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!
div(a,b), mod(a,b)
a*(b+c)

Trigonometric functions
sin(x), cos(x), tan(x)
asin(x), acos(x), atan(x)

Hyperbolic functions
sinh(x), cosh(x), tanh(x)
asinh(x), acosh(x), atanh(x)

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

Absolute value, sign
abs(x), sgn(x)

Square root
sqrt(x)

Conjugate
conjugate(x)

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

Implicit functions
f(x), f'(x), f'...'(x)
f(x1,...,xm), f{n1,...,nm}(x1,...,xm)

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

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

Vector algebra
grad(f(x1,...,xn),{x1,...,xn})
diverg({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})
laplacian({f1(x1,...,xn),...,fm(x1,...,xn)},{x1,...,xn})
jacobian({f1(x1,...,xn),...,fm(x1,...,xn)},{x1,...,xn})

Geometric differential operator
del({f1(x1,...,xn),...,fp(x1,...,xn)},{x1,...,xn})
p=2^n, n=1...4

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
geometric({a1,...,ap},{b1,...,bp})
p=2^n, n=0...4

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

Polynomial solving
solve(p(x),x,subscript_opt)

Variable substitution
subst(f(x),x,value)

Limit
lim(f(x),x,value,direction_opt)
value : -infin, a, infin
direction : <0, 0, >0

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

Number theory
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