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A one-dimensional function has a gradient which is defined as the slope of the tangent to the curve at . We wish to extend this idea to cover scalar fields in two and three dimensions.

Consider a two-dimensional scalar field , which is (say) the height of a hill. Let be an element of horizontal distance. Consider , where is the change in height after moving an infinitesimal distance . This quantity is somewhat like the one-dimensional gradient, except that depends on the direction of , as well as its magnitude. In the immediate vicinity of some point , the slope reduces to an inclined plane (see Fig. 16). The largest value of is straight up the slope. For any other direction

 (93)

Let us define a two-dimensional vector, , called the gradient of , whose magnitude is , and whose direction is the direction up the steepest slope. Because of the property, the component of in any direction equals for that direction. [The argument, here, is analogous to that used for vector areas in Sect. 2.3. See, in particular, Eq. (13).]

The component of in the -direction can be obtained by plotting out the profile of at constant , and then finding the slope of the tangent to the curve at given . This quantity is known as the partial derivative of with respect to at constant , and is denoted . Likewise, the gradient of the profile at constant is written . Note that the subscripts denoting constant- and constant- are usually omitted, unless there is any ambiguity. If follows that in component form

 (94)

Now, the equation of the tangent plane at is

 (95)

This has the same local gradients as , so
 (96)

by differentiation of the above. For small and , the function is coincident with the tangent plane. We have
 (97)

but and , so
 (98)

Incidentally, the above equation demonstrates that is a proper vector, since the left-hand side is a scalar, and, according to the properties of the dot product, the right-hand side is also a scalar, provided that and are both proper vectors ( is an obvious vector, because it is directly derived from displacements).

Consider, now, a three-dimensional temperature distribution in (say) a reaction vessel. Let us define , as before, as a vector whose magnitude is , and whose direction is the direction of the maximum gradient. This vector is written in component form

 (99)

Here, is the gradient of the one-dimensional temperature profile at constant and . The change in in going from point to a neighbouring point offset by is
 (100)

In vector form, this becomes
 (101)

Suppose that for some . It follows that

 (102)

So, is perpendicular to . Since along so-called isotherms'' (i.e., contours of the temperature), we conclude that the isotherms (contours) are everywhere perpendicular to (see Fig. 17).

It is, of course, possible to integrate . The line integral from point to point is written

 (103)

This integral is clearly independent of the path taken between and , so must be path independent.

In general, depends on path, but for some special vector fields the integral is path independent. Such fields are called conservative fields. It can be shown that if is a conservative field then for some scalar field . The proof of this is straightforward. Keeping fixed we have

 (104)

where is a well-defined function, due to the path independent nature of the line integral. Consider moving the position of the end point by an infinitesimal amount in the -direction. We have
 (105)

Hence,
 (106)

with analogous relations for the other components of . It follows that
 (107)

In physics, the force due to gravity is a good example of a conservative field. If is a force, then is the work done in traversing some path. If is conservative then

 (108)

where corresponds to the line integral around some closed loop. The fact that zero net work is done in going around a closed loop is equivalent to the conservation of energy (this is why conservative fields are called conservative''). A good example of a non-conservative field is the force due to friction. Clearly, a frictional system loses energy in going around a closed cycle, so .

It is useful to define the vector operator

 (109)

which is usually called the grad or del operator. This operator acts on everything to its right in a expression, until the end of the expression or a closing bracket is reached. For instance,
 (110)

For two scalar fields and ,
 (111)

can be written more succinctly as
 (112)

Suppose that we rotate the basis about the -axis by degrees. By analogy with Eqs. (7)-(9), the old coordinates (, , ) are related to the new ones (, , ) via

 (113) (114) (115)

Now,
 (116)

giving
 (117)

and
 (118)

It can be seen that the differential operator transforms like a proper vector, according to Eqs. (10)-(12). This is another proof that is a good vector.

Next: Divergence Up: Vectors Previous: Volume integrals
Richard Fitzpatrick 2006-02-02