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#### 10.1.5 Rearranging Formulas using Selections

The j R (`calc-commute-right`) command moves the selected sub-formula to the right in its surrounding formula. Generally the selection is one term of a sum or product; the sum or product is rearranged according to the commutative laws of algebra.

As with j ' and j <DEL>, the term under the cursor is used if there is no selection in the current formula. All commands described in this section share this property. In this example, we place the cursor on the ‘a’ and type j R, then repeat.

```     1:  a + b - c          1:  b + a - c          1:  b - c + a
```

Note that in the final step above, the ‘a’ is switched with the ‘c’ but the signs are adjusted accordingly. When moving terms of sums and products, j R will never change the mathematical meaning of the formula.

The selected term may also be an element of a vector or an argument of a function. The term is exchanged with the one to its right. In this case, the “meaning” of the vector or function may of course be drastically changed.

```     1:  [a, b, c]          1:  [b, a, c]          1:  [b, c, a]

1:  f(a, b, c)         1:  f(b, a, c)         1:  f(b, c, a)
```

The j L (`calc-commute-left`) command is like j R except that it swaps the selected term with the one to its left.

With numeric prefix arguments, these commands move the selected term several steps at a time. It is an error to try to move a term left or right past the end of its enclosing formula. With numeric prefix arguments of zero, these commands move the selected term as far as possible in the given direction.

The j D (`calc-sel-distribute`) command mixes the selected sum or product into the surrounding formula using the distributive law. For example, in ‘a * (b - c)’ with the ‘b - c’ selected, the result is ‘a b - a c’. This also distributes products or quotients into surrounding powers, and can also do transformations like ‘exp(a + b)’ to ‘exp(a) exp(b)’, where ‘a + b’ is the selected term, and ‘ln(a ^ b)’ to ‘ln(a) b’, where ‘a ^ b’ is the selected term.

For multiple-term sums or products, j D takes off one term at a time: ‘a * (b + c - d)’ goes to ‘a * (c - d) + a b’ with the ‘c - d’ selected so that you can type j D repeatedly to expand completely. The j D command allows a numeric prefix argument which specifies the maximum number of times to expand at once; the default is one time only.

The j D command is implemented using rewrite rules. See Selections with Rewrite Rules. The rules are stored in the Calc variable `DistribRules`. A convenient way to view these rules is to use s e (`calc-edit-variable`) which displays and edits the stored value of a variable. Press C-c C-c to return from editing mode; be careful not to make any actual changes or else you will affect the behavior of future j D commands!

To extend j D to handle new cases, just edit `DistribRules` as described above. You can then use the s p command to save this variable's value permanently for future Calc sessions. See Operations on Variables.

The j M (`calc-sel-merge`) command is the complement of j D; given ‘a b - a c’ with either ‘a b’ or ‘a c’ selected, the result is ‘a * (b - c)’. Once again, j M can also merge calls to functions like `exp` and `ln`; examine the variable `MergeRules` to see all the relevant rules.

The j C (`calc-sel-commute`) command swaps the arguments of the selected sum, product, or equation. It always behaves as if j b mode were in effect, i.e., the sum ‘a + b + c’ is treated as the nested sums ‘(a + b) + c’ by this command. If you put the cursor on the first ‘+’, the result is ‘(b + a) + c’; if you put the cursor on the second ‘+’, the result is ‘c + (a + b)’ (which the default simplifications will rearrange to ‘(c + a) + b’). The relevant rules are stored in the variable `CommuteRules`.

You may need to turn default simplifications off (with the m O command) in order to get the full benefit of j C. For example, commuting ‘a - b’ produces ‘-b + a’, but the default simplifications will “simplify” this right back to ‘a - b’ if you don't turn them off. The same is true of some of the other manipulations described in this section.

The j N (`calc-sel-negate`) command replaces the selected term with the negative of that term, then adjusts the surrounding formula in order to preserve the meaning. For example, given ‘exp(a - b)’ where ‘a - b’ is selected, the result is ‘1 / exp(b - a)’. By contrast, selecting a term and using the regular n (`calc-change-sign`) command negates the term without adjusting the surroundings, thus changing the meaning of the formula as a whole. The rules variable is `NegateRules`.

The j & (`calc-sel-invert`) command is similar to j N except it takes the reciprocal of the selected term. For example, given ‘a - ln(b)’ with ‘b’ selected, the result is ‘a + ln(1/b)’. The rules variable is `InvertRules`.

