Metric Differential Geometry

You recall with what delight you first discovered that many geometrical and physical situations can be analyzed, and problems solved, by means of diflerential and integral calculus. Fortunately this pleasure of discovery can be continued, for many parts of more advanced mathematics and physics use the concepts of calculus as basic tools. This is particularly true in the study of properties of curves. and surfaces in space. Just as in plane analytic geometry we ordinarily use equal scales on two directed axes at right angles to each other in locating points (x,y), in solid analytic geometry we use equal scales on three concurrent directed axes which are at right angles each to each in locating points (x,y,z). It is clear that if two of these directions are fixed, then there are exactly two possible directions for the third, as in Figures 1 and 2. In the first of these figures, an ordinary right-hand