advantageous to divide the integration area at points of discontinuity.

Indefinite integrals or symbolic evaluation are not possible with the built-in functions in SMath. Alternatively, the Maxima plugin can be used.

4.9.1 Integrals of piecewise continuous functions

The integration function int() fails when functions are defined using if(). The nonsensical error message then appears that the integration variable is not defined.

The way out is to use logical operators to define expressions that are only one in a certain range, but zero otherwise (integration accuracy set to 1000 here):

A convergence analysis shows the following dependency of the result on the set accuracy (Main menu> Extras> Settings> Calculation> Integrals: Accuracy):

Accuracy setting Value

Due to the discontinuity at t = 1, the standard setting (accuracy 100) results in an error of more than one percent.

4.9.2 Integrals with variable limits

You can define functions that contain integrations with variable limits. Make sure to use different variables for limits (here x) and integration (here y)!

However, expressions with variable limits cannot be plotted without further ado.

With the standard chart area, the plot only works if the expression is defined as a function and eval() is used when defining or calling it.

The X-plot diagram area (see section 5.2) can also only be displayed under certain conditions:
• a function is defined, whereby eval() is not required, or
• the integration variable used in the expression must not be undefined.
Otherwise, the plot area identifies two undefined variables in the expression and assumes an implicit function.

4.9.3 Integrals over quantities with units of measurement

For example, if you want to calculate the integral over a length, you must introduce a dimensionless coordinate ξ = x/ℓ:

You can also use a length unit LE directly as the reference length ℓ. The integral is then written:

In the following example, the meter is used as the unit of length, but the result is independent of this choice, only the dimension must match.

Length

Cross section
height

4.10 Matrices and vectors

Matrices in SMath can have any expressions as elements, including other matrices, character strings or lists.

Matrix elements can be addressed by address indices, which are generated with the square bracket [.

The "Insert matrix" dialog allows you to specify the number of rows and columns and creates a correspondingly large field of placeholders with Insert

The dialog is opened with
- Ctrl -M
- Menu > Insert > Matrix
- Side panel matrices >

A 3×3 placeholder is created by typing mat(
A 2×2 placeholder is created by typing mat ⇆
Change size: click on the left or right bracket, then a marker appears at the bottom right where the size can be dragged with the mouse.

The function matrix(m,n) generates an m ×n zero matrix.

By assigning a matrix element, a matrix of the required size is created if it does not yet exist, or enlarged if it was not large enough.

Address indices are generated with the el() function or with the keyboard shortcut [. Initially, only one index is created; a second index can be added by entering an argument separator (comma).
Matrix elements can be addressed by address indices that are generated with the square bracket [.

4.10.1 The SMathMatrixEditor

At the moment it is not possible to add rows or columns to matrices within SMath Studio, only vectors work the same way as lists and statement blocks.
To compensate for this shortcoming, Peter Streipardt has written a separate program, the SMathMatrixEditor. It is part of the unofficial SMath distribution.

4.10.2 Functions for generating matrices and vectors

identity matrix of a given dimension

Diagonal matrix from a vector

Zero matrix of a given dimension:

The range() function generates vectors with ascending or descending numerical values. The first value and the maximum value must be specified. Optionally, the second value can be specified, which then defines the increment. It is preset to +1 or -1, depending on whether the maximum value is greater or less than the final value.

4.10.3 Calculating with matrices

Multiplication with a number

Addition, subtraction

Transposition (input as transpose() or from the matrix panel of the sidebar)

Ctrl + 1

Matrix multiplication

Determinant (input as det() or from the matrix panel of the sidebar)

Rank of a matrix

The numerical calculation of determinant and rank is not robust, as the intermediate results are too large for the internal numerical representation. In individual cases, the symbolic evaluation can still provide meaningful values, but also reaches the limits of the numerical representation.

This problem is solved by the Maxima plug-in.

Inverse matrix (input as ^-1)

The minor is the value of the sub-determinant that results from deleting the i-th row and the j-th column. The input is made as minor() or with from the matrix panel of the sidebar. The function vminor() returns the corresponding matrix, unfortunately this cannot be distinguished from the operator representation.

