Welcome to Mathematica!
First things are first: you evaluate input by selecting a cell and pressing “Shift-Enter”.
You assign values to a variable using =; the names of a variables cannot start with a number.
x = 1
x1234 = 2
2 x
You cannot change the value of predefined variables, like Pi or E.
If Mathematica currently holds a value for a variable, it shows up in black; if it is undefined, then it shows up in blue.
Mathematica tries to be exact unless you specify otherwise; if you want the decimal approximation of a number, use N[number]
{E, e, Pi, applesauce}
N[{E, e, Pi, applesauce}]
You can remove the value of a variable using Clear[x]
Clear[x, x1234]
Square brackets are used for function arguments; Parentheses are used for order of operations; Curly brackets denote lists.
func = Sin[x]
poly = (x + y)^2
list = {w, x, y, z}
list2 = {a, b, c, d}
The built-in Help in Mathematica is amazing. If you don’t know what something does or how it’s used, either press F1 or use ? with the function name:
?LegendreP
You can find more information by using ?? before the function:
?? Plot
* is a wildcard; Plot*, for example, will return every function which starts with the characters “Plot”. You can use two of them at once as well:
?*Bessel*
As we stated, curly brackets denote lists.
list = {w, x, y, z}
list2 = {a, b, c, d}
Two square brackets take only the specified element of the list:
list[[2]]
Lists in Mathematica are much like vectors; nested lists (i.e. lists of lists) represent matrices.
Multiplication by a constant is pretty straightforward:
2 list
You can take the dot product of two lists:
list.list2
It’s sometimes useful to remove a piece of the list with Drop or select a specific part of a list with Take:
Drop[list, 1]
Drop[list, -1]
Take[list, 2]
To make two lists into one (Union also removes duplicated elements):
list3 = Union[list, list2]
To split a long list into a series of sublists:
Partition[list3, 2]
You can generate lists by hand using curly brackets, but Table does it automatically by evaluating an expression which runs over the index:
listInt = Table[i, {i, 1, 12}]
listSq = Table[i^2, {i, 1, 12}]
You can make two lists into a list of ordered pairs using Transpose:
Transpose[{listInt, listSq}]
list
You can temporarily set the value of a variable using the following commands:
list /. x -> 2
list /. {w -> 1, y -> 3}
(Semicolons suppress the output of a command, by the way.)
User-defined functions can take any number of arguments, but you have to specify them initially.
func2[x_] = Sin[x];
func3[a_,b_,c_] = a + b^2 + c^4;
(Note the necessary underscores!)
Evaluaion of functions is easy! ```Mathematica N[func2[1]]
N[func3[2,4,10] ```
If you don’t define the argument of a function, then you can still work with it using replacement rules:
func = Sin[x];
N[func /. x -> 1]
Visualization in Mathematica is an intricate subject; we will take it up in detail later. For now, only the basics.
Continuous functions can be plotted using Plot:
Plot[Sin[x]^2, {x, 0, 2 Pi}, ImageSize -> Large]
For lists, you use ListPlot:
ListPlot[pairs, ImageSize -> Large]
Usually, functions in Mathematica are fully spelled out. Not so for derivatives! The function is just a capital D.
D[Sin[w t], t]
D[BesselJ[n, x], x]
Mathematica can do both definite and indefinite integrals:
psi[x_] = A Sin[3 x];
Integrate[psi[x]^2, {x, -a, a}]
int = Integrate[(Sin[t^2]^4 - 10 Tan[2 t])/Csc[t], t]
N[int /. t -> 0]
As we all know, some integrals don’t have exact solutions (or they are really tough to find). You can do a numerical integration using NIntegrate.
NIntegrate[x^2/(E^x - 1), {x, 0, Infinity}]
(* ... *)