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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 = u(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  


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


or equivalently,


ole.gif


where 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


or equivalently,


             ole1.gif


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 F = F(x, y, z), then F is called a vector point function. Such a function would have a representation 


                        F = f1(x, y, z) i + f2(x, y, z) j + f3(x, y, z) k


or equivalently,


ole2.gif


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

 


                        F = f1(x, y, z, t) i + f2(x, y, z, t) j + f3(x, y, z, t) k


or equivalently,


ole3.gif



 

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.



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