Symbolic algebra

With =, Beermat works out a number. With the symbolic arrow →, it works on the expression itself and gives an expression back.

x/2 + x/3 -> ?
diff(x*sin(x), x) -> ?
int(x*sin(x), x) -> ? — As text
x/2 + x/3 -> ? diff(x*sin(x), x) -> ? int(x*sin(x), x) -> ?
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Type the expression, then Ctrl+. for the arrow. In a file the arrow is written ->, ending in -> ? where the result goes.

Keywords

A keyword between the expression and the result says what to do. Ctrl+Shift+. adds one, or type it after the arrow:

As text

(x + 1)^2 -> expand -> ?
x^2 - 1 -> factor -> ?
x^2 - 4 -> solve(x) -> ?
a*x^2 + b*x + c -> solve(x) -> ?
sin(x) -> series(x, 6) -> ?
π -> float(4) -> ?

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Keywords chain from left to right, each working on the one before:

As text

x^2 + 1 -> substitute(x == y + 1) -> expand -> ?
sqrt(x^2) -> assume(x > 0) -> simplify -> ?

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Calculus

int(x^2, x, 0, a) -> ?
lim(sin(x)/x, x, 0) -> ?
sum(1/k^2, k, 1, ∞) -> ?
1/(s^2 + 1) -> invlaplace(s) -> ? — As text
int(x^2, x, 0, a) -> ? lim(sin(x)/x, x, 0) -> ? sum(1/k^2, k, 1, ∞) -> ? 1/(s^2 + 1) -> invlaplace(s) -> ?
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Systems

As text
[x + y == 3; x - y == -1] -> solve(x, y) -> ?
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Each row of the result is one solution.

What the sheet has defined

Names defined above are put in: numbers by their value, and functions by their formula. Names that aren’t defined stay letters.

w := 3
w^2 + z -> ?
g_1(u) := u^2
diff(g_1(t), t) -> ? — As text
w := 3 w^2 + z -> ? g_1(u) := u^2 diff(g_1(t), t) -> ?
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A value with units goes in with its units, and they are checked:

q := 12 kN/m
L := 6 m
q*L^2/8 -> ? — As text
q := 12 kN/m L := 6 m q*L^2/8 -> ?
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The keywords

Keyword	Does
`simplify` (or no keyword)	Simplifies
`expand`, `factor`, `collect(x)`	Multiplies out; factors; gathers the powers of x
`solve(x)`, `solve(x, y)`	Solves an equation (`a == b`, or an expression equal to 0), or a system
`substitute(x == v)`	Puts v in place of x
`series(x, n)`, `series(x == a, n)`	The series in x (about a) up to below degree n
`parfrac(x)`, `coeffs(x)`	Partial fractions; the coefficients of a polynomial, from the constant up
`float`, `float(n)`	A decimal number, to n significant figures
`laplace(t)`, `invlaplace(s)`, `fourier(t)`, `invfourier(ω)`, `ztrans(n)`	Transforms
`assume(x > 0)`, `assume(x, real)`	Assumptions for the keywords after it

diff, int, lim, sum and prod work with the arrow too, as above.