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Section 7.1 Surfaces

Figure 7.1.1. The graph of a function of 2 variables.

There are many ways to describe a surface. Consider the following descriptions:

  • The unit sphere;

  • \(x^2+y^2+z^2=1\text{;}\)

  • \(r=1\) (where \(r\) is the spherical radial coordinate);

  • \(x=\sin\theta\cos\phi\text{,}\) \(y=\sin\theta\sin\phi\text{,}\) \(z=\cos\theta\text{;}\)

  • \(\rr(\theta,\phi) = \sin\theta\cos\phi\,\xhat + \sin\theta\sin\phi\,\yhat + \cos\theta\,\zhat\text{;}\)

all of which describe the same surface. Here are some more ways of describing surfaces: (Are these descriptions of the unit sphere?)

  • The graph of \(z=x^2+y^2\text{;}\)

  • The graph shown in Figure 7.1.1.

Which representation is best for a given problem depends on the circumstances. Often you will have to go back and forth between several representations.

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