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This is an example based course aiming to teach Mathematica at an understandable level to students in college. Advanced high school students, or students whose high school teaches Mathematica will also find this course invaluable.
It assumes no understanding of programming languages, although knowledge, even rudimentary, of C/C++/Java is a plus.
This course does NOT teach Mathematica as a programming language.
This course does NOT teach Mathematics in general. Although a textbook on Pure Mathematics will come in handy for reference.
Any high school textbook that teaches college level Pure Mathematics is recommended. College students may use their course textbooks recommended by their Professor.
Students are expected to:
1. Know basic arithmetic and algebra including basic transcendental functions (exponential and trigonometric)
2. Know what a function is, including function composition, inverse and the definition of onetoone functions
4. How to solve equations and inequalities.
3. Know common graphs of functions
4. Basic knowledge of set theory (preferably including partitioning and powersets)
5. Knowledge of matrices and vectors, preferably also knowledge of vector fields.
6. Rudimentary knowledge of 3D functions and functions on more than one variable
7. A broad knowledge of calculus including the basics of differential equations
8. Rudimentary knowledge of complex numbers
This course is in no way affiliated with Wolfram Research, Inc. The software, the Mathematica trademark and the associated logo belongs to Wolfram Research, Inc.
ScreenFlow, the software and associated trademarks belong to Telestream, Inc.
To contact me please email: 2shakilrafi@gmail.com
Follow me on Twitter @2shakilrafi
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Section 1: Bits and Bolts and Bolts and Bits  

Lecture 1  07:34  
This video, introduces who I am [Shakil Rafi] and to what Mathematica is. Here I explain the fact that chances are your college/university will have a site license. What that means is that your college has probably bought a bulk license for Mathematica, and its students can access it for the low, low price of zero, ask your academic advisor. If your college does not have a site license, you can still get it though, but you will need to verify your studenthood by entering your college email id: joesixpack@somerandomcollege.edu Music by: "Pamgaea" Kevin MacLeod (incompetech.com)
Licensed under Creative Commons: By Attribution 3.0 

Quiz 1 
Introduction

4 questions  
Lecture 2  07:21  
In this video I show you how the course is arranged: 1. Arithmetic (+,,*,/,N,Trig,Exp) 2. Algebra (Manipulation and how to solve algebraic expressions) 3. Lists (defining lists, changing them, getting information out of them) 4. Graphs (how to draw the most common types and how to adjust them) 5. Calculus (differentiation and integration, including differential equations and the laplace transform) 6. Matrices and Vectors (matrices and vectors and how to perform operations on them. There is a quiz after almost all of these lectures, so beware! Here I also introduce the concept of notebooks and how to save notebooks: Some basic keys that my software, ScreenFlow will record for me is also shown 

Quiz 2 
Getting the hang

4 questions  
Section 2: Arithmetic a.k.a. your TI82 has competition  
Lecture 3  18:04  
Here I teach about the Mathematica Environment. The fact that to evaluate something, you have to press [Shift]+[Enter] The fact that all builtin functions in Mathematica start with uppercase letters. The fact that constants such as π and e are entered as Pi and E The fact that whatever the function applies to is in square brackets [...] I show the basic arithmetic operations +,,* and / The trignometric and the log funtions, also the fact that the trigonometric functions are always evaluated in radians, and the log is always the natural log. 

Quiz 3  7 questions  
Basics of Arithmetic Using Mathematica 

Lecture 4  17:48  
Here I show you the fact that the N function can evaluate to any number of decimal figures. 

Quiz 4  5 questions  
This is a quiz on the "N function detailed lecture" 

Lecture 5  12:40  
Here I show you that Mathematica can, take log to practically any base. I show you that Mathematica handles exponents gracefully. I show you a fancy way of writing roots to a number. I also introduce the problem with taking the nth root of a number, where n > 3. 

Quiz 5 
Quiz on Log Function Detailed

5 questions  
Lecture 6 
Trig Functions detailed

10:19  
Lecture 7  04:30  
In this and the previous lecture I introduce the fact that Mathematica can also evaluate trigonometry in terms of degrees. I introduce the concept of entering symbols using the [Esc] key. I also introduce the fact that Mathematica can convert between degrees and radians. I finally also introduce the concept of inverse trigonometric functions. 

