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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 = f(x_{1}, x_{2}, ... , x_{n})

where x_{1}, x_{2}, ... , 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 = f_{1}(x_{1}, x_{2}, ... , x_{n}) i + f_{2}(x_{1}, x_{2}, ... , x_{n}) j + f_{3}(x_{1}, x_{2}, ... , x_{n}) k

or equivalently, in parametric form,

u_{1} = f_{1}(x_{1}, x_{2}, ... , x_{n})

u_{2} = f_{2}(x_{1}, x_{2}, ... , x_{n})

u_{3} = f_{3}(x_{1}, x_{2}, ... , x_{n})

where U = (u_{1}, u_{2}, u_{3}) and x_{1}, x_{2}, ... , 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 = u_{1}(x, y, z) i + u_{2}(x, y, z) j + u_{3}(x, y, z) k

or in parametric form,

u_{1} = u_{1}(x, y, z)

u_{2} = u_{2}(x, y, z)

u_{3} = u_{3}(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 = u_{1}(x, y, z, t) i + u_{2}(x, y, z, t) j + u_{3}(x, y, z, t) k

or in parametric form,

u_{1} = u_{1}(x, y, z, t)

u_{2} = u_{2}(x, y, z, t)

u_{3} = u_{3}(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} = f_{1}(x, y, z)

2] F_{y} = f_{2}(x, y, z)

F_{z} = f_{3}(x, y, z)

F_{1} = u_{1}(x, y, z)

3] F_{2} = u_{2}(x, y, z)

F_{3} = u_{3}(x, y, z)

F_{1} = u_{1}(x_{1}, x_{2}, x_{3})

4] F_{2} = u_{2}(x_{1}, x_{2}, x_{3})_{ }

F_{3} = u_{3}(x_{1}, x_{2}, x_{3})_{ }

7] F = u_{1}(x, y, z) i + u_{2}(x, y, z) j + u_{3}(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.

F_{1} = f_{1}(x, y, z, t)

2] F_{2} = f_{2}(x, y, z, t)

F_{3} = f_{3}(x, y, z, t)

F_{1} = u_{1}(x, y, z, t)

3] F_{2} = u_{2}(x, y, z, t)

F_{3} = u_{3}(x, y, z, t)

F_{1} = u_{1}(x_{1}, x_{2}, x_{3}, t)

4] F_{2} = u_{2}(x_{1}, x_{2}, x_{3}, t)_{ }

F_{3} = u_{3}(x_{1}, x_{2}, x_{3}, t)_{ }

7] F = u_{1}(x, y, z, t) i + u_{2}(x, y, z, t) j + u_{3}(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} = f_{1}(x, y, z, t)

1] v_{y} = f_{2}(x, y, z, t)

v_{z} = f_{3}(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.

v_{1} = f_{1}(x, y, z, t)

2] v_{2} = f_{2}(x, y, z, t)

v_{3} = f_{3}(x, y, z, t)

v_{1} = v_{1}(x, y, z, t)

3] v_{2} = v_{2}(x, y, z, t)

v_{3} = v_{3}(x, y, z, t)

v_{1} = v_{1}(x_{1}, x_{2}, x_{3}, t)

4] v_{2} = v_{2}(x_{1}, x_{2}, x_{3}, t)_{ }

v_{3} = v_{3}(x_{1}, x_{2}, x_{3}, t)_{ }

7] V = v_{1}(x, y, z, t) i + v_{2}(x, y, z, t) j + v_{3}(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} = f_{1}(x, y, z, t)

1] a_{y} = f_{2}(x, y, z, t)

a_{z} = f_{3}(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.

a_{1} = f_{1}(x, y, z, t)

2] a_{2} = f_{2}(x, y, z, t)

a_{3} = f_{3}(x, y, z, t)

a_{1} = a_{1}(x, y, z, t)

3] a_{2} = a_{2}(x, y, z, t)

a_{3} = a_{3}(x, y, z, t)

a_{1} = a_{1}(x_{1}, x_{2}, x_{3}, t)

4] a_{2} = a_{2}(x_{1}, x_{2}, x_{3}, t)_{ }

a_{3} = a_{3}(x_{1}, x_{2}, x_{3}, t)_{ }

7] A = a_{1}(x, y, z, t) i + a_{2}(x, y, z, t) j + a_{3}(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.

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