Overview:
- Mathematica has many many commands, we look at a sampler.
- Using these can get a feel for things which you do in your other classes.
- Operations don't mean anything till you become familiar with them, mathematica can help this process.
- Once you can use mathematica, you will be able to use the maths you need without having to become a mathematician, after all you've got more important things to do!
Typing commands is simple, doesn't mean that what it does is:
- Solving a quadratic is simple, a quartic? Mathematica makes it. Try solving
  
  x⁴-2x³+3x+7 = 0
  
  The exact solution if very messy, but using NSolve they are:
```
x -> -0.852175 - 0.841786i , x -> -0.852175 + 0.841786i,
x -> 1.85218 - 1.20339i , x -> 1.85218 + 1.20339i
```
- Likewise for various integrations. What is òx cos x³ dx? Again, just type it in and it's done! Using Integrate get a very messy solution involving Gamma functions, but NIntegrate from 0 to 1 gives 0.440408.
Note that while it is impressive, it cannot do everything. You need to be able to figure out what questions to ask it. Also we cannot solve all the questions which we would like to. Some because we just don't know whether their solutions exist or not, others since the work required is too much. Also in general we work with linear equations, since we can work with them. Many, many things are non-linear though and they are just too tricky!
In C programming where the traditional first program is to write something that just prints out ``hello world''. Being able to do this means that you have a C compiler, can write the program and basically everything that matters works. Once you've done that you are set!
So to work one thing out is 90This involves finding a pc, figuring out how to run it and typing the command in.
I can say the command to solve the quadratic x²+2x+1 is Solve[x²+2x+1 = = 0,x] but you'll still type in solve[x²+x = 0] and wonder why it doesn't work! You need to actually use the program to be able to use it. So will give you some things to work out using it, and will give you the solutions next week. Have a go, its help is excellent so it shouldn't be a problem.
Now back to some more commands. Taking a first look at them and then will look over them again later. So 3 things:
- Sum: If we want to 1+2+3...+100 then use the Sum command:
  
  Sum[n,{n,1,100}]
  
  which is equal to 5050. If wanted to add from a to b it would be
  
  Sum[n,{n,a,b}]
  
  Finally if wanted to add n+n² from 1 to 5 would be
  
  Sum[n+n²,{n,1,5}]
  
  which equals 70.
- Vectors and Matrices:Mathematics can also work with vectors and matrices which it represents using lists.
  - We can define
    
    x = {1,2,3}
    
    Then the first component which is 1 is referred to as x[[1]].
  - We can define a vector {3,5,7,9,11} as follows:
    
    x = Table[1+2i,{i,1,5}]
    
    Then x[[3]] = 7.
  - Matrices are then stored as lists of lists. So
    
    M = { {a,b},{ c,d }}
    
    would define a two by two matrix M. This can be viewed in a more familiar form by using the MatrixForm command.
    
    MatrixForm[M]
    
    produces the standard 2 by 2 form.
- Differential equations
  - Recall that to differentiate we use the command D. So
    
    D[x²,x] = 2x.
  - To solve a differential equation we say
    
    DSolve[D[y[x],x]-y[x]] = = 0,y[x],x]
    
    This is a classic problem. We could say that the rate of population growth is proportional to the current population. So D[P[t],t] = kP[t]. The solution to this is
    
    P[t] = P[0]e^kt.
    
    Of course this isn't a great model since if this was always true we would have have trillions of people at some stage. In reality k is not a constant.
  - The same equations can be used to describe radioactive decay. That is why we can look at a sample and determine how old it is. So in practice you need to be able to model a problem, this leads to maths problems and that's where mathematica comes in.

File translated from T_EX by T_TH, version 2.70.
On 6 Nov 2002, 14:27.