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Summary: A short introduction to writing Content MathML by hand. It covers tokens, prefix notation, and applying functions and operators. In addition it introduces writing derivatives, integrals, vectors, and matrices.
The authoritative reference for Content MathML is Section 4 of the MathML 2.0 Specification. The World Wide Web Consortium (W3C) is the body that wrote the specification for MathML. The text is very readable and it is easy to find what you are looking for. Look there for answers to questions that are not answered in this tutorial or when you need more elaboration. This tutorial is based on MathML 2.0.
In this document, the m prefix is used to
denote tags in the MathML namespace. Thus the
<apply> tag is referred to as
<m:apply>. Remember all markup in
the MathML namespace must be surrounded by
<m:math> tags.
The fundamental concept to grasp about Content MathML is that it consists of applying a series of functions and operators to other elements. To do this, Content MathML uses prefix notation. Prefix notation is when the operator comes first and is followed by the operands. Here is how to write "2 plus 3".
MathML Specification
Mozilla for Windows
<m:math>
<m:apply>
<m:plus/>
<m:cn>2</m:cn>
<m:cn>3</m:cn>
</m:apply>
</m:math>
There are three types of elements in the Content MathML
example shown above. First, there is the apply
tag, which indicates that an operator (or function) is about
to be applied to the operands. Second, there is the function
or operator to be applied. In this case the operator,
plus, is being applied. Third, the operands
follow the operator. In this case the operands are the
numbers being added. In summary, the apply tag applies the
function (which could be sin or
Content MathML has three tokens: ci,
cn, and csymbol. A
token is basically the lowest level element.
The tokens denote what kind of element you are acting on.
The cn tag indicates that the content of the
tag is a number. The ci tag indicates that the
content of the tag is an identifier. An
identifier could be any variable or function;
ci elements can contain
Presentation MathML. Tokens, especially ci and
cn, are used profusely in Content MathML.
Every number, variable, or function is marked by a token.
csymbol is a different type of token from
ci and cn. It is used to create a
new object whose semantics is defined externally. It can
contain plain text or Presentation MathML. If you find that
you need something, such as an operator or function, that is
not defined in Content MathML, then you can use csymbol to
create it.
Both ci and csymbol can use
Presentation MathML to determine how an identifier or a new
symbol will be rendered. To learn more about Presentation
MathML see Section 3
of the MathML 2.0 Specification. For example, to
denote "
<m:math>
<m:ci>
<m:msub>
<m:mi>x</m:mi>
<m:mn>2</m:mn>
</m:msub>
</m:ci>
</m:math>
The ci elements have a type attribute which
can be used to provide more information about the content of
the element. For example, you can declare the contents of a
ci tag to be a function
(type='fn'), or a vector
(type='vector'), or a complex number
(type='complex'), as well as any number of
other things. Using the type attribute helps encode the
meaning of the math that you are writing.
In order to apply a function to a variable, make the
function the first argument of an apply. The second
argument will be the variable. For example, you would use
the following code to encode the meaning, "the function
type='fn' on the
ci tag denoting
<m:math>
<m:apply>
<m:ci type='fn'>f</m:ci>
<m:ci>x</m:ci>
</m:apply>
</m:math>
There are also pre-defined functions and operators in
Content MathML. For example, sine and cosine are
predefined. These predefined functions and operators are
all empty tags and they directly follow the
apply tag. "The sine of
<m:math>
<m:apply>
<m:sin/>
<m:ci>x</m:ci>
</m:apply>
</m:math>
You can find a more thorough description of the different predefined functions in Chapter 4 of the MathML specification.
In addition to the predefined functions, there are also many
predefined operators. A few of these are plus
(for addition), minus (for subtraction),
times (for multiplication), divide
(for division), power (for taking the
Most operators expect a specific number of child tags. For example, the power operator expects two children. The first child is the base and the second is the value in the exponent. However, there are other tags which can take many children. For example, the plus operator merely expects one or more children. It will add together all of its children whether there are two or five. This is referred to as an n-ary operator.
Representing "the negative of a variable" and explicitly
representing "the positive of a variable or number" has
slightly unusual syntax. In this case you apply the plus or
minus operator to the variable or number, etc., in question.
The following is the code for "negative
<m:math>
<m:apply>
<m:minus/>
<m:ci>x</m:ci>
</m:apply>
</m:math>
In contrast to representing the negative of a variable, the negative of a number may be coded as follows:
<m:math><m:cn>-1</m:cn></m:math>
To create more complicated expressions, you can nest these
bits of apply code within each other. You can create
arbitrarily complex expressions this way.
