## 3.1 Definition of a Scalar Field

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.