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The terms arrow and vector will be used interchangeably.
This applet
(online only) illustrates the ideas in this chapter.
Map Addition Method
In navigation, drawing an arrow on a map from a point A to a point
B represents a linear
FOOTNOTE: Physically, the arrow from
A to C should lie on the path a taut line or
string would follow between the two points. displacement or movement between them,
that is the tail point A and the head point B. To show
a second displacement from the head point B, put the tail
of one second arrow at B. The result of these two
movements, is a nonlinear movement from a tail point A of the first
arrow to the head point, say C, of the second arrow.
The straight arrow joining the tail point A of the first
arrow to the head point C of the second is a third arrow
called the sum of the first two. It represents a linear
movement from the points A to the point C. The foregoing describes
the head to tail map addition method for adding two arrows
together when the head of one is at the tail of another.
Two of the solid arrows and dotted lines parallel
to them in the above diagram form the sides and diagonal of
a parallelogram. The three solid arrows form a triangle.
Another triangle is formed by the dotted lines and the
diagonal arrow of the parallelogram. The rotation of these
triangles and the parallelogram will have a deep consequence
in the following overlapping discussion of complex, sines
and cosines.
Reading Guide
Too much may be said in this chapter. If you get lost in
the details, read this chapter lightly or go on to the next
chapter. One aim of this chapter was to make fuss about some
technical details - gaps in the author's comprehension perhaps.
Parallelogram Addition Method
When two arrows have the same tail points, they determine a
parallelogram as well. They can be added by moving without
a change in direction, one the arrows to the head of the
other. This gives the parallelogram method for
adding or summing two arrows with the same tail points (or origins).
Observe the presence of a two triangles and a
parallelogram.
The dotted lines indicate
positions of two of the solid arrows,
after a movement to the head of the other without
a change of direction. We will describe each displacement as a
parallel movement of one along the other. The arrows before
and after movement altogether form the sides of a
parallelogram. The addition of one arrow to a second is
represented by the parallel movement of the tail point of
one to the
head of the other. The formation of the parallelogram
implies that which is added to which is immaterial, the
result will be the same. Either way, the solid arrow along the diagonal of the
parallelogram gives the (linear) arrow sum of the other two.
Arrow Components
The above diagram shows how the arrow from A to B can be
regarded as the parallelogram sum of a horizontal arrow
and a vertical arrow. The horizontal and vertical arrows are respectively
called the horizontal and vertical components of the initial
arrow from A to B. (These components depend on the choice of
directions for the so-called horizontal and vertical axes.)
The initial arrow is the map and parallelogram sum of the two
component arrows. In the representation of arrows,
an arrow can be viewed as the map addition of its vertical
component to horizontal component arrows. Here the tail of
the vertical component is moved to the head of the
horizontal component. (The arrow could be also be viewed as
the map addition of the horizontal component to the
vertical one.) Which map addition is shown on a diagram is
immaterial.
Component Addition Method
The following diagram shows the map addition of two solid arrows,
namely the tail to head addition of the arrow from B to
C to the arrow from A to B gives the same result as
parallelogram addition of (i) the sum of the vertical components to
(ii) the sum of the horizontal components.
and
(ca,cb) = (|c|r,q+180°) = (-|c|a,-|c|b)
if c < 0.
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Why Slopes
and
More Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
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Read slowly, Volumes 2 & 3 may ease or avoid
calculus difficulties. Take the risk.
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Chapters and Appendices
Content Guide
Foreword
2nd Content Guide
1. Introduction
Geometric Calculus Preview (1983)
2. Algebraic 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.
What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent
or Independent
Variable, a Matter of Choice
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For Parents & Teachers: Speaking
Skills, Reading
& Writing,
Preparing for Science, ends,
values and methods for work and study, parent- friendly mathematics booklets for ages 4-14.
-
Math
Education
Essays (opinions,
possibilities, references)
- POMME, a two
level program for future skill development in
schools and colleges worldwide. Address content &
motivation gaps with ends, values & methods for skill
development to say which way to go, how and why. -
Present Day Curriculum:
(A) Secondary
I Mathematics
consolidate fractions and measurement, skills and
sense consolidation,
(B)
Secondary II Mathematics
year of algebra and proportionality
(C) See too the following:
- Arithmetic
& Number Theory Practices (horribly put, but
useful)
- Algebra and
Logic SubProgram
(well put, extremely useful)
For Avid Readers in School & Out -
Online Books
1. Elements of
Reason. 1996
1A. Pattern
Based Reason 1995
1B. Math
Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3.Why
Slopes & More.Math
1995
Tour their forewords.
Calculus Prep or Help: See Volumes 2 & 3,
and this bigger
Calculus
Guide.
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Senior
High School &
Calculus Students
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Free Live Lesson
- Operations with Decimals - Comparison, Subtraction and Long Division
- Click here
to attend.
