Constants
pi=3.141592653589793

Variables
a, ai
a[i1]...[in]
a', a'...'

Vectors and matrices
{a1,...,an}
{{a11,...,a1p},...,{an1,...,anp}}

Expressions
a+b, a-b, a*b, a/b, a**b, a!
quotient(a,b), rem(a,b), factorof(a,b)
a*(b+c)

Trigonometric and hyperbolic functions
sin(x), cos(x), tan(x), cot(x)
asin(x), acos(x), atan(x), acot(x)
sinh(x), cosh(x), tanh(x), coth(x)
asinh(x), acosh(x), atanh(x), acoth(x)

Logarithm and exponential
log(x), exp(x)

Absolute value, conjugate
abs(x), conjugate(x)

Square root, cubic root
sqrt(x), cubic(x)

Implicit roots
root[i](a0,...,an)

Implicit functions
f(x), f'(x), f'...'(x)
f(x1,...,xm), f{n1,...,nm}(x1,...,xm)
f[i1]...[in](x)

Derivation
d(f(x),x,value_opt,order_opt)

Taylor series
taylor(f(x),x,value,order)

Integration
integral(f(x),x,a,b)
integral(f(x),x)

Vector algebra
grad(f(x1,...,xn),{x1,...,xn})
divergence({f1(x1,...,xn),...,fn(x1,...,xn)},{x1,...,xn})
curl({fx(x,y,z),fy(x,y,z),fz(x,y,z)},{x,y,z})
laplacian(f(x1,...,xn),{x1,...,xn})
dalembertian(f(t,x,y,z),{t,x,y,z})
jacobian({f1(x1,...,xn),...,fm(x1,...,xn)},{x1,...,xn})

Scalar and vector product
{a1,...,an}*{b1,...,bn}
vector({ax,ay,az},{bx,by,bz})

Matrix product
matrix({{a11,...,a1k},...,{an1,...,ank}},{{b11,...,b1p},...,{bk1,...,bkp}})
matrix({{a11,...,a1k},...,{an1,...,ank}},{b1,...,bk})
matrix({a1,...,ak},{{b11,...,b1p},...,{bk1,...,bkp}})

Tensor product
tensor({{a11,...,a1p},...,{an1,...,anp}},{{b11,...,b1q},...,{bm1,...,bmq}})

Complex and quaternion product
complex({a,b},{c,d})
quaternion({a,b,c,d},{e,f,g,h})

Geometric product and differential operator
geometric({a1,...,am},{b1,...,bm},algebra_opt)
del({f1(x1,...,xn),...,fm(x1,...,xn)},{x1,...,xn},algebra_opt)
m=2^n
algebra : cl(p,q), p+q=n

Transposition, trace, determinant
transpose({{a11,...,a1p},...,{an1,...,anp}})
trace({{a11,...,a1n},...,{an1,...,ann}})
determinant({{a11,...,a1n},...,{an1,...,ann}})

Binomial coefficient
C(n,p)

Polynomial coefficients and solving
coef(p(x),x)
solve(p(x),x,subscript_opt)

Variable substitution
subst(f(x),x,value)
subst(f(x1,...,xn),{x1,...,xn},{a1,...,an})

Graphing and various transforms
graph(f(x),x)
graph({f1(x),...,fn(x)},x)
expand(a)
factorize(a)
elementary(a)
simplify(a)
numeric(a)
quote(a)

Limit
limit(x),x,value,direction_opt)
value : -oo, a, oo
direction : <0, 0, >0

Sum, product
sum(a[i],i,n1,n2)
product(a[i],i,n1,n2)

Comparison
eq(a,b)
neq(a,b)
leq(a,b)
lt(a,b)
geq(a,b)
gt(a,b)
approx(a,b)

Boolean functions
and(a,b)
or(a,b)
xor(a,b)
not(a)
implies(a,b)

Number theory
mod(a,modulo)
modpow(a,exponent,modulo)
modinv(a,modulo)
eulerphi(a)
primitiveroots(a)

Groebner basis computation
groebner({p1(x1,...,xm),...,pn(x1,...,xm)},{x1,...,xm},ordering_opt,modulo_opt)
ordering : lex=lexicographic, tdl=total degree lexicographic, drl=degree reverse lexicographic, elim(k)=kth-elimination
modulo : prime integer