Jon's Mathematica Info & Examples 9/7/99

• 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.

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 1, the sine of /2.
• (2+I)*(3-I) multiplies the imaginary numbers 2+i and 3-i.
• ArcCos[1/2] computes 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[E^4.1,15] gives the value of e^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 3^rd 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.  Note that the order is somewhat unusual: the command given here evaluates in the order that would typically be written "dy dx".
• 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:

• 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:

• 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.

Generate graphical representations of vector fields:

• If you first load an optional package with the command <<Graphics`PlotField`, then the command PlotVectorField[{-y,x},{x,-3,3},{y,-3,3}] plots the vector field F(x,y) = -yi + xj for the specified domain.
• If you first load an optional package with the command <<Graphics`PlotField3D`, then the command PlotVectorField3D[{y, -x, 0}/z, {x, -1, 1}, {y, -1, 1}, {z, 1, 3}] produces a plot of the three dimensional vector field given for the specified domain.