Website owner: James Miller
CURL (OR ROTATION), CIRCULATION, CIRCULATION INTENSITY
Def. Curl (or Rotation). Let A(x, y, z) = A1 (x, y, z) i + A2 (x, y, z) j + A3 (x, y, z) k be a vector point function that is defined and differentiable at each point (x, y, z) in a certain region of space. The curl (or rotation) of A is a vector point function defined as
Curl. Physical interpretation.
Before giving the physical interpretation of the curl of a vector we define the concepts of “the circulation of a vector around a curve” and “the intensity of circulation at a point”.
Def. Circulation of vector A about curve C. Let A(x, y, z) = A1 (x, y, z) i + A2 (x, y, z) j + A3 (x, y, z) k be a vector point function (i.e. vector field) that is defined and differentiable at each point (x, y, z) in a certain region G of space. Let C be a simple closed curve (i.e a closed curve that doesn’t intersect itself anywhere) within G defined by r(t) = x(t) i + y(t) j + z(t) k . Then the circulation of vector A about C is defined as
is a measure of the tendency of A to flow along the curve C.
Def. Intensity of circulation at point P. Pass a plane through a point P in region G (any plane). Let ΔS be a small segment of the plane which contains point P and which is bounded by the simple closed curve C. See Fig. 1. Then the intensity of circulation at point P is defined as
where the limit is taken in such away that ΔS shrinks to P. As one can see, dividing by ΔS yields a circulation per unit area of bounded surface, which is in the nature of an intensity of circulation.
In general, for each plane that passes through point P, there will be a different circulation intensity. For some plane passing through point P, there will be a maximum circulation intensity.
Physical interpretation of curl. Curl A at point P is that vector the direction of which is normal to the plane of the algebraically maximum circulation intensity at P and the magnitude of which is the maximum circulation intensity.
Example. If v is the velocity at a point P(x, y, z) in a moving fluid, the vector angular velocity of an infinitesimal portion of the fluid about P is given by ω = ½ curl v.
If a vector field F represents a moving fluid, for example, a paddle wheel placed at various points in the field would tend to rotate in regions where curl F ≠ 0. In areas where curl F = 0 there would be no rotation.
Def. Irrotational field. A field in which the curl is everywhere zero.
Def. Vortex field. A field that is not irrotational.
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?