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3.1 Definition of a Scalar Field
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A scalar field is a name we give to a function defined in some sort
of space. Thus, in ordinary three dimensional space the following are examples
of scalar fields: sin xyz, cos z, x2 + y2 + z2.
A linear field is one of the form ax + by + cz + d for some constants a, b,
c and d.
In one dimension, an ordinary function is said to be differentiable at
a point, if, when plotted against the variable, it looks like a straight line
on a sufficiently small scale around that point.
A field defined in two dimensions is differentiable at the point (x',
y') if its plot agains x and y looks like a plane on a sufficiently small scale
around that point. The same statement is: the field is differentiable
at (x', y') if it looks like a linear field and is approximated by one as
closely as one wants at distances sufficiently close to (x', y'). In this form
differentiability is easy to visualize in any dimension.
A field can have singularities of many kinds:
Example
The straight line that the ordinary function, f (x), looks like at x = x' is called
the tangent line to f at x'.
In two dimensions, the analogous concept is that of the tangent plane.
The plane that f (x, y) resembles at (x', y') is called the tangent plane to f
at (x', y').
A similar, if perhaps less easily visualized concept exists in any dimension;
the (hyper)surface that describes the linear function that f resembles at a point
is the tangent hyperplane there.