Nov 16, 2022 · In this section we will give a quick review of some important topics about functions of several variables. In particular we will discuss finding the domain of a function of several variables as well as level curves, level surfaces and traces.

Sketch several traces or level curves of a function of two variables. If hikers walk along rugged trails, they might use a topographical map that shows how steeply the trails change. A topographical map contains curved lines called contour lines. Each contour line corresponds to the points on the map that have equal elevation (Figure 1).

As in this example, the points $(x,y)$ such that $f(x,y)=k$ usually form a curve, called a level curve of the function. A graph of some level curves can give a good idea of the shape of the surface; it looks much like a topographic map of the surface.

Example 1: The graph of $z=f(x,\,y)$ as a surface in $3$-space can be regarded as the level surface $w = 0$ of the function $w(x,\,y,\,z) = z - f(x,\, y)$. Example 2: Spheres $x^2+y^2+z^2 = r^2$ can be interpreted as level surfaces $w = r^2$ of the function $w = x^2+y^2+z^2$.

For the cone z = d m the level curves are c = d m (again circles - just square both sides). For the plane z = x + y the level curves are straight lines c = x +y (parallel to each other as c changes). The collection of level curves in the xy plane is a contour map. If you are climbing on the surface, the map tells you two important things:

LEVEL CURVES 2D: If f(x;y) is a function of two variables, then f(x;y) = c = constis a curve or a collection of curves in the plane. It is called contour curve or level curve. For example, f(x;y) = 4x2 + 3y2 = 1 is an ellipse. Level curves allow to visualize functions of two variables f(x;y). LEVEL SURFACES. We will later see also 3D ana-

Level curves Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. 28. z =2 x-y; @-2, 2Dµ@-2, 2D 29. z = x2 +4 y2; @-8, 8Dµ@-8, 8D 30. z =e-x 2-2 y2; @-2, 2Dµ@-2, 2D 31. z = 25 -x2-y2; @-6, 6Dµ@-6, 6D 32. z = y-x2-1 ; @-5, 5Dµ@-5, 5D 33. z =3 cos H2 x +yL; @-2 ...

For any surface, sketch a number of level curves (for different values of z), whose contours then form a contour map of the surface. With practice, one can locate minimum points, maximum points, saddle points, ridges, valleys, and other “terrain” forms.

Level curves of a cone, hyperboloid, and ellipsoid

Level curves are sets of points (x,y) (x, y) where f (x,y) = k f (x, y) = k, for some chosen constant number k k. When we lift the level curves up onto the graph, we get "horizontal traces."