From Tensor Analysis
5.7 Implicit Representation of a Curve; Contact of Curves
Our work in Chapter 5 has centered around technical topics needed to derive equations describing certain natural objects. We now turn to some less technical but quite important questions that can further our understanding of differential geometry.
We have described a curve by the representation r= r (t). From this point of view, a curve is the image of some one-dimensional domain of the parameter t. Similarly, a surface was an image of a two-dimensional domain, given by a formula
However, we know that in Cartesian coordinates a sphere is described by the equation
and this cannot be represented in the form (5.38) for all points simultaneously. This is an example of the implicit form of description of a surface. We can describe any small part of a sphere in an explicit form of the type (5.38). This is a point from which the theory of manifolds arose: a surface such as a sphere can be divided into small portions, each of which can be described in the needed way.
However, implicit form descriptions of surfaces and curves are quite common in practice. Let us suppose that the equation of a surface is
In Cartesian coordinates this would look like
Earlier we described coordinate curves in space as sets given by the equation r= r( ? 1, ? 2, ? 3) when two of the three coordinate parameters ? 1, ?
Products & Services
Topics of Interest
5.8 Osculating Paraboloid When considering the structure of a surface at a point, it is often helpful to approximate the surface using another surface whose behavior is more easily visualized. A...
The majority of terms relating to food chemistry are explained throughout the text of this book, but the following general definitions may be of use. Diet a description of the overall...
2.4 Other Energy-Yielding Components In addition to proteins, fats and carbohydrates, there are three other classes of compounds routinely found or used in foods that yield energy in the diet:...
The chem-ID is a precision chemical analysis laboratory in a small box designed to be easily used by someone with no analytical chemical background. It measures a detailed signature of an air sample...
CHAPTER LIST Chapter 4: Building Blocks of Biomolecules Structure and Dynamics Chapter 5: Structure and Function of Macromolecules Chapter 6: Biochemical Techniques PART OVERVIEW...