Jon's Mathematica Info & Examples 10/16/98
Some general tips on getting along with Mathematica
-
Capitalize all commands and use square brackets around their arguments
(e.g., Factor[x^2+2x+1]).
-
Use only parentheses as grouping symbols in your expressions -- Mathematica
reserves brackets and braces for other purposes.
-
Press Shift-Enter, not just Enter, when you want Mathematica to
evaluate an expression.
Some things Mathematica can do
Normal calculator functions and lots more:
-
2+2 gives 4.
-
Sqrt[2]^4 gives the square root of 2 to the fourth power, or 4.
-
Sin[Pi/2] gives 0, the sine of pi.
-
(2+I)*(3-I) multiplies the imaginary numbers 2+i and 3-i.
-
ArcCos[1/2] computer the inverse cosine of 1/2 (as an exact value).
-
Tanh[Log[2]] computes the hyperbolic tangent of the natural logarithm
of 2.
-
N[Sqrt[2],20] gives the square root of 2 to 20 decimal places.
-
N[Log[4.1],15] gives the natural log of 4.1 to 15 decimal places.
-
Expand[(x+2)(x+3)(x-7)(x-1)] produces 42 - 13 x - 27 x2
- 3 x3 + x4.
-
Factor[x^3+3x^2-4x-12] produces (-2 + x)(2 + x)(3 + x)).
-
Solve[x^2-4==0,x] solves the equation x2-4=0 for x.
-
FactorInteger[2434500] produces {{2, 2}, {3, 2}, {5, 3}, {541, 1}}
meaning 22, 32, 53, and 541 are the factors
of 2434500.
-
PrimeQ[5757] gives False because 5757 is not prime.
Graph functions:
-
Plot[Sin[3x]+Cos[5x],{x,0,20}]
produces a graph of the function for x values from 0 to 20.
-
Plot[Sin[3x],{x,0,20},PlotStyle
-> RGBColor[1,0,0]] produces a graph in red.
-
Show[%,%%] displays
the last two graphs on the same set of axes (note "%3" refers to the
3rd output. % and %% refer to the last output and the output
before that).
-
ParametricPlot[{Sin[2t],Sin[3t]},{t,0,2Pi}]
produces a parametric graph.
Perform standard Calculus tasks:
-
Limit[Sin[x]/x, x->0] produces 1, the limit of the specified function
as x approaches 0.
-
Limit[E^x/x, x-> -Infinity] produces 0, the limit of this function
as x grows large.
-
D[x^2,x] produces 2 x, the derivative of x2.
-
D[x^4,{x,3}] produces 24 x, the third derivative of x4.
-
Integrate[Sin[.5x],x] produces - 0.2 Cos [0.5 x], the antiderivative
of the given function.
-
Integrate[6x^2,{x,0,1}] produces 2, the value of the definite integral.
-
Integrate[E^x,{x,0,Infinity}] produces 1, the value of the improper
integral.
-
Integrate[Cos[x]Cos[y],{x,-Pi/2,Pi/2},{y,-Pi/2,Pi/2}] produces 4,
the value of the double integral.
-
Sum[i,{i,1,100}] produces 5050, the sum of the integers from 1 to
100.
-
N[Sum[(1/2)^i,{i,1,Infinity}] produces 1.
Produce several varieties of three dimensional
graphs:
-
Plot3D[Sin[x] Sin[y],
{x,-6,6}, {y,-6,6}] produces a graph for the specified domain.
-
ContourPlot[(Sin[x])^2+1/4*y^2,
{x,0,7Pi/2}, {y,-2,2}] produces a contour map (darker is lower).
-
DensityPlot[E^(-x^2-y^2),{x,-2,2},{y,-2,2}]
produces a density plot (darker is higher).
-
ParametricPlot3D[{Sin[t],Cos[t],t/Pi},{t,0,2Pi}]
produces a parametric curve plot given the three coordinate functions.
-
ParametricPlot3D[{5Sin[u]
Cos[v], 5Sin[u] Sin[v], 5Cos[u]}, {u,0,Pi}, {v,0,2Pi}] produces
a parametric surface plot given the three coordinate functions.
More advanced options for three dimensional graphs:
(These options can be included just within the final bracket of the Plot3D
or other command)
-
Axes->None eliminates the axes from the plot.
-
AxesLabel->{"x","y","z"} labels the three axes with the names given.
-
BoxRatios->{1,1,1} changes the ratios of length, width, and height
from the squished default.
-
Framed->False eliminates the box normally plotted around the graph.
-
Mesh->False eliminates the trace lines Mathematica normally
includes on the surface.
-
PlotPoints->30 samples 30 points along each axis, rather than the
default of 15. This can produce a much smoother graph.
-
PlotRange->{-.3,.3} specifies the vertical range to be plotted.
-
Shading->False produces a graph without any shading of the surface.
-
Ticks->{Automatic,None,{0,Pi/2,Pi,3Pi/2,2Pi}} puts the tick marks
on the x axis in the default locations, puts no tick marks on the y axis,
and forces the tick marks on the z axis to take the specified locations.
-
ViewPoint->{2,2,2} specifies the point from which to view the surface.
Generate basic geometric objects:
(use these commands to produce objects, then Show them with each
other or with a Plot3D output)
-
Graphics3D[Point[{Pi/3,0,1/2}]] creates a point at the coordinates
given.
-
Graphics3D[Line[{{-2,-3,0},{5,1,2}}]] creates a line segment connecting
the two points given.