SolitaryRoad.com

Website owner:  James Miller


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

DIVERGENCE THEOREM, STOKES’ THEOREM, GREEN’S THEOREM AND RELATED INTEGRAL THEOREMS




Divergence Theorem. Let S be a closed surface in space enclosing a region V and let A(x, y, z) be a vector point function, continuous, and with continuous derivatives, over the region. Then


ole.gif


where n is the positive (outward drawn) normal to S.


Also called: Gauss’ Theorem, Gauss’ Divergence Theorem, Green’s Theorem in Space, Ostrogradski’s Theorem.


Proof


Physical interpretation. The left member


             ole1.gif


represents the algebraic sum of all flux inside surface S (flux from all sources minus outgoing flux at sinks) and the right member


             ole2.gif


represents the net flux emanating from the surface.


 


Stokes’ Theorem. Let S be a two-sided surface in space and let R be a region of S enclosed by a simple closed curve C ( i.e. a closed, non-intersecting curve). Let A(x, y, z) be a vector function of position defined and with continuous derivatives over R and C. Then


ole3.gif


where r(t) = x(t) i + y(t) j + z(t) k is the position vector defining curve C, n is the positive normal to R, and the curve C is assumed to be traversed in the positive (counterclockwise) direction.


Note. The positive direction is that direction in which an observer, walking on the boundary of S, with his head pointing in the direction of the positive normal of S, always has the surface on his left.




Green’s Theorem in the Plane. If R is a closed region of the xy plane bounded by a simple closed curve C and M(x, y) and N(x, y) are continuous functions of x and y, having continuous derivatives in R, then


ole4.gif


where C is traversed in the positive (counterclockwise) direction.


In vector form this theorem is


ole5.gif  


where A = M i + N j and r = x i + y j.


Proof


Green’s theorem in the plane is a special case of Stokes’ theorem. In addition, the Divergence theorem represents a generalization of Green’s theorem in the plane where the region R and its closed boundary C in Green’s theorem are replaced by a space region V and its closed boundary (surface) S in the Divergence theorem. Because of this the Divergence theorem is often called Green’s Theorem in space.




Green’s Identities.



Green’s first identity. Let S be a closed surface in space enclosing a region V and let Φ(x, y, z) and ψ(x, y, z) be scalar functions of position with continuous second derivatives over the region. Then


  ole6.gif


where n is the positive (outward drawn) normal to S.



Green’s second identity. Let S be a closed surface in space enclosing a region V and let Φ(x, y, z) and ψ(x, y, z) be scalar functions of position with continuous second derivatives over the region. Then



ole7.gif


where n is the positive (outward drawn) normal to S. Note the symmetry.


Equation 6) can be obtained from 5) by permuting Φ and ψ and subtracting. Equation 5) can be obtained from the Divergence Theorem by letting A of 1) be Φ ole8.gif ψ so that


             ole9.gif







Other related theorems.


Theorem 1. Let S be a closed surface in space enclosing a region V and let A(x, y, z) be a vector function of position defined, continuous, and with continuous derivatives, over the region. Then


ole10.gif


where n is the positive (outward drawn) normal to S. Note that the form is the same as the divergence theorem with the dot product of the divergence theorem replaced by the cross product.



Theorem 2. Let S be a surface enclosed by a simple closed curve C and let Φ(x, y, z) be a scalar function of position defined and with continuous derivatives over S and C. Then


ole11.gif


where r(t) = x(t) i + y(t) j + z(t) k is the position vector defining curve C, n is the positive normal to S, and the curve C is assumed to be traversed in the positive (counterclockwise) direction.


Note the similarity in form to Stokes’ theorem with A of Stokes’ theorem replaced by Φ.





References

  Spiegel. Vector Analysis.

  Hsu. Vector Analysis.

  Angus Taylor. Adv. Calculus.



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 ]