SolitaryRoad.com

Website owner:  James Miller


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

SCALAR AND VECTOR FUNCTIONS, POINT FUNCTIONS, SCALAR POINT FUNCTIONS, VECTOR POINT FUNCTIONS, SCALAR AND VECTOR FIELDS



In vector analysis we deal with scalar and vector functions.


Def. Scalar function. A scalar function is a function that assigns a real number (i.e. a scalar) to a set of real variables. Its general form is


                                    u = f(x1, x2, ... , xn)


where x1, x2, ... , xn are real numbers.



Def. Vector function. A vector function is a function that assigns a vector to a set of real variables. Its general form is  


                        U = f1(x1, x2, ... , xn) i + f2(x1, x2, ... , xn) j + f3(x1, x2, ... , xn) k


or equivalently, in parametric form,


            u1 = f1(x1, x2, ... , xn)

            u2 = f2(x1, x2, ... , xn)

            u3 = f3(x1, x2, ... , xn)


where U = (u1, u2, u3) and x1, x2, ... , xn are real numbers..



Example 1. Function defining a space curve. Let


            R(t) = x(t) i + y(t) j + z(t) k


be a radius vector to a point P(x, y, z) in space which moves as t increases in value. It traces out a curve in space. The parametric representation of space curves is


            x = x(t)

            y = y(t)

            z = z(t) .


Example 2. Function defining a surface in space. The function


            R(u,v) = x(u,v) i + y(u,v) j + z(u,v) k


represents a surface in space. Surfaces are represented by parametric equations of the type


            x = x(u, v)

            y = y(u, v)

            z = z(u, v)


If v is regarded as a parameter, u a variable, then this system describes a space curve. For each value of v there is another space curve, thus generating a surface.



Def. Point function. A point function u = f(P) is a function that assigns some number or value u to each point P of some region R of space. Examples of point functions are scalar point functions and vector point functions.



Def. Scalar point function. A scalar point function is a function that assigns a real number (i.e. a scalar) to each point of some region of space. If to each point (x, y, z) of a region R in space there is assigned a real number u = Φ(x, y, z), then Φ is called a scalar point function.


Examples. 1. The temperature distribution within some body at a particular point in time. 2. The density distribution within some fluid at a particular point in time.


Syn. scalar function of position



Scalar field. A scalar point function defined over some region is called a scalar field. A scalar field which is independent of time is called a stationary or steady-state scalar field.


A scalar field that varies with time would have the representation


            u = Φ(x, y, z, t) .



Def. Vector point function. A vector point function is a function that assigns a vector to each point of some region of space. If to each point (x, y, z) of a region R in space there is assigned a vector U =U(x, y, z), then U is called a vector point function. Such a function would have a representation


            U = u1(x, y, z) i + u2(x, y, z) j + u3(x, y, z) k


or in parametric form,


            u1 = u1(x, y, z)

            u2 = u2(x, y, z)

            u3 = u3(x, y, z)




Syn. vector function of position



Vector field. A vector point function defined over some region is called a vector field. A vector field which is independent of time is called a stationary or steady-state vector field.


A vector field that varies with time would have the representation


            U = u1(x, y, z, t) i + u2(x, y, z, t) j + u3(x, y, z, t) k


or in parametric form,


            u1 = u1(x, y, z, t)

            u2 = u2(x, y, z, t)

            u3 = u3(x, y, z, t)



Examples. 1. Gravitational field of the earth. 2. Electric field about a current-carrying wire. 3. Magnetic field generated by a magnet. 3. Velocity at different points within a moving fluid. 4. Acceleration at different points within a moving fluid.







Examples of vector point functions.


Force fields.


1] gravitational field


2] magnetic field near a magnet or current-carrying wire


3] electric field generated by a charge or current-carrying wire



Velocity field


1] Velocity at different points in space within a moving fluid (e.g. wind velocities in the atmosphere; particle velocities in a stream)



Acceleration field.