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For Senior
High School Mathematics & Calculus
Intro to Solving
Linear Equations
- a different paths for junior and even senior high
school students.
5
wordy Logic
Chapters
4 curious Algebra
Chapters
Words before & besides symbols. A Key Algebra
forward & backwards Chapter
First Calculus
Preview (1st intro)
Four Calculus
Chapters
(2nd intro)
Intro to Complex
Numbers (long)
Intro to Mathematical
Induction (romantic & wordy at first)
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Many More Topics
1. Decimal
Arithmetic Reference!
2. Integers
- Intro to Signed No.s
3. Fractions
- fully explained.
4. Fractions
with Units
5. Number
Theory,
6. Solving
Linear Equations
7 Formulas
Use Forward & backwards -
8. Proportionality,
Back- & For-wards.
9. Logic
Chapters:
10. Euclidean-Geometry
11. Slopes
& Equations of Straight Lines. (Take
I. See take II below)
12. Why
Study Slopes.
13. Maps,
Plans, Similarity & Trig,
(Take II included here)
14. Quadratics:
Starter lessons
15. Polynomials:
Starter lessons
16 Why
Factor Polynomials:
17 Functions
- Forwards & Backwards.
18. Exponents,
Radicals & logs.
19. Complex
Numbers before trig (new advance/ starter lesson)
20. DC
Electric
Circuits Etc
21. Real
Analysis
22. The
Olde Complex No, Trig
& Vector Section.
23. More
Calculus Stuff
- written after Volumes 2 and 3.
More For Instructors
-
Education
Essays
(opinions,
possibilities, references)
- POMME, a two
level program for instruction K1-14
- Math & Logic How-TOs
1. Arithmetic
2. Algebra
3. More Algebra
4. Beginner Geometry
5. More Geometry
6. Calculus
7. Show Work or Logic
These may be too dense for students.
Level I Material: New Stuff
Time and Date Matters
Level I Arithmetic.
Money Matters
Measurement Matters
Matters of Chance (Risk Control)
Logic
Chapters
(leave what's not clear in Level I to Level II)
Using/Making Maps and Plans.
(A variant of
Maps,
Plans, Similarity & Trig, to
appear here).
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Skill Development Tips
For All
Standards: (A) Take
care to avoid the domino effect of errors & approximations; (B) Do and
record steps in an manner that allows skill mastery to be seen or
corrected. Anything represent substandard work.
Key Numerical Methods
- To multiply signed numbers, prefix the product of their signs to the product
of their lengths or unsigned parts. The product is negative if the no of
negative sign in it is odd.
- To add signed numbers with like signs, prefix the common sign to the sum of the
lengths.
- To add signed numbers with opposite signs, prefix the sign of the longest to
the difference: length of longest minus length of shortest.
- Should we study roots and powers of real numbers with formulas involving exponential and log.
- How does adding and multiplying points in the plane and rotating the midpoint
of a line segment lead to mastery of complex numbers and the thought-based
development of their properties, all before trig?
- New Axioms for High School Mathematics: In accounting, totals of assets
and debts may be calculated by dividing the assets and debts into
non-overlapping (disjoint) groups and then adding subtotals. In general, sums
(and products) of counts and numbers, positive and negative numbers
included, can be obtained by adding subtotals (and multiplying
subproducts, respectively). These practices may be cast as axioms in
secondary mathematics. Then operations on polynomials are easily implied
justified by these "axioms" and the geometric introduction of column
methods for expanding a products of two sums. While set theory in pure
mathematics may imply the above axioms in university mathematics programs
instruction, an earlier and more accessible explanation based on easily accepted
and understood geometric and counting practices derivation of the above
axioms is possible at the high school for students heading for college programs
in science.
In Volume 2: Prep for Calculus
- What is the difference between saying A if B and saying A if and
only if B. Being aware of the difference will sharpen ye wits.
- What is a chain of reason?
-Are your arithmetic skills OK?
-Have words been missing in the introduction of algebra?
- Can ye talk about numbers & quantities varying apart from or before the
use of letters & functions?
- Do ye know about the forward & backward use of formulas?
-Contrapositive: is that backward use of A if B?
-What is a variable x? Answer before speaking of function f(x) = x.
-What a twist! There are no rules of algebra for subtraction and division. But
if you replace them by addition of -x and multiplication by 1/x, rules of
algebra (properties of arithmetic) can be used.
In Volume 3: Calculus Slowly?
-Why are slopes studied and polynomials factored in high school?
- Volume 3 suggests how to ease or delay algebra shock in
calculus *& beyond. In Calculus, derivatives and integrals introduced and defined by limits, but calculated
without when possible by using differentiation rules forwards and backwards. The
second site calculus section may help in differential calculus.
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