(1) One could just define S as a set, or perhaps a set with positive, finite area. The best definition of area is two-dimensional \u201cHausdorff measure\u201d <\/span>H<\/span>2<\/sup>.\u00a0<\/span> This definition of surface is too general, because it includes some fractal curves, for example. It is common to restrict to rectifiable sets, which have \u201capproximate tangent planes\u201d at almost all points.<\/span><\/p>\n (2) One could define S locally as the graph of say a C1<\/sup> function f from R<\/strong>2<\/sup> to <\/span>R<\/strong> or <\/span>R<\/strong>2<\/sup>. Then S will have tangent planes everywhere. This definition is fine for smooth surfaces, but it excludes even rather mild singularities. For example, in <\/span>R<\/strong>4<\/sup> = <\/span>C<\/strong>2<\/sup>, {w2<\/sup>-z3 <\/sup>= 0} is not the graph of a function in any neighborhood of (0,0).<\/span><\/p>\n (3) Often it is more convenient to define S as the level set {g = c} of say a C1<\/sup> function. When the gradient of g is nonvanishing, this is equivalent to the graph definition (2). If the gradient of g is allowed to vanish, you can pick up singularities such as {w2<\/sup>-z3 <\/sup>= 0}.<\/p>\n (4) In calculus, you often define a surface S locally as the image of a\u00a0C1<\/sup> function from R<\/strong>2<\/sup> to R<\/strong>3<\/sup> or R<\/strong>4<\/sup>. Again, when the Jacobian is nonvanishing, this is equivalent to definition (2), and the surface is called a\u00a0 C1<\/sup><\/span>\u00a0manifold.<\/em> If the Jacobian vanishes, you can pick up singularities\u00a0such as\u00a0 {w2<\/sup>-z3 <\/sup>= 0}<\/span>.\u00a0<\/p>\n (5) Any measurable set S yields a measure \u00a0\u00a0 \u00a0 One could define a surface as a measure with certain properties. Measures have nice compactness properties, but are much too general.<\/p>\n (6) If a surface S has a tangent plane (7) Similarly, if a surface S has a tangent plane at all points, one can integrate a smooth differential form over S and thus view S as a linear functional on differential forms, called a current.<\/em>\u00a0Currents have nice compactness properties and include surfaces with multiplicities. Currents and varifolds are the fundamental surfaces of geometric measure theory.<\/span><\/p>\n
