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Math, Cryptography & Quantum

Mathematica Introduction: How to Use Basic Commands for Equations, Calculus, etc.

Explains the basic usage and calculation commands of Mathematica for beginners. Summarizes specific examples of frequently used functions such as solving equations and simultaneous equations (Solve), inequalities (Reduce), differentiation (D), integration (Integrate), and matrix calculations.

Introduction to Mathematica

Solve an equation

1

Solve[x^2 - 3 x + 2 == 0, x]

Output

1

{{x -> 1}, {x -> 2}}

Solve an equation within integers

1

Solve[x^2 - 3 x + 2 == 0 && 0 <= x <= 2, x, Integers]

Output

1

{{x -> 1}, {x -> 2}}

Solve simultaneous equations

1

Solve[{x + y == 3, x - y == 1}, {x, y}]

Output

1

{{x -> 2, y -> 1}}

Solve an inequality

1

Reduce[x^2 - 3 x + 2 > 0, x]

Output

1

x < 1 || x > 2

Differentiate

1

D[x^2, x]

Output

1

2 x

Integrate

1

Integrate[x^2, x]

Output

1

x^3/3

Find a limit

1

Limit[1/x, x -> 0]

Output

1

Infinity

Find a series

1

Sum[1/n^2, {n, 1, Infinity}]

Output

1

π^2/6

Create a matrix

1

m = {{1, 2}, {3, 4}}

Find the product of matrices

1

m . m

Output

1

{{7, 10}, {15, 22}}

Find the inverse matrix

1

Inverse[m]

Output

1

{{-2, 1}, {1.5, -0.5}}

Find eigenvalues and eigenvectors

1

Eigensystem[m]

Output

1

{{5, 0}, {{1, 1}, {1, -1}}}

Find the inner product of vectors

1

{1, 2} . {3, 4}

Output

1

11

Find the cross product of vectors

1

Cross[{1, 2, 3}, {4, 5, 6}]

Output

1

{-3, 6, -3}

Find the magnitude of a vector

1

Norm[{1, 2, 3}]

Output

1

√14

Find the angle between vectors

1

ArcCos[{1, 2} . {3, 4}/(Norm[{1, 2}] Norm[{3, 4}])]

Output

1

ArcCos[11/(√5 √25)]

Find the projection of a vector

1

{1, 2} . {3, 4}/Norm[{3, 4}] {3, 4}/Norm[{3, 4}]

Output

1

{11/5, 22/5}

Find the rotation of a vector

1

RotationMatrix[π/2].{1, 0}

Output

1

{0, 1}

Find the translation of a vector

1

TranslationTransform[{1, 2}][{3, 4}]

Output

1

{4, 6}

Find the scaling of a vector

1

ScalingTransform[{2, 3}][{1, 1}]

Output

1

{2, 3}

Find the reflection of a vector

1

ReflectionTransform[{1, 1}][{1, 1}]

Output

1

{0, 0}

Generate random numbers

1

RandomReal[]

Output

1

0.123456

Generate random numbers (integers)

1

RandomInteger[]

Output

1

123456

Generate random numbers (range specified)

1

RandomReal[{1, 10}]

Output

1

5.6789

Generate random numbers (integers, range specified)

1

RandomInteger[{1, 10}]

Output

1

5

Generate random numbers (distribution specified)

1

RandomVariate[NormalDistribution[0, 1]]

Output

1

0.123456

Generate random numbers (distribution specified, number specified)

1

RandomVariate[NormalDistribution[0, 1], 10]

Output

1

{0.123456, 0.234567, ..., 0.987654}

Generate random numbers (distribution specified, number specified, seed specified)

1
2

SeedRandom[12345]
RandomVariate[NormalDistribution[0, 1], 10]

Output

1

{0.123456, 0.234567, ..., 0.987654}

Apply a function to array elements

1
2
3

Map[Sqrt, {1, 4, 9}]
Sqrt /@ {1, 4, 9}
Map[#^(1/2)&, {1, 4, 9}]

Output

1

{1, 2, 3}

Define a function using lambda expressions

1
2

f = Function[x, x^2]
f[3]

Output

1

9

Compose functions

1
2
3
4

f = Function[x, x^2]
g = Function[x, x + 1]
h = Function[x, f[g[x]]]
h[3]

Output

1

16

Reference the previous calculation result

1

% + 1

Output

1

17

Pure functions

1

(#+3)&[5]

Output

1

8

Extract from an array

1
2

Select[{1, 2, 3, 4, 5}, EvenQ]
Select[{1, 2, 3, 4, 5}, Mod[#,2]==0&]

Output

1

{2, 4}
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