"Though this be madness, yet there is method in 't." - Shakespeare

A HYPERGUIDE: RELATIONSHIPS AMONG TOPICS

This page demonstrates some features of our proposed "hyperguide" to math. The main focus will be to highlight patterns within the curriculum.

We hope that organizing material in a manner which emphasizes the big picture will make review or assimilation of topics more manageable.

The main point is that broader topics in pre-calculus algebra fit into a basic pattern of "do it again" and "un-do it", and the calculus curriculum

follows a similar pattern after another ingredient is included. Many of the fundamental formulas of algebra and calculus may be understood in

terms of the interactions of each operation obtained from these procedures with others. The streams of geometry and trigonometry begins

in parallel to the algebraic world, until calculus ultimately unifies them through the exponential function in the complex number plane.

Links to and from historical development and scientific motivations, and extensions to linear algebra and multivariable calculus, and

more advanced topics are being developed elsewhere.

ALGEBRA: THE NUMBER LINE

 ADDITION UNDO IT SUBTRACTION DO IT AGAIN DO IT AGAIN MULTIPLICATION UNDO IT DIVISION DO IT AGAIN DO IT AGAIN POWER AND EXPONENTIATION UNDO IT ROOT AND LOGARITHM

PLANE GEOMETRY AND TRIGONOMETRY: THE NUMBER PLANE

 COMPLEX ADDITION UNDO IT COMPLEX SUBTRACTION DO IT AGAIN DO IT AGAIN COMPLEX MULTIPLICATION UNDO IT COMPLEX DIVISION DO IT AGAIN DO IT AGAIN COMPLEX POWER AND EXPONENTIATION UNDO IT COMPLEX ROOT AND LOGARITHM PLANE TO LINE CIRCULAR/POLAR  TO  RECTANGULAR r exp (it) = r cos t + i r sin t LINE TO PLANE RECTANGULAR TO CIRCULAR/POLAR TRIGONOMETRIC/CIRCULAR UNDO IT ARC-TRIGONOMETRIC/CIRCULAR

CALCULUS: THE APPROXIMATED VALUES ("LIMITS") DEFINE CUMULATIVE AND INSTANTANEOUS CHANGE

AND UNIFY ALGEBRA AND GEOMETRY THROUGH THE COMPLEX EXPONENTIAL. POWER FUNCTIONS AND

THE BINOMIAL FORMULA EXPAND TO ALL REAL EXPONENTS.

 SUMMATION/INTEGRATION UNDO IT DIFFERENCING/DIFFERENTIATION DO IT AGAIN DO IT AGAIN MULTIPLE INTEGRALS UNDO IT HIGHER DERIVATIVES

 OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES Addition, m+n, sum Subtraction, x=y-n solves x+n=y, difference Natural or whole numbers, solving requires and definesexpansion to negative. Expand to rational, real, complex, matrix, function, etc. DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS Multiplication, x+x+...+x=nx See other operations (Multiplication, Power, Exponential, etc.) Shifting,shifted functions, f(x)=x+n,  f(x+n) VISUALIZATIONS INTERPRETATIONS PROPERTIES Shifting/translation Combined quantity or effect, shift x+y=y+x, (x+y)+z=x+(y+z), etc.

RELATIONSHIPS INVOLVING MULTIPLICATION

 OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES Multiplication, mn, m*n, product Division, x=y/n solves nx=y, quotient Natural or whole numbers, solving requires and defines expansion to rational. Expand to real, complex, function, etc. DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS Power/Exponential xx...x=xn x(y+z)=xy+xz, x(y-z)=xy-xz, (see alsoPower, Exponential, etc.) Linear, scaling, scaled functions, f(x)=nx, f(nx) Linear behavior/models f(x)=mx+b Polynomials p(x)=a0+ a1x +...+ anxn VISUALIZATIONS INTERPRETATIONS PROPERTIES Scaling, zoom, stretching Repeated additive combination or effect, scaling xy=yx (for "numbers"), (xy)z=x(yz)

RELATIONSHIPS INVOLVING SUBTRACTION

 OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES Subtraction Natural or whole numbers.Expand to integers, rational, real, complex, matrix, function, etc. DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS Division VISUALIZATIONS INTERPRETATIONS PROPERTIES

RELATIONSHIPS INVOLVING DIVISION

 OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES Division Natural or whole numbers,Expand to integer, rational, real, complex, matrix, function, etc. DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS Logarithm Inverse linear/scaling f(x)=x/n, f(x/n) n != 0 Rational functions VISUALIZATIONS INTERPRETATIONS PROPERTIES Whole part of x/y is the number of times you can subtract y from x with a result greater than 0.

