Comments on Colley, Section 7.1 We defined curves in terms of parametrizations, and in principle we would like to define surfaces likewise. However, there are some complications because we want to have a concept of surface which is broad enough to include most if not all the objects which are generally called surfaces in ordinary contexts. One simple way to define a parametrized surface in coordinate 3 – space is by means of a 3 – dimensional vector valued function X(u, v) of two variables. (Source: http://math.etsu.edu/MultiCalc/Chap5/Chap5-5/index.htm ) This definition includes the graphs of some function f (u, v) of two variables, in which case the parametrization is given by X(u, v) = (u, v, f (u, v) ) and in order to define tangent planes we need to assume that f has continuous partial derivatives. In order to avoid complicated discussions about defining partial derivatives at boundary points of regions, we also need to stipulate that the function f is defined on an open set as defined in the previous course. More generally, the parametric equations for a smoothly parametrized surface are assumed to have the form X(u, v) = ( x(u, v), y(u, v), z(u, v) ) where, as before, the coordinate functions x, y, z are assumed to have continuous partial derivatives over some open set. However, in order to obtain a decent notion of tangent plane we need an additional condition, just as we needed to assume the tangent vectors to parametrized curves were nonzero in order to define tangentlines. The condition may be stated as follows: Define the vector valued partial derivative functions , . Not only do we want these two partial derivatives to be nonzero everywhere, but we also want them to be linearly independent everywhere, so that neither is a nonzero scalar multiple of the other. This is conveniently summarized by the cross product condition . This condition automatically holds for the graph of a function, in which case the cross product is given by ( – fu , – fv , 1) ; the reader should check this directly using the definitions of partial derivatives and cross products. If we compare this with Theorem 2.3.3 on page 112 of the text, we see that this cross product is an upward normal vector to the tangent plane at a given point p = (u0 , v0 , f (u0 , v0) ). Finally, for the sake of simplicity it is extremely useful to eliminate surfaces with self – intersections, so our default assumption will be that the parametrization is one – to – one: Namely, the parametrization function X takes different points in its domain to distinct points in space. More generally, the tangent plane to a parametrized surface at some point (u0, v0) is defined to be the plane through the surface point X(u0 , v0) for which the standard normal vector N(u0 , v0) is equal to the cross product . In other words, the equation of the tangent plane to X at p is given by the set of all vectors w satisfying the equation N(u0 , v0) · (w – p) = 0. Here are some basic examples beyond graphs of functions. Surfaces of revolution. Suppose that we are given a curve which is the graph of a smooth (continuously differentiable) function y = f (x), where f is always positive valued and (say) 0 ≤ a < x < b. We can use such a curve to define surfaces of revolution about either the x – axis or the y – axis.Revolution about the y – axis (Source: http://curvebank.calstatela.edu/arearev/arearev.htm ) Revolution about the x – axis (Source: http://www.mathresources.com/products/mathresource/maa/surface_of_revolution.html ) Here is a link to an animated graphic: http://curvebank.calstatela.edu/arearev/rev3cont.gif In the first case (around the y – axis) the most straightforward paremetrization is given by X(u, v) = (u cos v, f (u), u sin v ) and in the second (around the x – axis) the most straightforward parametrization is given by X(u, v) = (u, f (u) cos v, f (u) sin v ). The coordinate functions satisfy the smoothness condition, and the cross products of the partials of X with respect to u and v are given by (1) N(u, v) = (u f ′ (u) cos v , – u , u f ′ (u) sin v ) in the y – axis case, (2) N(u, v) = ( f (u) f ′ (u) , – f (u) cos v , – f (u) sin v ) in the x – axis case.These formulas show that the lengths of the normal vectors are equal to u and f (u) times the square root of 1 + f ′ (u) 2 respectively; in each case the length of the cross product is given as a product of two positive numbers and hence is positive. Examples of surfaces of revolution about the x – axis include a sphere of radius 1 and center (1, 0, 0) with the poles on the x – axis removed, in which case the curves are given by y = SQRT(1 – (x – 1)2) where 1 < x < 2, right circular cylinders, in which case the curves are given by y = r for some positive constant r (and the x – limits are arbitrary nonnegative numbers), and right circular cones with the top vertices removed, in which case the curves are given by y = m x for some positive constant m (and the x – limits are 0 and some positive value h). More generally, if ΓΓΓΓ is a parametrized regular smooth curve in the first quadrant of the coordinate plane, it is possible to extend the preceding constructions to obtain surfaces of revolution with respect to the x – and y – axes. However, we shall pass on describing such generalizations explicitly and move to another crucial family of surfaces which arise naturally. Piecewise smooth surfaces There are still other examples beyond the smooth surfaces considered just far; just as it is often convenient to consider piecewise smooth curves, we also want to consider piecewise smooth surfaces in many situations. One major complication is that it is definitely more difficult to put the pieces of such a surface together than it is to do the same thing for curves, and because of this we have to set up the definitions carefully. Recall that an elementary region is one which is bounded by a pair of vertical lines x = a, x = b, and the graphs of two functions y = g(x), y = h(x). This concept figures in the definition of piecewise smooth surfaces which appears in Definition 7.1.3 on page 413 of the course text. Intuitively, one expects that a cube should be an example of a piecewise smooth surface, and a detailed verification of this point is given