Quiz 6 
Quiz on Trig Functions detailed

6 questions  
Section 3: Algebra a.k.a now your professor has competition  
Lecture 8  11:20  
In this lecture I show you how to define a variable, and the fact that practically anything can be a variable. I show you some conventions for naming variables. I show you how to find out information about a variable. I show you the fact that variables are global and what that really means in practice. 

Quiz 7 
Variables

6 questions  
Lecture 9  08:09  
Here I show you how to substitute one or more variables into an expression that has already been typed. 

Quiz 8 
Variable substitution

3 questions  
Lecture 10  11:44  
Here I show you the common algebra functions that Mathematica can do: I show you Expand to expand a binomial expression I show you, how that can be used with Coefficient to find the coefficient of a monomial in a Binomial expression.
I show you Simplify 

Quiz 9 
Algebraic Manipulation

6 questions  
Lecture 11  09:47  
Here I show you about the Solve[...] function, and the fact that it can solve almost anything. I also show you how Mathematica can solve generic equations 

Lecture 12  07:20  
Here I generalize the solve function to solve multiple equations 

Quiz 10 
Solving

5 questions  
Lecture 13  12:35  
Here I show you the notation for defining a function and some intricacies of functions. I also define the InverseFunction [...] 

Quiz 11 
Functions

5 questions  
Lecture 14  07:32  
Here I demonstrate the fact that Solve[...] does not solve quintic (fifth order) equations algebraiccally, and how to work around that. 

Quiz 12 
More about solve

2 questions  
Lecture 15  11:42  
Here I show you how to conditionally solve an equation, i.e. how to solve for Reals and Complexes and Integers 

Quiz 13 
Conditional Solve

2 questions  
Lecture 16  14:28  
Here I show Reduce as used to solve inequalities. I also show Reduce [...] used to: Solve equations of any degree Solve systems of Linear Equations Solve inequations and systems of inequalities Reduce logical expressions to True or False 

Quiz 14 
Reduce

5 questions  
Section 4: Lists a.k.a. more than your average shopping list  
Lecture 17  06:04  
In this video I introduce the concept of lists: that lists can contain, numbers, images, algebraic expressions, pictures etc that lists can contain other lists that lists can be named the naming conventions behind naming lists 

Quiz 15 
Intro to Lists

5 questions  
Lecture 18  04:04  
Very short vide. Here I show the fact that almost any function taken on a list applies to each and every element of said list 

Quiz 16 
Functions on Lists

2 questions  
Lecture 19  03:12  
Here I show that you can, add, subtract, multiply and divide two lists. 

Quiz 17 
Functions between two lists

5 questions  
Lecture 20  11:05  
Here I show you how to get basic information from lists: like length, max, min, etc. I also show you how to get parts of a list. 

Quiz 18 
Getting info out of lists

5 questions  
Lecture 21  18:35  
Here I show you how to search for information out of lists:
I show you how to search for elements of a list that meet a criteria or a combination of criteria. 

Quiz 19 
Searching Lists

5 questions  
Lecture 22  11:08  
Here I show you how to join two lists. I also show you how to insert, replace and delete parts of lists. 

Quiz 20 
Manipulating Lists

5 questions  
Lecture 23  03:48  
Here I show you the difference between the Split[...] and Partition[...] function 

Quiz 21 
Partitioning Lists

2 questions  
Lecture 24  17:40  
Here I show you how to use lists to do set calculations: I show you how to intersect and unionize(?) sets. I show you how to delete duplicates in a list and how to take the powerset of a list. Finally I show you how to take the Cartesian product of two sets, using the Combinatorica package. 

Quiz 22 
Lists as Sets

5 questions  
Lecture 25  20:34  
Here I show you how make tables: I show you how make tables using a basic equation. How to loop using the Table[...] function How make lists with depth n, where n >2 

Quiz 23 
Tables

2 questions  
Section 5: Drawing Graphs a.k.a. making calculations pretty  
Lecture 26  06:42  
Here I show you how to draw basic plots using Mathematica. I show you how to draw 2D Cartesian Graphs. 

Quiz 24 
2D Cartesian Graphs

4 questions  
Lecture 27  09:35  
Here I show you how to plot Parametric2D. I show you common Parametric graphs. 

Quiz 25 
2D Parametric

2 questions  
Lecture 28  05:09  
Here I generalize the concept of plotting graphs to plotting multiple graphs in Mathematica. 