The j E (`calc-sel-jump-equals`) command moves the selected term from one side of an equation to the other. Given ‘a + b = c + d’ with ‘c’ selected, the result is ‘a + b - c = d’. This command also works if the selected term is part of a ‘*’, ‘/’, or ‘^’ formula. The relevant rules variable is `JumpRules`.

The j I (`calc-sel-isolate`) command isolates the selected term on its side of an equation. It uses the a S (`calc-solve-for`) command to solve the equation, and the Hyperbolic flag affects it in the same way. See Solving Equations. When it applies, j I is often easier to use than j E. It understands more rules of algebra, and works for inequalities as well as equations.

The j * (`calc-sel-mult-both-sides`) command prompts for a formula using algebraic entry, then multiplies both sides of the selected quotient or equation by that formula. It performs the default algebraic simplifications before re-forming the quotient or equation. You can suppress this simplification by providing a prefix argument: C-u j *. There is also a j / (`calc-sel-div-both-sides`) which is similar to j * but dividing instead of multiplying by the factor you enter.

If the selection is a quotient with numerator 1, then Calc's default simplifications would normally cancel the new factors. To prevent this, when the j * command is used on a selection whose numerator is 1 or -1, the denominator is expanded at the top level using the distributive law (as if using the C-u 1 a x command). Suppose the formula on the stack is ‘1 / (a + 1)’ and you wish to multiplying the top and bottom by ‘a - 1’. Calc's default simplifications would normally change the result ‘(a - 1) /(a + 1) (a - 1)’ back to the original form by cancellation; when j * is used, Calc expands the denominator to ‘a (a - 1) + a - 1’ to prevent this.

If you wish the j * command to completely expand the denominator of a quotient you can call it with a zero prefix: C-u 0 j *. For example, if the formula on the stack is ‘1 / (sqrt(a) + 1)’, you may wish to eliminate the square root in the denominator by multiplying the top and bottom by ‘sqrt(a) - 1’. If you did this simply by using a simple j * command, you would get ‘(sqrt(a)-1)/ (sqrt(a) (sqrt(a) - 1) + sqrt(a) - 1)’. Instead, you would probably want to use C-u 0 j *, which would expand the bottom and give you the desired result ‘(sqrt(a)-1)/(a-1)’. More generally, if j * is called with an argument of a positive integer n, then the denominator of the expression will be expanded n times (as if with the C-u n a x command).

If the selection is an inequality, j * and j / will accept any factor, but will warn unless they can prove the factor is either positive or negative. (In the latter case the direction of the inequality will be switched appropriately.) See Declarations, for ways to inform Calc that a given variable is positive or negative. If Calc can't tell for sure what the sign of the factor will be, it will assume it is positive and display a warning message.

For selections that are not quotients, equations, or inequalities, these commands pull out a multiplicative factor: They divide (or multiply) by the entered formula, simplify, then multiply (or divide) back by the formula.

The j + (`calc-sel-add-both-sides`) and j - (`calc-sel-sub-both-sides`) commands analogously add to or subtract from both sides of an equation or inequality. For other types of selections, they extract an additive factor. A numeric prefix argument suppresses simplification of the intermediate results.

The j U (`calc-sel-unpack`) command replaces the selected function call with its argument. For example, given ‘a + sin(x^2)’ with ‘sin(x^2)’ selected, the result is ‘a + x^2’. (The ‘x^2’ will remain selected; if you wanted to change the `sin` to `cos`, just press C now to take the cosine of the selected part.)

The j v (`calc-sel-evaluate`) command performs the basic simplifications on the selected sub-formula. These simplifications would normally be done automatically on all results, but may have been partially inhibited by previous selection-related operations, or turned off altogether by the m O command. This command is just an auto-selecting version of the a v command (see Algebraic Manipulation).

With a numeric prefix argument of 2, C-u 2 j v applies the default algebraic simplifications to the selected sub-formula. With a prefix argument of 3 or more, e.g., C-u j v applies the a e (`calc-simplify-extended`) command. See Simplifying Formulas. With a negative prefix argument it simplifies at the top level only, just as with a v. Here the “top” level refers to the top level of the selected sub-formula.

The j " (`calc-sel-expand-formula`) command is to a " (see Algebraic Manipulation) what j v is to a v.

You can use the j r (`calc-rewrite-selection`) command to define other algebraic operations on sub-formulas. See Rewrite Rules.