Lecture 2.pptxLecture 2.pptxLecture 2.pptxLecture 2.pptx

Lecture 2.pptxLecture 2.pptxLecture 2.pptxLecture 2.pptxLecture 2.pptxLecture 2.pptxLecture 2.pptxLecture 2.pptxLecture 2.pptxLecture 2.pptx

1. Introduction to Mathematica

2. What is Mathematica? Overview : Mathematica is a powerful computational software used for symbolic and numerical calculations, data visualization, and technical computing. Developer : Created by Wolfram Research, it has been a cornerstone in computational software since its release in 1988. Usage : Widely used in academia, research, engineering, and industries requiring sophisticated mathematical computations.

3. Why Mathematica? Symbolic and Numerical Computation : Combines symbolic and numerical computation seamlessly, making it versatile for a range of mathematical problems. Interactive Notebooks : Provides an interactive environment for creating dynamic documents that combine code, text, and graphics. Integrated System : Features a fully integrated system with built-in functions covering various fields such as calculus, algebra, statistics, and more. Automation : Automates many aspects of computation, allowing users to focus on problem-solving rather than coding intricacies.

4. Applications Education : Essential tool in education for teaching complex mathematical concepts through interactive and visual means. Research : Facilitates advanced research in mathematics, physics, engineering, economics, and other fields by providing robust computational tools. Industry Applications : Used in finance, biotechnology, software development, and more to solve real-world problems with high precision. Innovation : Drives innovation by enabling rapid prototyping and testing of mathematical models and algorithms.

5. Getting Started Notebook and Text-Based Interfaces To start Mathematica on SCC cluster, type mathematica (or Mathematica ) at the prompt: % mathematica To start Mathematica Kernel ( text-base interface), type math at the prompt: % math

6. Getting Started Notebook and Text-Based Interfaces To start Mathematica on Windows: Start -> Wolfram Mathematica -> Wolfram Mathematica 9 To start Mathematica Kernel ( text-base interface) on Windows: Start -> Wolfram Mathematica -> Wolfram Mathematica 9 Kernel

7. Getting Started Notebook and Text-Based Interfaces Notebook Interface Text-Based Interface Start mathematica math Execute command Shift-Enter Enter Exit Choose the Quit menu item Cntr -D or Quit[]

8. Numerical Calculations 21.7 + 19.94 In[1] := 21.7 + 19.94 Out[1] := 41.64 Press Shift + Enter

9. Numerical Calculations x+y+z add x -y subtract x /y divide x y z or x*y* z multiply x^y power x *( y+z ) control grouping by parentheses Note : You can use space or a * sign for multiplication

10. Numerical Calculations You get exact result with Mathematica unless you request otherwise. In[1] := 2 ^ 100 (* get exact result *) Out[1] := 1267650600228229401496703205376 In[2] := 2 ^ 100 //N (* get approximation *) Out[2] := 1.26765x10 30 In[3] := 1/3 + 2/7 (* get exact result *) Out[3] := In[4] := 1/3 + 2/7 //N (* get approximation *) Out[4] := 0.619048  

11. Numerical Calculations If an input number contains an explicit decimal point, Mathematica produces an approximate numerical result. In[5] := 11/3 + 2/7 (* exact result *) Out[5] := In[6] := 1 . 1/3 + 2/7 (* approximation *) Out[6] := 0.652381  

12. Numerical Calculations Sqrt [x] Exp [x] e x Log[x] ln x Log[ b,x ] log b x Sin[x], Cos[x], Tan[x] trigonometric functions ArcSin [x], ... inverse trigonometric functions n! factorial FactorInteger [n] prime factors of n Abs[x] |x| Round[x] closest integer to x Max[ x,y ,...], Min[ x,y ,...] maximum and minimum of a set Mod[ n,m ] remainder of division of n by m Random[] random number between and 1 Sqrt [x] Exp [x] e x Log[x] ln x Log[ b,x ] log b x Sin[x], Cos[x], Tan[x] trigonometric functions ArcSin [x], ... inverse trigonometric functions n! factorial FactorInteger [n] prime factors of n Abs[x] |x| Round[x] closest integer to x Max[ x,y ,...], Min[ x,y ,...] maximum and minimum of a set Mod[ n,m ] remainder of division of n by m Random[] random number between and 1 Common Mathematical Functions

