Website owner: James Miller
Mapping of figures
1) u = u(x, y)
v = v(x, y)
represent a mapping from the xy plane into the uv plane. Let
2) f(x, y) = 0
be the equation of some curve C in the xy plane. See Fig. 1.
Q. What is the equation of the image C' of C in the uv plane?
A. The equation of the image of f(x, y) = 0 in the uv plane is obtained by solving the system
3) u = u(x, y)
v = v(x, y)
f(x, y) = 0
simultaneously and eliminating x and y, to produce an equation g(u, v) = 0. The equation g(u, v) = 0 is the equation of the image C' of C.
Proof. Let S be the solution set of system 3). It consists of all quadruplets (x, y, u, v) that satisfy the system. Every point on C is mapped into the uv plane and is thus included in the solution set. Now in every quadruplet (xi, yi, ui, vi) of S the pair (xi, yi) satisfies 2), which means it lies on C. Thus S contains those points and only those points which lie on C. Every quadruplet (xi, yi, ui, vi) satisfies system 1) which means that the pair (xi, yi) is mapped into (ui, vi). Every pair (u, v) that satisfies g(u, v) = 0 is in S and is the image of a point on C.
Note. The above theorem can be generalized to mappings in 3-space or n-space.
The Way of Truth and Life
God's message to the world
Jesus Christ and His Teachings
Words of Wisdom
Way of enlightenment, wisdom, and understanding
Way of true Christianity
America, a corrupt, depraved, shameless country
On integrity and the lack of it
The test of a person's Christianity is what he is
Who will go to heaven?
The superior person
On faith and works
Ninety five percent of the problems that most people have come from personal foolishness
Liberalism, socialism and the modern welfare state
The desire to harm, a motivation for conduct
The teaching is:
On modern intellectualism
On Self-sufficient Country Living, Homesteading
Principles for Living Life
Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.
America has lost her way
The really big sins
Theory on the Formation of Character
You are what you eat
People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest
Cause of Character Traits --- According to Aristotle
These things go together
We are what we eat --- living under the discipline of a diet
Avoiding problems and trouble in life
Role of habit in formation of character
The True Christian
What is true Christianity?
Personal attributes of the true Christian
What determines a person's character?
Love of God and love of virtue are closely united
Walking a solitary road
Intellectual disparities among people and the power in good habits
Tools of Satan. Tactics and Tricks used by the Devil.
On responding to wrongs
Real Christian Faith
The Natural Way -- The Unnatural Way
Wisdom, Reason and Virtue are closely related
Knowledge is one thing, wisdom is another
My views on Christianity in America
The most important thing in life is understanding
Sizing up people
We are all examples --- for good or for bad
Television --- spiritual poison
The Prime Mover that decides "What We Are"
Where do our outlooks, attitudes and values come from?
Sin is serious business. The punishment for it is real. Hell is real.
Self-imposed discipline and regimentation
Achieving happiness in life --- a matter of the right strategies
Self-control, self-restraint, self-discipline basic to so much in life
We are our habits
What creates moral character?