"
<m:math>
<m:apply>
<m:times/>
<m:ci>a</m:ci>
<m:apply>
<m:plus/>
<m:ci>b</m:ci>
<m:ci>c</m:ci>
</m:apply>
</m:apply>
</m:math>
The eq operator is used to write equations. It
is used in the same way as any other operator. That is, it
is the first child of an apply. It takes two (or more)
children which are the two quantities that are equal to each
other. For example, "
<m:math>
<m:apply>
<m:eq/>
<m:apply>
<m:plus/>
<m:apply>
<m:times/>
<m:ci>a</m:ci>
<m:ci>b</m:ci>
</m:apply>
<m:apply>
<m:times/>
<m:ci>a</m:ci>
<m:ci>c</m:ci>
</m:apply>
</m:apply>
<m:apply>
<m:times/>
<m:ci>a</m:ci>
<m:apply>
<m:plus/>
<m:ci>b</m:ci>
<m:ci>c</m:ci>
</m:apply>
</m:apply>
</m:apply>
</m:math>
The operator for an integral is int. However,
unlike the operators and functions discussed above, it has
children that define the independent variable that you
integrate with respect to (bvar) and the interval
over which the integral is taken (use either
lowlimit and uplimit, or
interval, or condition).
lowlimit and uplimit (which go
together), interval, and condition
are just three different ways of denoting the integrands.
Don't forget that the bvar, lowlimit,
uplimit, interval, and
condition children take token elements as well.
The following is "the integral of
<m:math>
<m:apply>
<m:int/>
<m:bvar><m:ci>x</m:ci></m:bvar>
<m:lowlimit><m:cn>0</m:cn></m:lowlimit>
<m:uplimit><m:ci>b</m:ci></m:uplimit>
<m:apply>
<m:ci type='fn'>f</m:ci>
<m:ci>x</m:ci>
</m:apply>
</m:apply>
</m:math>
The derivative operator is diff. The derivative
is done in much the same way as the integral. That is, you
need to define a base variable (using bvar). The
following is "the derivative of the function
<m:math>
<m:apply>
<m:diff/>
<m:bvar>
<m:ci>x</m:ci>
</m:bvar>
<m:apply>
<m:ci type="fn">f</m:ci>
<m:ci>x</m:ci>
</m:apply>
</m:apply>
</m:math>
To apply a higher level derivative to a function, add a
degree tag inside of the bvar tag.
The degree tag will contain the order of the derivative. The
following shows "the second derivative of the function
<m:math>
<m:apply>
<m:diff/>
<m:bvar>
<m:ci>x</m:ci>
<m:degree><m:cn>2</m:cn></m:degree>
</m:bvar>
<m:apply><m:ci type="fn">f</m:ci>
<m:ci>x</m:ci>
</m:apply>
</m:apply>
</m:math>
Vectors are created as a combination of other elements using
the vector tag.
<m:math>
<m:vector>
<m:apply>
<m:plus/>
<m:ci>x</m:ci>
<m:ci>y</m:ci>
</m:apply>
<m:ci>z</m:ci>
<m:cn>0</m:cn>
</m:vector>
</m:math>
Matrices are done in a similar manner. Each
matrix element contains several
matrixrow elements. Then each
matrixrow element contains several other
elements.
<m:math>
<m:matrix>
<m:matrixrow>
<m:ci>a</m:ci>
<m:ci>b</m:ci>
<m:ci>c</m:ci>
</m:matrixrow>
<m:matrixrow>
<m:ci>d</m:ci>
<m:ci>e</m:ci>
<m:ci>f</m:ci>
</m:matrixrow>
<m:matrixrow>
<m:ci>g</m:ci>
<m:ci>h</m:ci>
<m:ci>j</m:ci>
</m:matrixrow>
</m:matrix>
</m:math>
There are also operators to take the determinant and the transpose of a matrix as well as to select elements from within the matrix.
MathML defines its own entities for many special characters used in mathematical notation. While the entity references have the advantage of being mnemonic with respect to the characters they stand for, they also entail some technical limitations, and so their use in Connexions content is deprecated. Please use the UTF-8-encoded Unicode characters themselves where possible, or, failing that, the XML Unicode character references for the characters. At some time in the future, the Connexions repository system will likely convert entity references and character references silently to the UTF-8-encoded Unicode characters they stand for. See 6.2.1 Unicode Character Data from the XML Specification for more information. The MathML specification contains a list of character entities with their corresponding Unicode code points.
There are character picker utilities available to help you select and paste UTF-8 characters into applications like Connexions. If you are running Microsoft Windows, the Windows accessory Character Map can help you. The "Lucida Sans Unicode" font seems to have a good selection of mathematical operators and special characters. Under Linux, the charmap utility and GNOME applet provide access to all Unicode characters.
There is a lot more that can be done with Content MathML. Especially if you are planning on writing a lot of Content MathML, it is well worth your time to take a look at the MathML specification.
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This work is licensed by Sarah Coppin and Brent Hendricks under a Creative Commons Attribution License (CC-BY 1.0), and is an Open Educational Resource.
Last edited by Chuck Bearden on Nov 2, 2005 5:13 pm US/Central.
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