Quiz 26 
Plotting Multiple Graphs

2 questions  
Lecture 29  10:27  
In this course, I show you how to Polar equations, and show you the plots of common polar equations. 

Quiz 27 
2D Polar

2 questions  
Lecture 30  11:50  
In this video, I show you how to draw 3D functions, given three parameters, basically I draw space curves and show you how to do it. 

Lecture 31  10:30  
In this equation I show you how to draw surfaces, given two parameters and how to plot surfaces in general. 

Quiz 28 
3D Parametric and Surfaces

6 questions  
Lecture 32  07:33  
Here I introduce the concept of contour plotting, both 3d and 2d using ContourPlor[ ] and ContourPlot3D[ ] respectively. 

Quiz 29 
Contour Plotting

3 questions  
Lecture 33  04:43  
Here I introduce spherical plots. 

Quiz 30 
Spherical Graphs

1 question  
Lecture 34  13:12  
Here I show you two common options for plotting graphs: PlotRange and AspectRatio and how you can use them to adjust a plot. 

Quiz 31 
PlotRange & AspectRatio

4 questions  
Lecture 35  05:21  
Here I introduce the concept of shading graphs and how you can shade top or bottom and shade between two graphs. 

Quiz 32 
Shading 2D Graphs

2 questions  
Lecture 36  13:56  
Here I show you how to label a graph, including its axes and how to adjust the ticks on a graph. 

Quiz 33 
Labeling and Ticks

5 questions  
Lecture 37  03:22  
Here I show you how to change the PlotStyle of a graph. Basically how to colour individual graphs, how to adjust thickness of plots and how to adjust the style (dotted vs solid) of plots. 

Quiz 34 
PlotStyle

2 questions  
Lecture 38  09:10  
Here i show you some 3D options for plots, including the BoxRatio, the ViewPoint and how to change the color of graphs. 

Quiz 35 
Some 3D options

5 questions  
Section 6: Calculus a.k.a the coolest sounding branch of Mathematics  
Lecture 39  16:40  
Here I introduce the Limits capability of Mathematica. I show you how it can approach a limit from two sides, I also show you assumptions in limits. 

Quiz 36 
Limits

5 questions  
Lecture 40  10:25  
Here I show you how Mathematica takes derivatives, including some fairly complex ones. 

Quiz 37 
Differenciation

6 questions  
Lecture 41  19:20  
Here I show how Mathematica can calculate the antiderivative of an expression, including definite, indefinite and improper. 

Quiz 38 
Integration

4 questions  
Lecture 42  05:53  
Here I show you how Mathematica can perform the Laplace transform, of constant, and expressions, including some fairly complex ones. 

Quiz 39 
Laplace Transform

3 questions  
Lecture 43  11:37  
Here I show how Mathematica can solve differential equations, first, second, linear, ordinary and partial. 

Quiz 40 
Differential Equations

1 question  
Section 7: Matrices and Vectors a.k.a Neo and a Mosquito walks into a bar...  
Lecture 44  06:55  
Here I show you how Mathematica handles matrices. The proper way to input matrices into Mathematica. 

Lecture 45  06:23  
Here I show you how to enter special matrices into Mathematica. This includes, diagonal, identity, using the Table[ ] function and using sparsearrays. 

Quiz 41 
Intro to Matrices

5 questions  
Lecture 46  10:05  
Here I show you simple stuff like finding the determinant and the inverse of a 2x2 matrix. 

Quiz 42 
More about Matrices

3 questions  
Lecture 47  05:31  
Here I show you even more advanced stuff like row reducing, finding the rank and the eigenvalues and eigenvectors of a matrix 

Quiz 43 
Even more about Matrices

3 questions  
Lecture 48  07:33  
Here I give a brief introduction to vectors and how Mathematica handles them. Basically the conventions Mathematica uses when handling vectors. 

Quiz 44 
Introduction to Vectors

5 questions  
Lecture 49  10:11  
Here I show common functions on vectors like norm, projection, orthogonalization etc. 
Hi there!
I am a Math Major at Troy University in Alabama. I am currently a Junior and I have been using software to do my math for me for quite a while. Partly because its easy and partly because my course requires it. I will be instructing you on Mathematica 9. Have a blast !