The algebraic complement is equal to the minor, multiplied by (-1)i+j. Input with alg() or from the matrix panel of the sidebar. Caution, the function is displayed in the same way as address indices on a matrix A

Trace of a matrix (sum of the diagonal elements)

Symbolic arithmetic with matrices and vectors

Be extremely careful with symbolic evaluation (optimization) for matrices and vector operations. SMath simplifies arithmetic expressions according to the rules for scalar values (distributivity and commutativity of multiplication). These do not generally apply to vectors and matrices. If in doubt, switch to numerical evaluation.

One example is the multiplication of three vectors, where a scalar product is formed with two vectors and the result is multiplied by the third vector. The placement of brackets is crucial here. The vectors a, b and c are not yet defined here (SMath does not yet know that the variables are to become vectors). Therefore, they must be evaluated symbolically.

As you can see, the brackets disappear. The expression is processed from left to right during the calculation. The product a -b or b -a is calculated first. Let's look at what happens when we assign vectors to the variables:

Symbolic evaluation

If SMath realizes that both factors are vectors, then the scalar product is calculated. This results in a number that is then multiplied by the vector c in the second step. However, this is definitely not what is meant. The expressions marked in red are calculated incorrectly because SMath performs the symbolic simplifications of the expressions before inserting the concrete values. If you switch to numerical evaluation, you will get the correct results:

Numerical optimization

4.10.4 Manipulation of matrices

Determine the dimensions of matrices (number of rows, number of columns, number of elements) :

Access to rows, columns and elements:

Address indices are generated with the el() function or with the keyboard shortcut [. Initially, only one index is created; a second index can be added by entering an argument separator (semicolon).

Matrix elements can be defined individually:

If the index values exceed the size of the matrix, it is expanded dynamically:

Linear addressing: Matrix elements can also be addressed via a single index that runs from 1 to the length of the matrix. However, this does not work on the left-hand side of assignments.

Matrices can be connected on top of or next to each other if the dimensions are suitable:

An important application is the row-by-row or column-by-column structure of matrices in program loops. Matrices to be constructed row by row can be initialized with zero rows and the appropriate number of columns:

Access to sub-matrices:

Search and sort

The function findrows(matrix, expression, i) picks out all rows from a matrix whose value in column i is equal to the expression. If no matching entry is found, the function returns the value zero.

With csort(Matrix, i) the rows of a matrix are sorted according to the i-th column:

Accordingly, rsort(Matrix, i) sorts the columns of a matrix based on the i-th row:

The sort() function sorts the elements of a column matrix (a vector) by size.

reverse() reverses the order of the rows in a matrix:

You can use this to sort in descending order:

The maximum and minimum of a matrix are determined using the following functions:

The Custom Functions plugin (see section 10.2.3) contains extended versions of the maximum and minimum functions, which can also cope with multiple arguments of different shapes (lists, matrices, numbers).

4.10.5 Applying functions to vector or matrix elements

Matrices can be understood as data tables, similar to Excel. If you want to link columns with functions to new columns, loops are usually required across all data rows.

Generate data vectors with a value range:

The range(2) function is used for this purpose:
x:rang ⇆ 1 → 5 ↵

Alternatively, an increment can be specified using the range(3) function

s:rang ↓ ⇆ 0 → 2* p Ctrl+G → p Ctrl+G /4 ↵

Calculations with for loops

Typically, you need a loop whose counter runs from 1 to the length of the vector. This is created like this:
for ⇆ i → rank ⇆ 1 → len ⇆ s ↵

in the loop body, the desired value is then assigned to the ith element of the result vector. Typically, the i-th element of one or more other vectors is used.
y[i space. :sin ⇆ s[i ↵

4.10.6 Totals and products

Sum and product symbols can be inserted into the spreadsheet in three ways:

•via the functions panel of the sidebar

•via the menu Insert> Functions> All> sum(4) or product

•via the keyboard
sum ↓ ⇆
pr ⇆

The sum of all numbers from 1 to 100 is calculated as follows:
sum ↓ ⇆ i → i → 1 → 100=

Quantities that you want to address in a sum with their index must be defined as a vector. Make sure to use address indices (instead of text indices). They are created with the el() function from the functions panel in the sidebar or by using the [ (square bracket open) keyboard character.

In the following example, the total center of gravity of three surfaces Ai is calculated with the partial centers of gravity xi. The quantities are defined

A[1 →:20cm⇆^2

x.S:sum ↓ ⇆ A[j Space. x[j → → → → j → 1 → n Space. /sum ⇆ A=

The denominator contains an abbreviated form of the sum sign, which sums all elements of a vector and can be entered via the "Insert function" dialog or the keyboard; sum(1) must be selected.

