Math Help

Level Curves and Visualization of Surfaces

In a Nut Shell: Let z = f(x,y) represent the graph of a function. Then level curves are

the set of all points (x,y) such that f(x,y) = c where c is a given constant. Each value

of c provides a new level curve.

If the function represents temperature, T = f(x,y), then each level curve represents an

isothermal. If the function represents altitude, Z = g(x,y), then each level curve

represents an isocline. If the function represents pressure, P = h(x,y), then each level curve

represents an isobar.

Interpretation of Level Curves Suppose the level curves represent changes in elevation of

a surface such as a mountain. Further suppose the level curves change by equal increments

(equal changes in elevation. Then in steep areas of the mountain the level curves are closely

spaced whereas in fairly level valleys, the level curves may be much farther apart. Closely

spaced level curves are an indication of a large gradient.

Strategy in Visualization of Surfaces

A trace is an intersection of the surface with planes parallel to the coordinate planes.

Plot traces on planes. i.e. For the function z = f(x,y). Set x = 0 and plot the trace in the

yz-plane. Set y = 0 and plot the trace in the xz-plane.

Check the value of f(x,y) at the origin. i.e. x = y = 0.

Does f(x,y) = f(˗ x, ˗ y)? Symmetric about both axes

Check for symmetry Does f(x,y) = f(˗ x, y)? Symmetric about x-axis

Does f(x,y) = f( x, ˗ y)? Symmetric about y-axis

Combine methods of level curves, traces, values of f(x,y), and symmetry.

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