This notebook "MathematicaDemo.nb" can be downloaded from the course web page.

Transcription

1 Mathematica demo This notebook "MathematicaDemo.nb" can be downloaded from the course web page. Basics Evaluate cells by pressing "shift-enter" / shift-return In[]:= + 3 Out[]= 5 Square brackets for function arguments! Round brackets for grouping. In[]:= Sin[0.7] Out[]= There is a lot of documentation, including a help page for each function, tutorials, etc. To solve the exercises, it is easiest to copy examples from the documentation (or from the notebook below) and then to modify them. Functional (rule-based) programming Mathemtica is an example of a rule based programming language. As opposed to imperative languages, the code is not a series of instructions, but consists of a series of functions (or substitution rules) acting on expressions. Mathematica works using replacement rules In[3]:= expression = (x + ) ; To numerically evaluate for x = 3, we replace x with 3 In[4]:= expression /. x 3 Out[4]= 6 Alternatively, we can define a function The quantity x_ is a pattern. A placeholder for the argument that will later be passed to the function. In[5]:= f[x_] := (x + ) ; In[6]:= f[3] Out[6]= 6 In[7]:= f[cos[θ]] Out[7]= ( + Cos[θ]) Nontrivial example

2 MathematicaDemo.nb In[8]:= Fact[x_] := (x - ) Fact[x - ]; In[9]:= Fact[] := ; The function calls itself! (Recursion) In[0]:= Fact[0] Out[0]= In[]:= Fact[9.5] $RecursionLimit::reclim : Recursion depth of 04 exceeded during evaluation of B Out[]= Hold[(9.5 - ) Fact[9.5 - ]] (... disaster could have been avoided with better programming.) Same with rules In[]:= In[3]:= Out[3]= factrule = fact[], fact[n_] (n - ) fact[n - ] ; fact[5] /. factrule 4 fact[4] In[4]:= Out[4]= fact[5] /. factrule /. factrule fact[3] The command //.factrule applies the rule until the result no longer changes. In[5]:= fact[5] //. factrule Out[5]= 4 Substitution rules are quite natural for mathematics In[6]:= expression = Sin[x] 0 + Cos[x] 0 ; In[7]:= expression = expression /. Sin[x] - Cos[x] Out[7]= Cos[x] Cos[x] 5 In[8]:= expression3 = Expand expression Out[8]= - 5 Cos[x] + 0 Cos[x] 4-0 Cos[x] Cos[x] 8... obviously Mathematica has built in functions to do similar manipulations In[9]:= Out[9]= In[0]:= expression4 = TrigReduce expression ( Cos[4 x] + 5 Cos[8 x]) 8 Simplify expression4 - expression3 Out[0]= 0 Solving equations The solution is given as a list.

3 MathematicaDemo.nb 3 In[]:= Solve[x^ + * x + 4 == 0, x] Out[]= x - - i 3, x - + i 3 Note the difference between == and = a == b means "a is equal to b" a = b means "a is defined to be b" In[]:= Out[]= False In[3]:= = Out[3]= Set::setraw : Cannot assign to raw object. B Differentiation and integration Define a function f In[4]:= fun = Sin[x] Cos[x]; Note the square brackets for function arguments! 4th derivative of f In[5]:= res = D fun, {x, 4} Out[5]= -4 Sec[x] + Sin[x] Tan[x] - Sec[x] Tan[x] + 4 Cos[x] Sec[x] Sec[x] Tan[x] + Sin[x] 6 Sec[x] 4 Tan[x] + 8 Sec[x] Tan[x] 3 In[6]:= res // FullSimplify Out[6]= -Cos[x] + Sec[x] - 0 Sec[x] Sec[x] 5 Taylor series around x=0 to 0th order: In[7]:= Series fun, {x, 0, 0} Out[7]= x + x x x x O[x] Integrals In[8]:= Out[8]= intf = Integrate fun, x -Log Cos x - Sin x + Log Cos x + Sin x - Sin[x] In[9]:= intf = Integrate fun, x, 0, Pi 4 Out[9]= - + ArcTanh Tan π 8 In[30]:= N intf Out[30]= Numerical value of the integral, to 800 digits accuracy

