Scalar and vector point functions. Scalar and vector fields

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, ... , 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 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:

F_x = F_x(x, y, z)

1] F_y = F_y(x, y, z)

F_z = F_z(x, y, z)

where F_x, F_y, and F_z are the x, y and z components of the force F at point (x, y, z).

F_x = f₁(x, y, z)

2] F_y = f₂(x, y, z)

F_z = 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)

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:

F_x = F_x(x, y, z, t)

1] F_y = F_y(x, y, z, t)

F_z = F_z(x, y, z, t)

where F_x, F_y, and F_z 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)

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:

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.

References.

James and James. Mathematics Dictionary.

Murray R. Spiegel. Vector Analysis.

Angus E. Tayor. Advanced Calculus.