Wolfram: Mathematica Notation Basics

Here's a list of the most commonly used Mathematica (Wolfram language) syntax.

Arithmetic Operator

• x ^ y means x raised to power of y.
• space is implicit multiplication. e.g. a b means a * b.

Brackets

• Square brackets means function argument. e.g. Sin[x]
• Parenthesis means order of operation. e.g. (3 + 5) * 2
• Curly braces means list. e.g. {a,b,c}, also for vector. Nested list represent a matrx or tree.
• Double Square brackets means getting an element from a list. e.g. {x,y,z}[[1]] returns x.

Vector and Matrix

• {a,b} is a 2D vector. {a,b,c} is 3D vector.
• {{a,b},{c,d}} is a matrix, with 1st row {a,b}, second row {c,d}.

• {a,b} . {x,y} means dot product.
• {{a,b},{c,d}} . {x,y} means matrix multiplication with the vector {x,y}.

• {a,b} + {x,y} means vector addition {a + x, b + y}.
• {a,b} + c means {a + c, b + c}.
• {x,y} * c means {x c, y c}.
• {x,y} / c means {x/z, y/z}.

• Norm[v] is the length of a vector v.
• Normalize[v] is the unit vector of v.

Function

• f[x] is a function f applied on x.
• f@x is the prefix notation for f[x]. e.g. Cos@x means Cos[x].
• x//f is the postfix notation for f[x]. e.g. x//Cos means Cos[x].

Function Definition

• Function[{x},x+1] is a lambda function, adding 1 to arg.
• Function[{x,y},x+y] is a lambda function, sum of the args.

• # the hash sign stand for parameter of a function. # or #1 is first parameter, #2 second, etc. For example, Function[#1+#2] is equivalent to Function[{x,y},x+y].
• & (ampersand) means the expression before is a function, with hash sign # meaning its parameters. e.g. (#1 + #2 &) is the same as Function[ {x, y}, x + y ]

• f[x_,y_] := x + y is a definition of function using pattern matching. Here it is a sum of 2 args.

Complex Number

• I (Capital letter I) is the imaginary number.