13. Numerical Calculations Functions in Mathematica The arguments of ALL Mathematica functions are enclosed in square brackets ; The names of built-in Mathematica functions begin with capital letters ; Unless //N option or decimal point is present, Mathematica tries to output exact value

14. Numerical Calculations Pi E e Degree I i = Infinity ∞ Pi E e Degree I Infinity ∞ Common Mathematical Constants The names of all built-in constants begin with capital letters .

15. Numerical Calculations Specify the degree of precision In[1] := N[Pi, 30] (* approximation *) Out[1] := 3.14159265358979323846264338328 In[2] := N[ Sqrt [7], 10] Out[2] := 2.645751311

16. Numerical Calculations Using Previous Results - use with care! In[1] := 7 + 3 Out[1] := 10 In[2] := % + 1 Out[2] := 11 % the last result generated %% the next-to-last result %n the result on output line Out[n] Note: % is always defined to be the last result that Mathematica generated. It can be anywhere in the script!

17. Numerical Calculations Variables definition x = value assign a value to the variable x x = y = value assign a value to both x and y x = . or Clear[x] remove any value assigned to x Notes: Mathematica is case-sensitive; To avoid confusion with built-in functions, choose names that start with lower-case letters; x y means x times y; xy with no space means variable name xy ; 5x means 5 times x;

18. Numerical Calculations Lists of Objects List is a collection of several objects in Mathematica In[1] := vec = {2, 4, 1.8} Out[1] := {2, 4, 1.8} In[2] := vec^2 Out[2] := {4, 16, 3.24} In[3] := vec /(vec-1) Out[3] := {1, , 2.25} In[4] := vec [[2]] (* extract second element *) Out[4] := 4 In[5] := Part[vec,1] (* extract first element *) Out[5] := 2  

19. Functions Function Definition Use underscore _ after a variable name (function argument) f([x_] := x^2 + 4 x + 4 ; To execute a function, simply call it with a given value: f[1] Mathematica’s built-in functions start with upper-case letter. Start with lower case letter for the user defined function.

20. Functions Using Functions Show function definition: ?f Expand function: Expand[f[x + y + 1]] Find derivative of a function: D[f[x], x] Find integral of a function: Integrate[f[x] ,x] Find definite integral of a function: Integrate[f[x] ,{x,0,1}] Clear the definition of a function: Clear[f]

21. Equations Solving equations Define equation using double equal sign == Solve[4 x^2 + 4 x + 1 == 0] Mathematica can solve an equation for one variable in terms of another Solve[5 x^2 -2 Log[y] == 3 x,y ] Solve system of equations: Solve[{x + 2 y == 5, 7 x – 5 x == -3},{ x,y }]

22. Equations Solving equations Mathematica can solve algebraic equations in one variable for power less than 5 and sometimes even higher. But there are some equations for which it is impossible to find the root(s) algebraically. Mathematica will use Root object to represent the solution. Use N[%] to evaluate the solution numerically. In some cases Mathematica can solve equations involving other functions: In[1] := Solve[Sin[x] == a, x] Out[1] := {{ x -> ArcSin [a]}}

23. Equations Solving equations You can also find an approximate numerical solution using FindRoot[] : In[1] := FindRoot[Cos[x] == x, {x,0}] Out[1] := { x -> 0.739085} Mathematica can solve system of simultaneous equations. It can eliminate a variable in a system, using Eliminate[] function, or simplify the system using Reduce[] function.

24. Programs Programming Constructs Assignments: = += ++ *= AppendTo Loops: Do While For Table Nest Conditionals: If Which Switch And(&&) Equal(==) Less(<) … Flow Control: Return Throw Catch TimeConstrained Scope Constructs: Module With Block I/O: Print Input Pause Import OpenRead …