Plotting a surface defined by a 2-variable function.

Plot3D[ x^2 + y^2, {x,-2,2}, {y,-2,2}]

The "AxesLabel" option to “Plot3D” lets you label the axes for easier recognition.

Plot3D[ x^2 + y^2, {x,-3,3}, {y,-3,3}, AxesLabel-> {"X","Y Axis","Axis 3"}]

Mathematica plots may be visually distorted because Mathematica doesn’t by default use equal amounts of screen space for equal distances on the 3 axes.

Plot3D[ Sqrt[9 - x^2 - y^2], {x,-5,5},{y,-5,5}]

Plot3D[ Sqrt[9 - x^2 - y^2], {x,-3,3},{y,-3,3}]

A function that looks sort of like an egg carton:

Plot3D[ Sin[x] * Cos[y], {x,-4Pi, 4Pi}, {y,-4Pi,4Pi}]

Plotting level curves or contour lines with the “ContourPlot” function.

ContourPlot[ x^2 + y^2, {x,-2,2},{y,-2,2}]

The “Contours” option lets you select which level curves to show.

ContourPlot[ x^2+y^2,{x,-2,2},{y,-2,2}, Contours->{1,3}]

Mathematica contour plots can have an almost artistic effect…

ContourPlot[ Sin[x]*Cos[y], {x,-4Pi, 4Pi}, {y,-4Pi,4Pi}]

In the following plot, the roughly square regions are the regions on each side of the z = 0 level curve; the circles are the z = 0.4 level curve. Darker/cooler colors are regions where the function is negative, while lighter colors are positive. Level curves are the boundaries between colors.

ContourPlot[ Sin[x]*Cos[y], {x,-4Pi, 4Pi}, {y,-4Pi,4Pi}, Contours->{0,0.4}]