Elementary Mathematics and Science
For each conceptual goal develop at least
one page of notes, for each capability come
up with at least three examples.
A Note About Computer Algebra Systems: I
strongly advise the purchase of Mathematica Home Edition, it is about $300 and is an important addition
to any amateur scientist's toolbox. From
here on I will assume that you are using
Mathematica. There are free computer algebra
systems that you can use.
The main topics covered here are:
- Elementary mathematics through calculus.
- Elementary physics.
- Elementary chemistry.
Unit 1: Introduction to Mathematics and Science
This is the foundational unit, and many will
be able to just breeze through it. I do recommend
that you take some time, though. The topics
are not technically difficult, but they do
form the basis of a philosophy of science,
from which all of your work will be based.
The stronger this foundation, the stronger
will be your work.
Goals:
- The scientific method.
- The nature of mathematics.
Useful Sources of Study
Unit 2: Mathematics
This unit is designed to give you a quick
overview of mathematics.
Requirements:
Unit 1 Elementary Mathematics and Science:
Introduction to Mathematics and Science,
and Unit 1 Elementary Practical Science:
Introduction to Doing Science, and Unit 1
Independent Study: Introduction to Studying
Mathematics and Science.
Goals:
- Arithmetic operations: addition, subtraction,
multiplication, division, exponentiation,
radicals, and logarithms.
- The basics of set theory: sets, elements,
lists, set-builder notation, subsets, equal
sets, the empty set, union, intersection,
difference, and symmetric difference.
- The number system: natural numbers, integers,
rational numbers, irrational numbers, decimals
as approximations, real numbers, imaginary
numbers, and complex numbers.
- Geometric ideas: points, sets of points,
lines, planes, intersecting lines, angles,
triangles, other polygons, circles, solid
figures, area, surface area, and volume.
- Algebra: symbolic representation, algebraic
expressions, equations, inequalities, representing
equations and inequalities geometrically,
and types of equations and inequalities.
- Functions: Cartesian products, relations,
equivalence relations, functions, common
algebraic functions, and representing functions
geometrically.
- Calculus: Sequences, limits of sequences,
limits of functions, the derivative, differentiation
rules, the antiderivative, the integral,
the fundamental theorem of calculus, and
series.
Capabilities:
- Perform each arithmetic operation on any
two numbers.
- Represent a set in set-builder notation,
including subsets, union, intersection, difference,
and symmetric difference.
- Demonstrate sets using Venn diagrams: subsets,
union, intersection, difference, and symmetric
difference.
- Explain the idea of completeness of the number
system.
- Explain how the number system evolved from
natural numbers to complex numbers to gaurantee
completeness.
- Be able to make diagrams of each type of
geometric figure.
- Be able to calculate lengths, angles, areas,
surface areas, and/or volumes for all major
geometrical figures.
- Be able to convert a written statement concerning
quantities into an algebraic expression.
- Be able to perform arithmetic operations
on algebraic expressions.
- Be able to factor, expand, and simplify algebraic
expressions as needed.
- Derive, solve, and plot the solutions of
linear, rational, radical, quadratic, cubic,
and quartic equations and inequalities.
- Explain Cartesian products as they relate
to graphs.
- Explain relations, equivalence relations,
and functions in terms of sets.
- Represent common algebraic functions as equations,
tables, and Cartesian graphs. The functions
include linear, polynomial, rational, exponential,
and logarithmic.
- Explain sequences and convergence.
- Be able to calculate the limit of a sequence.
- Extend the idea of the limit of a sequence
to a function.
- Be able to calculate the limits of functions.
- Explain the idea of a derivative, higher-order
derivatives, the constant rule, the constant
multiple rule, the sum rule, the power rule,
the product rule, the quotient rule, and
the chain rule.
- Be able to calculate the derivatives of functions.
- Explain the idea of the integral.
- Be able to calculate the integrals of functions.
- Use the fundamental theorem of calculus to
calculate definite integrals.
Useful Sources of Study
- US Navy Training Manual, Mathematics, Basic
Math and Algebra, search for NAVEDTRA 14139 in your search
engine.
- US Navy Training Manual. Mathematics, Pre-Calculus
and Introduction to Probability, search for NAVEDTRA 14141 in your search
engine.
- David A. Santos, (2008), Andragogic Propaedeutic Mathematics. This is a free download for the website:
http://faculty.ccp.edu/faculty/dsantos/lecture_notes.html. This book does not cover complex numbers,
but it does have an introduction to sets.
- David A. Santos, (2008), Ossifrage and Algebra. This is a free download for the website:
http://faculty.ccp.edu/faculty/dsantos/lecture_notes.html. This book does not cover complex numbers.
- David A. Santos, (2008), Precalculus. This
is a free download for the website: http://faculty.ccp.edu/faculty/dsantos/lecture_notes.html.
- CK-12 Foundation, (2010), Math Analysis, CK-12 Flexbook. This is a free download
from the website: www.ck12.org
- HELM Workbooks (Helping Engineers Learning
Mathematics), Available as a free download from the website:
http://www.maths.sussex.ac.uk/Staff/RML/HELM/
- Jerrold E. Marsden, Alan Weinstein, (1985),
Calculus I. Available as a free download from the website:
http://caltechbook.library.caltech.edu/view/person-az/Marsden-J-E.html
- CK-12 Foundation, (2010), Single-Variable Calculus, CK-12 Flexbook. This is a free download
from the website: www.ck12.org
Unit 3: Science
This unit is designed to give you a quick
overview of chemistry and physics. This is
a very long unit.
Requirements:
Unit 2 Elementary Mathematics and Science:
Mathematics, and Unit 2 Elementary Practical
Science: Problem-Solving.
Goals:
- Basic Science: Units of measurement, scientific
notation, unit conversions, dimensional analysis
- Physics: Motion, position, displacement,
velocity, mass, kinetic energy, acceleration,
potential energy, the conservation of energy,
momentum, the conservation of momentum.
- Particles: Newton's laws of motion, work,
power, angular momentum, and the conservation
of angular momentum.
- Matter: Collisions, accretion, explosions,
systems of particles, center of mass, moment
of inertia, rigid bodies, statics, elastic
solids, fluid statics, fluid dynamics, and
the zeroth, first, second, and third laws
of thermodynamics.
- Chemistry: Atoms, molecules, chemical elements
and isotopes, chemical formulas and equations,
chemical reactions, reaction yields, kinetic
theory, thermochemistry, and chemical equilibrium.
Capabilities:
- Identify the primary SI units of length,
time, mass, temperature, and current..
- Identify the derived SI units for area, volume,
density.
- Given a density, find the mass or the volume.
- Use the prefixes for units from pico to mega
in scientific notation.
- Be able to treat units as algebraic symbols.
- Convert from SI to cgs and English systems
of units.
- Use dimensional analysis to derive formulas.
-
Useful Sources of Study
- CK-12 Foundation, (2010), People's Physics Book Version 2, CK-12 Flexbook. This is a free download
from the website: www.ck12.org
- CK-12 Foundation, (2010), 21st Century Physics Flexbook, CK-12 Flexbook. This is a free download
from the website: www.ck12.org
- Benjamin Crowell, (2009), Simple Nature. This is a free download for the website:
http:// www.lightandmatter.com
- CK-12 Foundation, (2010), Chemistry, CK-12 Flexbook. Specifically chapter 1.
This is a free download from the website:
www.ck12.org
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