1] Acceleration at different points in space within a moving fluid




I Functional representation of a force field (static case).     All of the following are equivalent ways of expressing the functional representation of a static force field:



            Fx = Fx(x, y, z)

1]        Fy = Fy(x, y, z)

            Fz = Fz(x, y, z)


where Fx, Fy, and Fz are the x, y and z components of the force F at point (x, y, z).



            Fx = f1(x, y, z)

2]        Fy = f2(x, y, z)

            Fz = f3(x, y, z)


            F1 = u1(x, y, z)

3]        F2 = u2(x, y, z)

            F3 = u3(x, y, z)


            F1 = u1(x1, x2, x3)

4]        F2 = u2(x1, x2, x3)

            F3 = u3(x1, x2, x3)


ole.gif  


ole1.gif  

 

7]        F = u1(x, y, z) i + u2(x, y, z) j + u3(x, y, z) k


where i, j and k are unit vectors along the coordinate axes.




● A vector field which is independent of time is called a stationary or steady-state vector field.

 




II Functional representation of a force field (dynamic case).        All of the following are equivalent ways of expressing the functional representation of a dynamic force field:



            Fx = Fx(x, y, z, t)

1]        Fy = Fy(x, y, z, t)

            Fz = Fz(x, y, z, t)


where Fx, Fy, and Fz are the x, y and z components of the force F at point (x, y, z) at time t.



            F1 = f1(x, y, z, t)

2]        F2 = f2(x, y, z, t)

            F3 = f3(x, y, z, t)


            F1 = u1(x, y, z, t)

3]        F2 = u2(x, y, z, t)

            F3 = u3(x, y, z, t)


            F1 = u1(x1, x2, x3, t)

4]        F2 = u2(x1, x2, x3, t)

            F3 = u3(x1, x2, x3, t)


ole2.gif  


ole3.gif  

 

7]        F = u1(x, y, z, t) i + u2(x, y, z, t) j + u3(x, y, z, t) k


where i, j and k are unit vectors along the coordinate axes.



III Functional representation of a velocity field in a moving, turbulent fluid. The functional representation of the velocities at various points within a moving, turbulent fluid can be expressed in the following equivalent ways:


            vx = f1(x, y, z, t)

1]        vy = f2(x, y, z, t)

            vz = f3(x, y, z, t)


where vx, vy, and vz are the x, y and z components of the velocity V at point (x, y, z) at time t.



            v1 = f1(x, y, z, t)

2]        v2 = f2(x, y, z, t)

            v3 = f3(x, y, z, t)


            v1 = v1(x, y, z, t)

3]        v2 = v2(x, y, z, t)

            v3 = v3(x, y, z, t)


            v1 = v1(x1, x2, x3, t)

4]        v2 = v2(x1, x2, x3, t)

            v3 = v3(x1, x2, x3, t)


ole4.gif  


ole5.gif  

 

7]        V = v1(x, y, z, t) i + v2(x, y, z, t) j + v3(x, y, z, t) k


where i, j and k are unit vectors along the coordinate axes.




IV Functional representation of an acceleration field in a moving, turbulent fluid. The functional representation of the accelerations at various points within a moving, turbulent fluid can be expressed in the following equivalent ways:


            ax = f1(x, y, z, t)

1]        ay = f2(x, y, z, t)

            az = f3(x, y, z, t)


where ax, ay, and az are the x, y and z components of the acceleration A at point (x, y, z) at time t.



            a1 = f1(x, y, z, t)

2]        a2 = f2(x, y, z, t)

            a3 = f3(x, y, z, t)


            a1 = a1(x, y, z, t)

3]        a2 = a2(x, y, z, t)

            a3 = a3(x, y, z, t)


            a1 = a1(x1, x2, x3, t)

4]        a2 = a2(x1, x2, x3, t)

            a3 = a3(x1, x2, x3, t)


ole6.gif  


ole7.gif  

 

7]        A = a1(x, y, z, t) i + a2(x, y, z, t) j + a3(x, y, z, t) k


where i, j and k are unit vectors along the coordinate axes.





References.


  James and James. Mathematics Dictionary.

  Murray R. Spiegel. Vector Analysis.

  Angus E. Tayor. Advanced 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 ]