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# Content MathML

Module by: Sarah Coppin, Bill Wilson. E-mail the authors

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.

Note: You are viewing an old version of this document. The latest version is available here.

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 refered to as <m:apply>.

## The Fundamentals of Content MathML: Applying Functions and Operators

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".


<m:apply>
<m:plus/>
<m:cn>2</m:cn>
<m:cn>3</m:cn>
</m:apply>


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 f, etc.) or operator (which could be plus or minus, etc.) to the elements that follow it.

### Tokens

Content MathML has three tokens: ci, cn, and csymbol. A token is basically the lowest level element; it contains no other Content MathML. 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; x, y, and f are examples of identifiers. In addition, 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 that is defined externally. It can take 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 "x with a subscript 2", where the 2 does not have a more semantic meaning, you would use the following code.


<m:ci>
<m:msub>
<m:mi>x</m:mi>
<m:mn>2</m:mn>
</m:msub>
</m:ci>


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.

### Functions and Operators

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 f of x". (Note that you have to include the attribute type='fn' on the ci tag denoting f.)


<m:apply>
<m:ci type='fn'>f</m:ci>
<m:ci>x</m:ci>
</m:apply>


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 x" is similar to the example above.


<m:apply>
<m:sin/>
<m:ci>x</m:ci>
</m:apply>


You can find a more thorough description of the different predefined functions in the MathML specification.

In addition to the predefined functions, there are also many predefined operators. A few of these are <m:plus/> (for addition), <m:minus/> (for subtraction), <m:times/> (for multiplication), <m:divide/> (for division), <m:power/> (for taking the n-power of something), and <m:root/> (for taking the n-root of something).

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.

Representing "the negative of a number" and explicitely representing "the positive of a number" has slightly unusual syntax. In this case you apply the plus or minus operator to the number or variable, etc., in question. The following is the code for "negative x."


<m:apply>
<m:minus/>
<m:ci>x</m:ci>
</m:apply>


To create more complicated expressions, you can nest these bits of apply code within each other. "a times the quantity b plus c" would be written as follows.


<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>


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 children which are the two quantities that are equal to each other. For example, "a times b plus a times c equals a times the quantity b plus c" would be written as shown.


<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>


## Integrals

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 f of x with respect to x from 0 to b."


<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>


## Derivatives

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 f of x, with respect to x."


<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>


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 f of x, with respect to x."


<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>


## Vector and Matrices

Vectors are created as a combination of other elements using the vector tag.


<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>


Matrices are done in a similar manner. Each matrix element contains several matrixrow elements. Then each matrixrow element contains several other elements.


<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>


There are also operators to take the determinant and the transpose of a matrix as well as to select elements from within the matrix.

## Entities

MathML predefines its own entities for many characters that you might need to use (Greek letters for example). They are also very useful when you need to imbed Presentation MathML within Content MathML. A list of these entities is found in the MathML 2.0 specification. It is better to use these entities than the Unicode character that they stand for, because these entities can be redefined as necessary.

## Other Resources

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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