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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) height above sea-level in a hilly region. 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 Figure A.111. The largest value of is straight up the slope. It is easily shown that for any other direction

 (1346)

where is the angle shown in Figure A.111. Let us define a two-dimensional vector, , called the gradient of , whose magnitude is , and whose direction is the direction of steepest ascent. The variation exhibited in the above expression ensures that the component of in any direction is equal to for that direction.

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

 (1347)

Now, the equation of the tangent plane at is

 (1348)

This has the same local gradients as , so
 (1349)

by differentiation of the above. For small and , the function is coincident with the tangent plane, so
 (1350)

But, and , so
 (1351)

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

 (1352)

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
 (1353)

In vector form, this becomes
 (1354)

Suppose that for some . It follows that
 (1355)

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 Figure A.112.

It is, of course, possible to integrate . For instance, the line integral of between points and is written

 (1356)

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

Consider a vector field . In general, the line integral depends on the path taken between the end points, 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

 (1357)

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
 (1358)

Hence,
 (1359)

with analogous relations for the other components of . It follows that
 (1360)

In Newtonian dynamics, the force due to gravity is a good example of a conservative field. Now, if is a force-field then is the work done in traversing some path. If is conservative then

 (1361)

where corresponds to the line integral around a closed loop. The fact that zero net work is done in going around a closed loop is equivalent to the conservation of energy (which 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 .

Next: Grad Operator Up: Vector Algebra and Vector Previous: Volume Integrals
Richard Fitzpatrick 2011-03-31