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2.2.4 Applying Functions to Parts of Expressions
If you have a list of elements, it is often important to be able to apply a function separately to each of the elements. You can do this in Mathematica using Map.
This applies f separately to each element in a list.
In[1]:= Map[f, {a, b, c}]
Out[1]= 
This defines a function which takes the first two elements from a list.
In[2]:= take2[list_] := Take[list, 2]
You can use Map to apply take2 to each element of a list.
In[3]:= Map[take2, {{1, 3, 4}, {5, 6, 7}, {2, 1, 6, 6}}]
Out[3]= 
Applying a function to each element in a list.
What Map[f, expr] effectively does is to "wrap" the function f around each element of the expression expr. You can use Map on any expression, not just a list.
This applies f to each element in the sum.
In[4]:= Map[f, a + b + c]
Out[4]= 
This applies Sqrt to each argument of g.
In[5]:= Map[Sqrt, g[x^2, x^3]]
Out[5]= 
Map[f, expr] applies f to the first level of parts in expr. You can use MapAll[f, expr] to apply f to all the parts of expr.
This defines a  matrix m.
In[6]:= m = {{a, b}, {c, d}}
Out[6]= 
Map applies f to the first level of m, in this case the rows of the matrix.
In[7]:= Map[f, m]
Out[7]= 
MapAll applies f at all levels in m. If you look carefully at this expression, you will see an f wrapped around every part.
In[8]:= MapAll[f, m]
Out[8]= 
In general, you can use level specifications as described in Section 2.1.7 to tell Map to which parts of an expression to apply your function.
This applies f only to the parts of m at level 2.
In[9]:= Map[f, m, {2}]
Out[9]= 
Setting the option Heads->True wraps f around the head of each part, as well as its elements.
In[10]:= Map[f, m, Heads->True]
Out[10]= 
Ways to apply a function to different parts of expressions.
Level specifications allow you to tell Map to which levels of parts in an expression you want a function applied. With MapAt, however, you can instead give an explicit list of parts where you want a function applied. You specify each part by giving its indices, as discussed in Section 2.1.4.
Here is a  matrix.
In[11]:= mm = {{a, b, c}, {b, c, d}}
Out[11]= 
This applies f to parts {1, 2} and {2, 3}.
In[12]:= MapAt[f, mm, {{1, 2}, {2, 3}}]
Out[12]= 
This gives a list of the positions at which b occurs in mm.
In[13]:= Position[mm, b]
Out[13]= 
You can feed the list of positions you get from Position directly into MapAt.
In[14]:= MapAt[f, mm, %]
Out[14]= 
To avoid ambiguity, you must put each part specification in a list, even when it involves only one index.
In[15]:= MapAt[f, {a, b, c, d}, {{2}, {3}}]
Out[15]= 
Applying a function to specific parts of an expression.
Here is an expression.
In[16]:= t = 1 + (3 + x)^2 / x
Out[16]= 
This is the full form of t.
In[17]:= FullForm[ t ]
Out[17]//FullForm= 
You can use MapAt on any expression. Remember that parts are numbered on the basis of the full forms of expressions.
In[18]:= MapAt[f, t, {{2, 1, 1}, {2, 2}}]
Out[18]= 
Applying a function to parts and their indices.
This applies f to each element in a list, giving the index of the element as a second argument to f.
In[19]:= MapIndexed[f, {a, b, c}]
Out[19]= 
This applies f to both levels in a matrix.
In[20]:= MapIndexed[f, {{a, b}, {c, d}}, 2]
Out[20]= 
Map allows you to apply a function of one argument to parts of an expression. Sometimes, however, you may instead want to apply a function of several arguments to corresponding parts of several different expressions. You can do this using MapThread.
Applying a function to several expressions at once.
This applies f to corresponding pairs of list elements.
In[21]:= MapThread[f, {{a, b, c}, {ap, bp, cp}}]
Out[21]= 
MapThread works with any number of expressions, so long as they have the same structure.
In[22]:= MapThread[f, {{a, b}, {ap, bp}, {app, bpp}}]
Out[22]= 
Functions like Map allow you to create expressions with parts modified. Sometimes you simply want to go through an expression, and apply a particular function to some parts of it, without building a new expression. A typical case is when the function you apply has certain "side effects", such as making assignments, or generating output.
Evaluating functions on parts of expressions.
Map constructs a new list in which f has been applied to each element of the list.
In[23]:= Map[f, {a, b, c}]
Out[23]= 
Scan evaluates the result of applying a function to each element, but does not construct a new expression.
In[24]:= Scan[Print, {a, b, c}]
Scan visits the parts of an expression in a depth-first walk, with the leaves visited first.
In[25]:= Scan[Print, 1 + x^2, Infinity]
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