SolitaryRoad.com

Website owner:  James Miller


[ Home ] [ Up ] [ Info ] [ Mail ]

                               DIVERGENCE




Def. Divergence of a vector. 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 divergence of A is a scalar point function defined as


  ole.gif


The divergence can also be written, using the del operator, as


             ole1.gif


                         ole2.gif



Def. Flux strength. A measure of the flow of a fluid. It is equal to the amount of fluid passing through one square unit of cross-sectional area perpendicular to the direction of flow per unit of time.


Example. Water is flowing in a river at a rate of 3 feet per second. The amount passing through a one square foot cross-section perpendicular to the direction of flow is 3 cubic feet per second. The measure of water flow, viewed as flux, would then be given as 3 cu ft/sec per sq. ft.




Divergence. Physical interpretation. Let A be a vector point function defined throughout some region R of space. Then the divergence at any point P is given by



ole3.gif



ole4.gif

where A represents flux strength at point P, ΔV is the volume enclosed by the surface ΔS centered at P, n is the outward drawn unit normal at dS, and the limit is obtained by shrinking ΔV to point P. See Fig. 1 where ΔV is the volume of the depicted cube centered at P, ΔS is the surface of the cube, and ds is a differential element on ΔS. Physically


             ole5.gif


represents the net flux outflow per unit volume of the flux A from the surface ΔS. The quantity A∙n ds measures the rate of flow of the flux through the surface element dS.


Proof


If the integral


             ole6.gif


vanishes, there is no net outward flow. In this case any outward flux of A over part of the surface is balanced by an equal inward flux flow over the rest of the surface. If the integral is positive, the flux flow out of ΔV exceeds that into ΔV, and one says that there are sources within the volume. A negative value of the integral indicates the existence within ΔV of sinks, or points at which flux is being destroyed. If there are both sources and sinks within ΔV, the integral gives a measure of their algebraically combined strength. Thus div A can be taken as a measure of the source or sink strength at point P. If in a region there are no sources or sinks, then div A = 0 and we call A a solenoidal vector field.


Equation 2) can be taken as a definition of the divergence of A , and all the properties may be derived from it.


Continuity equation of an incompressible fluid. The equation


            div A = 0


is called the continuity equation of an incompressible fluid. It states the condition that fluid is neither created nor destroyed.



More from SolitaryRoad.com:

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 Homosexuality

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

Moral Perversion

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

Television

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-discipline

Self-control, self-restraint, self-discipline basic to so much in life

We are our habits

What creates moral character?


[ Home ] [ Up ] [ Info ] [ Mail ]