4.10.7 Reading matrices from files or writing them to files

By default, only the importData() function is available, which allows data to be read from text files. The decimal separator and the column separator as well as the start and end rows and columns can be specified. However, only numbers or internal SMath expressions can be read.

The importCell() function from the Excel Collaboration plugin reads a single cell from an xls (Excel) file, but this does not work for floating point numbers, only for integers or strings.

The Data Exchange plugin offers the functions ImportData.ODS() and ImportData.XLSX(), which can import entire matrices at once. ods is the file format of OpenOffice Calc, xlsx is the format of MS Excel.

The Mathcad File Access plugin offers functions for reading pixel data from audio and graphic files as well as for reading and writing audio and binary files.

File names can be provided with relative or absolute paths. The rules for the current directory are described in section 3.15.

Data can be output using the functions

4.10.8 Vector and tensor calculus in Cartesian coordinates

Vectors are represented by column matrices of their Cartesian coordinates. Similarly, tensors are represented as square coordinate matrices.

Base, vector components and vector coordinates

Base vectors in the Cartesian coordinate system:

You can also define base vectors as a function. The ith basis vector is the ith column in a unit matrix of suitable dimension (here 3):

A physical vector can be represented as the sum of the products of unitized coordinates and the corresponding basis vectors. For example, Fx, Fy, Fz are the coordinates of a force vector:

Then the products of these coordinates with the corresponding basis vectors are the so-called components of the force vector in the direction of the coordinate axes. Note: Coordinates are (possibly unitized) numbers, components are vectors and therefore have a direction.

The sum of the components gives the complete force vector:

You can also write the coordinates directly in a column matrix:

SMath can be used to address the coordinates of a vector with the index function:

Scalar product

The scalar product of two vectors a and b is a number that is equal to the product of the magnitude |a| of one, the magnitude |b| of the other and the cosine of the included angle. In Cartesian coordinates, this is equal to the sum of the products of the coordinates of the two vectors:

(symbolic evaluation)

The base vectors of Cartesian coordinate systems have a length of 1 and are perpendicular to each other. Therefore, the scalar product of a basis vector with itself has the value 1 and with another basis vector the value 0. Such a basis is called orthonormal (perpendicular and length 1).

Value

The absolute value of a vector is equal to the square root of the scalar product with itself.

(symbolic evaluation)

Cross product

The cross product of two vectors a and b is equal to a vector that is perpendicular to both vectors and whose magnitude is equal to the trapezoidal area spanned by a and b.
This area in turn is equal to the product of the magnitudes of a and b and the sine of the included angle. The cross product can be calculated very easily from the Cartesian coordinates of the two vectors. In SMath, however, a separate function is available for this, accessible via the button in the "Matrices" panel of the sidebar or Ctrl -8

The cross products of two basis vectors provide the third basis vector, which is perpendicular to both:

Spat product

The spar product of three vectors is the volume of the parallelepiped (spar) spanned by these three vectors.

(symbolic evaluation)

4.11 Lists

Lists are one-dimensional fields that are displayed in SMath as a column with a curly bracket on the left.
Lists are used in different contexts:

• Representation of several expressions in a diagram
• Simultaneous use of multiple values for a variable in expressions, set object.

In English, these objects are also called multiple values objects or systems.

4.11.1 Create list objects

Lists can be created with the button button in the "Functions" side panel or by entering
sys ⇆

Elements can be inserted by entering a semicolon before or after an expression. Empty placeholders are deleted by pressing ⇐ (backspace).
As with matrices, the length of lists can be increased or decreased by clicking on the bracket and dragging the bottom right-hand corner.

Some functions or arithmetic operations also return lists as results, such as the plus/minus operator (found on the arithmetic side panel) and the minus/plus operator (by entering the % sign):

insert > Operator

The example with undefined variables and symbolic evaluation clearly shows the difference between the two operators.

The list entries can be addressed for display or definition with address indices. If necessary, the list is extended. In the following example
- the list s is defined with the elements 1 and 2.5,
- element number 2 is displayed,
- element number 4 is set to the value 3
- the entire list is displayed.

Another display option is the question of whether values that occur more than once are summarized. This can change the length of the list. The setting is made in Main menu> Tools> Options> Calculation> Results that are multiple values. With "Show all values", all generated list elements are displayed: