Content Guide
Foreword
Chapter Descriptions
1. Introduction
Calculus Appetizer (1983)
2. Second Preview Begins
2 Skier in Motion (V)
2 The Skier (V)
2. Position Dependent (V)
3 Slope & Extrema (V)
4 Single Factor Analysis (V)
4 Two Factor (V)
4 More Factors (V)
4 With Divisors (V)
5 Maxima & Minima Tests
6 Jumps & Discontinuities
8 Review  (optional)
9 On Calculus Studies
11 Slope of Slope
13  Acceleration
14 Limits & Error Control (V)
14 Limit of a Fn.
14. Limited Error Control
14 Signif. Digits
14 Cauchy Limits
14 Sequence Limits
14 Decimal Arith.
15 What is Slope (V)
15 Slope Calculation (V)
15 Slope, a Limit
15 Tangent Lines
15 Linear Approx.,
15 Limits via Algebra (V)
15 Recap.
PS.Chain Rule for Polys
PS Chain Rule- General  (V) -
PS More Chain Rule (V)
PS - Sign Analysis (V)
16 What is Velocity
17  What is Area
18 Integration
18 Area Calculation
18  Fn DefN, 6 Ways
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc

Units in Calculations:
7 Velocity
7 Varying Velocity Example
7. Velocity Calculation
7 Changing Units
7 Same Velocity  Motions
10 Slopes without Units.
10 Units & Slopes
10  Units in Cost vs. Quantity
10  How Units  Appear
10 Unit  Elimination
10 Partial Elimination
10 Interest & Units
12 More on Units

Appendices:

Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side Theorem
Integration & Lipschitz
 Continuity

These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
 Range Theorem is a postscript,
not in printed version.

Elimination of Units

Let u₁ = the unit of measurement of y. Let u₂ = the unit of measurement of x. Then y = Y·u₁ and x = X·u₂ where X and Y are real numbers. The latter provide unit-free description of the two quantities x and y provided the units are known.

The expression y = f(x) is equivalent to an expression Y = F(X) involving no units. In particular y = f(x) holds if and only if

Y·u1 = f(X·u2)

This is equivalent to Y = F(X) provided

F(X) = 1 u1 ·f(X·u2)

Evaluation of Y = F(X) for a given real-value of X results in a real number Y. Somehow, the units in the expression for F(X) cancel. Now calculations involving a formula f(x) can be expressed in terms of a unit-free (unit canceling) formula F(X). Thus calculations can be done and discussed with no mention of units - after their elimination. The instruction to write quantities y = Y·u₁, x = X·u₂ etc lead to calculations involving numbers X, Y, etc., in which units of measurement are absent. They been canceled or factored out. Thus calculations and education in mathematics can proceed without any further mention of units.

The elimination of units has an effect on slope calculation. Slopes to the graph of a unit-free Y = F(X) curve have the units

units of Y units of X = 1 1 = 1

That is, it is unit free, or it involves the improper unit 1.

The elimination of units and the formulation of unit-free equations Y = F(X), for further manipulation or computation, require replacements such as y = Y·u₁ and x = X·u₂. This demands an explicit choice of units of measurement and algebraic representation for all the quantities present in the problem. It further ties the further computations to the choice of units. This formality represents extra work and an extra burden in computations or the solving of equations. In particular, equations y = f(x) involving quantities, can be manipulated, in the first instance, without a selection of units of measurement and algebraic representations for every quantity mentioned in the statement of a problem. Moreover, when an algebraic solution is found in terms of a subset of the quantities present in a problem, substitution of quantities into the formula leads to computations involving both numbers and units of measurements. Here different units or measures of length, mass, time etc may appear in various combinations, for instance [cm/( meter²)]. Such odd ratios can be converted as needed at the end of computations, and not necessarily before.

For complicated equations, eliminating units and obtaining a dimensionless (i.e unit) free formulation has some advantages. It may indicate some similarity between frequently done computations. But for simple once-only computations the benefits may be minimal, and the elimination of units could be an unnecessary step. Here others may disagree.

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