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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(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

F = 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,

where 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

or equivalently,

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

or equivalently,

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

or equivalently,

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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