4 4 MathematicaDemo.nb In[3]:= N intf, 800 Out[3]= Vectors and matrices In[3]:= vector = a, b, c ; In[33]:= vector = {,, 3}; (Because of the semicolons at the end of the lines, the output is suppressed.) Third element In[34]:= Out[34]= vector[[3]] c Scalar product In[35]:= Out[35]= vector.vector a + b + 3 c In[36]:= matrix = {{,, 3}, {4, 5, 6}, {7, 8, 9}}; In[37]:= matrix // MatrixForm Out[37]//MatrixForm= In[38]:= matrix[[3, ]] Out[38]= 7 In[39]:= Matrix times vector matrix.vector Out[39]= {4, 3, 50} Egenvectors, Eigenvalues In[40]:= Eigensystem matrix Out[40]= , , 0, , ,, , ,, {, -, }

5 MathematicaDemo.nb 5 In[4]:= values = Eigensystem matrix [[]] Out[4]= , , 0 In[4]:= mat = Eigensystem matrix [[]] // MatrixForm Out[4]//MatrixForm= In[43]:= Inverse[Transpose[mat]].matrix.Transpose[mat] // Simplify // MatrixForm Out[43]//MatrixForm= Vector analysis: Curl[f,{x,y,z}] = f Exercise 3. a In[44]:= r = x + y + z ; In[45]:= vecf = {x, y, z} f[r] r ; In[46]:= Curl vecf, {x, y, z} Out[46]= {0, 0, 0} In[47]:= pot = Grad[g[r], {x, y, z}] Out[47]= x g x + y + z x + y + z, y g x + y + z z g x + y + z, x + y + z x + y + z Differential equations Analytical solution In[48]:= Out[48]= DE = D[x[t], {t, }] - D[x[t], t] - Sin[t] -Sin[t] - x [t] + x [t] Note again: A=B means A is defined to be B, while A B represents the equation A=B: In[49]:= DSolve[0 DE, x[t], t] Out[49]= x[t] e t C[] + C[] + Cos[t] - Sin[t] This second order equation involves two integration contants C[] and C[]:

6 6 MathematicaDemo.nb The boundary conditions x(0) = and dx/dt(0) = are specified as follows: In[50]:= soln = DSolve[{0 DE, x[0], x [0] }, x[t], t] Out[50]= x[t] et + Cos[t] - Sin[t] Mathematica gives the solutions as a list, even if it is unique. To obtain the solution itself, use soln, In[5]:= solutionwithboundary = x[t] /. soln, Out[5]= et + Cos[t] - Sin[t] In[5]:= Plot solutionwithboundary, {t, 0, 4} Out[5]= Numerical solution First order In[53]:= firstorder = NDSolve[{y'[x] x * y[x] * ( - y[x]), y[0] }, y[x], {x, 0, }] Out[53]= y[x] InterpolatingFunction Domain: {{0.,.}} [x] The solution is computed for a set of points, and Mathematica provides an interpolation for arbitrary values in the specified interval, in our case x [0,]. In[54]:= yfun = y[x] /. firstorder, Out[54]= InterpolatingFunction Domain: {{0.,.}} [x]

7 MathematicaDemo.nb 7 In[55]:= Plot yfun, {x, 0, }.4.3 Out[55]= Second order In[56]:= a =.0; In[57]:= secondorder = NDSolve {y''[x] -y[x]^3 + Cos[a x], y[0], y'[0] 0}, y[x], {x, 0, 600}, MaxSteps 0 5 Out[57]= y[x] InterpolatingFunction [x] In[58]:= yfun = y[x] /. secondorder, Out[58]= InterpolatingFunction [x] In[59]:= Plot yfun, {x, 0, 60} Out[59]= In[60]:= vel = D yfun, x Out[60]= InterpolatingFunction [x] Complicated, but periodic:

8 8 MathematicaDemo.nb In[6]:= ParametricPlot yfun, vel, {x, 0, 00}, PlotRange All Out[6]= Animation

9 MathematicaDemo.nb 9 In[6]:= Animate Graphics Disk., yfun /. x t, 0., PlotRange {{0.5,.5}, {-.3,.3}}, {t, 0, 30}, AnimationRate t Out[6]= Chaotic oscillation Same as before, but we change the frequency In[63]:= a =.8; In[64]:= secondorder = NDSolve {y''[x] -y[x]^3 + Cos[a x], y[0], y'[0] 0}, y[x], {x, 0, 600}, MaxSteps 0 5 Out[64]= y[x] InterpolatingFunction [x]

10 0 MathematicaDemo.nb In[65]:= yfun = y[x] /. secondorder, Out[65]= InterpolatingFunction [x] In[66]:= Plot yfun, {x, 0, 60} Out[66]= In[67]:= vel = D yfun, x Out[67]= InterpolatingFunction [x] In[68]:= ParametricPlot yfun, vel, {x, 0, 600}, PlotRange All 6 4 Out[68]= This is no longer periodic...