Integral

fun = 1/( 1 + x^2 + y^2 + z^2)^2

ans = NIntegrate[ fun, {x,0,1},{y,0,1},{z,0,1}] ;

N[ ans, 9]

N[ Pi^2/32, 9]

Pendulum

theta''[t] + g/l Sin[theta[t]] == 0

eq = % ;

Series[ #, {theta[t],0,1} ] & /@ eq

eqLinear = Normal[%]

NDSolve[ {eqLinear, theta[0]==Pi/2, theta'[0]==0}
          /. g/l -> 1, theta, {t,0,20} ] ;

plotlin = Plot[ Evaluate[ theta[t] /. % ], {t,0,20},
                PlotStyle -> Dashing[{0.01,0.01}] ]

NDSolve[ {eq, theta[0]==Pi/2, theta'[0]==0}
          /. g/l -> 1, theta, {t,0,20} ] ;

plotnon = Plot[ Evaluate[ theta[t] /. % ], {t,0,20} ]

Show[ plotnon, plotlin]

Lorenz Attractor

<<Programm`RungeKutta`

RungeKutta[ { - 3 (x-y),
              - x z + 26.5 x - y,
                x y - z            },
            {x, y, z}, {0,1,0}, {50, 0.01} ];

sol = % ;

?RungeKutta

Show[ Graphics3D[ Line[sol] ], Axes -> Automatic ]

?Line

?Graphics3D

Fields

<<Graphics`PlotField`

<<Graphics`PlotField3D`

pot2d[x_,y_] := 1/Sqrt[x^2+y^2+0.1]

plot1 = ContourPlot[ pot2d[x,y], {x,-2,2},{y,-2,2},
                       ContourShading -> False,
                       ContourSmoothing -> True ]

??ContourPlot

plot2 = PlotGradientField[ -pot2d[x,y], 
                              {x,-2,2},{y,-2,2} ]

?PlotGradientField

pot3d[x_,y_,z_] := 1/Sqrt[x^2+y^2+z^2+0.1]

PlotGradientField3D[ -pot3d[x,y,z],
                     {x,-2,2},{y,-2,2},{z,-2,2},
                     PlotPoints -> 6,
                     VectorHeads -> True,
              Graphics`PlotField3D`ScaleFunction->(1&)];

?PlotGradientField3D