Website owner:  James Miller

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, ... , x_n)

where x₁, x₂, ... , x_n 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, ... , x_n) i + f₂(x₁, x₂, ... , x_n) j + f₃(x₁, x₂, ... , x_n) k

or equivalently, in parametric form,

            u₁ = f₁(x₁, x₂, ... , x_n)

            u₂ = f₂(x₁, x₂, ... , x_n)

            u₃ = f₃(x₁, x₂, ... , x_n)

where U = (u₁, u₂, u₃) and x₁, x₂, ... , x_n 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 7) 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.

References.

  James and James. Mathematics Dictionary.

  Murray R. Spiegel. Vector Analysis.

  Angus E. Tayor. Advanced Calculus.

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:

            u_x = f₁(x, y, z)

1]        u_y = f₂(x, y, z)

            u_z = f₃(x, y, z)

where u_x, u_y, and u_z are the x, y and z components of the force U at point (x, y, z).

            u₁ = f₁(x, y, z)

2]        u₂ = f₂(x, y, z)

            u₃ = f₃(x, y, z)

            u₁ = u₁(x, y, z)

3]        u₂ = u₂(x, y, z)

            u₃ = u₃(x, y, z)

            u₁ = u₁(x₁, x₂, x₃)

4]        u₂ = u₂(x₁, x₂, x₃)

            u₃ = u₃(x₁, x₂, x₃)

7]        U = 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:

            u_x = f₁(x, y, z, t)

1]        u_y = f₂(x, y, z, t)

            u_z = f₃(x, y, z, t)

where u_x, u_y, and u_z are the x, y and z components of the force U at point (x, y, z) at time t.

            u₁ = f₁(x, y, z, t)

2]        u₂ = f₂(x, y, z, t)

            u₃ = f₃(x, y, z, t)

            u₁ = u₁(x, y, z, t)

3]        u₂ = u₂(x, y, z, t)

            u₃ = u₃(x, y, z, t)

            u₁ = u₁(x₁, x₂, x₃, t)

4]        u₂ = u₂(x₁, x₂, x₃, t)

            u₃ = u₃(x₁, x₂, x₃, t)

7]        U = 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:

            v_x = f₁(x, y, z, t)

1]        v_y = f₂(x, y, z, t)

            v_z = f₃(x, y, z, t)

where v_x, v_y, and v_z 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)

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:

            a_x = f₁(x, y, z, t)

1]        a_y = f₂(x, y, z, t)

            a_z = f₃(x, y, z, t)

where a_x, a_y, and a_z 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)

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.