Book contents
- Frontmatter
- Foreword
- Preface
- Contents
- Short Biography of H. S. Wall
- 1 Numbers
- 2 Ordered Number Pairs
- 3 Slope
- 4 Combinations of Simple Graphs
- 5 Theorems about Simple Graphs
- 6 The Simple Graphs of Trigonometry
- 7 The Integral
- 8 Computation Formulas Obtained by Means of the Integral
- 9 Simple Graphs Made to Order
- 10 More about Integrals
- 11 Simple Surfaces
- 12 Successive Approximations
- 13 Linear Spaces of Simple Graphs
- 14 More about Linear Spaces
- 15 Mechanical Systems
- Integral Tables
- Index of Simple Graphs
- Glossary of Definitions
11 - Simple Surfaces
- Frontmatter
- Foreword
- Preface
- Contents
- Short Biography of H. S. Wall
- 1 Numbers
- 2 Ordered Number Pairs
- 3 Slope
- 4 Combinations of Simple Graphs
- 5 Theorems about Simple Graphs
- 6 The Simple Graphs of Trigonometry
- 7 The Integral
- 8 Computation Formulas Obtained by Means of the Integral
- 9 Simple Graphs Made to Order
- 10 More about Integrals
- 11 Simple Surfaces
- 12 Successive Approximations
- 13 Linear Spaces of Simple Graphs
- 14 More about Linear Spaces
- 15 Mechanical Systems
- Integral Tables
- Index of Simple Graphs
- Glossary of Definitions
Summary
The statement that f is a simple surface means that f is a collection, each element of which is an ordered pair (P, z), whose first member P is a point and whose second member z is a number such that no two ordered pairs in f have the same first member. The second member of that ordered pair in f whose first member is P is denoted by f(P) (read f of P) or, if P = (x, y), by f(x, y) (read f of x and y).
Problem. Generalize to simple surfaces some of the ideas concerning simple graphs such as slope, property S, length, and integral.
To picture a simple surface, we regard the XY-plane as horizontal and represent the ordered pair (P, z) of f as a dot on the vertical line containing P at a distance z above the XY-plane if z > 0, at P if z = 0, and below the XY-plane a distance |z| if z < 0 (see Figure 11.1).
The XY-projection of f is the point set to which P belongs only if P is the first member of an ordered pair in f. In Figure 11.1 the XY-projection of f is the rectangular interval [ab;cd].
Gradient
To generalize to simple surfaces the notion of slope of a simple graph, we first reformulate our definition of slope of a simple graph.
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- Information
- Creative Mathematics , pp. 93 - 118Publisher: Mathematical Association of AmericaPrint publication year: 2009