## 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]=

 Map[f, {a, b, ... }] apply f to each element in a list, giving {f[a], f[b], ... }

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.
 Out[10]=

 Map[f, expr] or f /@ expr apply f to the first-level parts of expr MapAll[f, expr] or f //@ expr apply f to all parts of expr Map[f, expr, lev] apply f to each part of expr at levels specified by lev

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]=

 MapAt[f, expr, {, , ... }] apply f to specified parts of expr

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]=

 MapIndexed[f, expr] apply f to the elements of an expression, giving the part specification of each element as a second argument to f MapIndexed[f, expr, lev] apply f to parts at specified levels, giving the list of indices for each part as a second argument to f

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.

 MapThread[f, {, , ... }] apply f to corresponding elements in each of the MapThread[f, {, , ... }, lev] apply f to parts of the at the specified level

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.

 Scan[f, expr] evaluate f applied to each element of expr in turn Scan[f, expr, lev] evaluate f applied to parts of expr on levels specified by lev

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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