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 = 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
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
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
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
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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