RELATIONSHIPS INVOLVING EXPONENTIALS

 OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES Exponential Logarithm Natural or whole numbers,Expand to integer, rational, real, complex, matrix, function, etc. DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS with addition:            ax+y= ax ay with multiplication:  axy = (ax)y Exponential function f(x)=ax Exponential behavior/models f(x)=Cax VISUALIZATIONS INTERPRETATIONS PROPERTIES

RELATIONSHIPS INVOLVING LOGARITHMS

 OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES Logarithm Exponential Natural or whole numbers,Expand torational, real, complex, matrix, function, etc. DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS VISUALIZATIONS INTERPRETATIONS PROPERTIES Whole part of logy x is the number of times you can divide y from x with a result greater than 1.

RELATIONSHIPS INVOLVING POWERS

 OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES Power Natural or whole numbers,Expand to integer, rational, real, complex, matrix, function, etc. DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS with addition:    (x+y)n = xn+nxn-1y+...+yn with multiplication: (xy)n = xnyn VISUALIZATIONS INTERPRETATIONS PROPERTIES

RELATIONSHIPS INVOLVING ROOTS

 OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES Root Power Natural or whole numbers,Expand to integer, rational, real, complex, matrix, function, etc. DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS VISUALIZATIONS INTERPRETATIONS PROPERTIES

RELATIONSHIPS INVOLVING DIFFERENTIATION

 OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES Differentiation, f'(x), df/dx, derivative Integration Differentiable functions, expand to distributions DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS Higher derivatives f(n)(x), (d/dx)nf with addition:             (f + g)' = f' + g' with subtraction:        (f - g)' = f' - g' with multiplication:     (cf)'=cf', (fg)'=f'g+fg'  (product rule)            (see integration by parts) with division:              (f/g)'=(gf'-fg')/g2 (quotient rule) with composition:        (f o g)'(x)=f'(g(x))g'(x) (chain rule)                     (see substitution) with inversion:            f-1 '(x)=1/f'(f-1(x))   with power:                 d/dx(xn/n!)=xn-1/ (n-1)! with exponential:         d/dx(ex)=ex with circular functions:                             cos'(x)=-sin(x),sin'(x)=cos(x) VISUALIZATIONS INTERPRETATIONS PROPERTIES Slope of tangent line Recover instantaneous change from cumulative change, e.g., speedometer from tripmeter

RELATIONSHIPS INVOLVING INTEGRATION

 OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES Integration, integral Differentiation Integrable functions, distributions DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS Multiple integrals with addition:             I(f+g)= I(f)+ I(g) with subtraction:        I(f-g)= I(f)  - I(g) with multiplication:   I(cf)=cI(f),I(f'g)=fg-I(fg') (integration by parts)  (see product rule) with composition:      I(f(u(x)u'(x))= (substitution)                    (see chain rule)   with power:                 I(xn/n!)=xn+1/ (n+1)! with exponential:         I(ex)=ex with circular functions:                       I(cos(x))=sin(x),I(sin(x))=-cos(x) VISUALIZATIONS INTERPRETATIONS PROPERTIES Area between graph and independent variable axis Recover cumulative change from instantaneous change, e.g., tripmeter from speedometer

RELATIONSHIPS INVOLVING TRIGONOMETRIC/CIRCULAR FUNCTIONS

 OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES Horizontal and vertical projection: cosine,sine arccosine, arcsine Angles, Arclength DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS with addition: Trigonometric Addition Formulas   with subtraction:   with multiplication: cos (2x)=cos^2 x - sin^2 x                              sin (2x)=2 sin x cos x cosx cos y = 1/2 (cos (x+y) + cos (x-y)) sin x cos y = 1/2 (sin (x+y) - sin (x-y)) sin nx sin mx =1/2 (cos (x-y) - cos (x+y)) with division: tangent, cotangent, secant, cosecant VISUALIZATIONS INTERPRETATIONS PROPERTIES Relationship between uniform circular motion and the graphs of natural cosine and sine cos (x) = cos (-x), cos (x+2 \pi)= cos (x) sin (x) = -sin (-x), sin (x+2 \pi)= sin (x) sin (x) = cos (x- {\pi \over 2})

RELATIONSHIPS INVOLVING ARC-TRIGONOMETRIC/CIRCULAR FUNCTIONS

 OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES Angles from horizontal or vertical projections, arccosine, arcsine Horizontal and vertical projections  (trigonometric/circular) DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS with addition: with subtraction: with multiplication: with division: arctangent, arccotangent, arcsecant, arccosecant VISUALIZATIONS INTERPRETATIONS PROPERTIES

 OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS VISUALIZATIONS INTERPRETATIONS PROPERTIES

MORE TO COME INCLUDING

DIFFERENCING            SUMMATION

LIMIT                            VARIATION

LINEAR TRANSFORMATIONS

Warnings/Pitfalls, Alternate notation and terminology