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
simplifies each side with a s
(calc-simplify) 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 normal
default simplifications on the selected sub-formula. These are
the simplifications that are normally done automatically on all
results, but which 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 a s (calc-simplify) command
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.