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DIFFERENTIAL OPERATIONS 31
Keeping higher-order terms in this expansion will not change the results of this deriva- tion. Substituting Equation 2.54 into Equation 2.53 and performing the integration gives
(2.55)
Next the integration along C3, the top section of the loop, is performed. Along this
path, y is held fixed at y = yo + A y , while x varies from xo + A x to x,. Therefore,
L, d f. V = l:Axd x V,. (2.56)
Again, we expand V x ( x ,yo + A y ) to first order with a Taylor series:
Substituting Equation 2.57 into Equation 2.56 and performing the integration gives
Combining Equations 2.55 and 2.58 gives
L , d? * V + 1 , d i. V ~ -31 A x A y. (2.59)
After performing similar integrations along the C, and C, paths, we can combine all the results to obtain
ay ( X 0 , Y O )
The error of Equation 2.60 vanishes as the dimensions of the loop shrink to the infinitesimal, A x ----f 0 and A y + 0. Also, using Equation 2.51, the term in the
brackets on the RHS of Equation 2.60 can be identified as the z component of v X v.
Therefore, we can write
(2.61)
where C is the contour that encloses S and duz = dx d y is a differential area of that
surface. What does all this tell us about the curl? The result in Equation 2.61 can be reorganized as:
32 DIFFERENTIAL AND INTEGRAL OPERATIONS
C
Figure 2.10 Fields with Zero (a) and Nonzero (b) Curl
This says the z component of V x V at a point is the line integral of V on a loop around that point, divided by the area of that loop, in the limit as the loop becomes vanishingly small. Thus, the curl does not directly tell us anything about the circulation on a macroscopic scale. The situations in Figure 2.8 can now be understood. If the “curved” field shown in Figure 2.10(a) has a magnitude that drops off as l/r, exactly enough to compensate for the increase in path length as r increases, then the integral around the closed differential path shown in the figure will be zero. Thus, the curl at that point is also zero. If the magnitude of the “straight” vector field in Figure 2.10(b) varies as indicated by the line density, the integral around the differential path shown cannot be zero and the field will have a nonzero curl. We derived Equation 2.61 in two dimensions and only picked out the z-component of the curl. The generalization of this result to three dimensions, and for any orienta- tion of the differential loop, is straightforward and is given by
In this equation, S is still the area enclosed by the path C. Note that the direction of diF is determined by the direction of C and a right-hand convention.
23. 4 Differential Operator Identities
Subscriptlsummationnotation greatly facilitates the derivationof differentialoperator identities. The relations presented in this section are similar to the vector identities discussed in Chapter 1, except now care must be taken to obey the rules of differential calculus. As with the vector identities, a Cartesian coordinate system is used for the derivations, but the final results are expressed in coordinate-independent vector notation.
Example 2. 1
subscriptlsummationnotation, make the substitution
Consider the operator expression, 7.(7@). To translate this into
