Mathematics of Physics and Engineering

146 Functions of Several Variables

the points with the Q coordinates {c\ ± (a/2),C2 ± (a/2),c 3 ± (a/2)). The
volume of this box is approximately a^3 |r' 1 (0) • (r 2 (0) x r 3 (0))| = a^3 hih 2 h 3 ;
draw a picture and see equations (3.1.35). By (3.1.36) and (3.1.38), The
basis vector qk is normal to two of the faces so that the area of each of
those faces is approximately a^2 hmhn, where k ^ m ^ n. Consider a
continuously differentiable vector field F written in the Q coordinates as
F(qi,Q2,q3) = Fi(qi,q 2 ,q 3 )q 1 + ^2(91,92,93)92 +-^3(91,92,93) <Z 3 - Tnen
the flux of this vector field through the pair of faces with the normal vector
q 1 is approximately

a^2 (h 2 (ci + (a/2),C2,c 3 )h 3 (ci + (a/2),c 2 ,c 3 )F 1 (a + (a/2),c 2 ,c 3 )

-h 2 (ci - (a/2),C2,c 3 )/i 3 (c 1 - (a/2),c 2 ,c 3 )Fi(ci - (a/2),c 2 ,c 3 )J

3 d(/i 2 /t 3 Fi)

09l

with the approximation getting better as a —> 0. Similar expressions hold

for the fluxes across the other two pairs of faces. Summing up all the fluxes,

dividing by the volume of the box, and passing to the limit a —> 0, we get

the formula for the divergence of F in Q coordinates:

ay*^1 = nhr (d{htFl) + d{htF2) + d{htF*]) • (3-1-43)

hih 2 h 3 V OQi 9q 2 093 /

EXERCISE 3.1.30.B (a) Verify (3.1.43)- (b) Verify that, in cylindrical coor-

dinates,

divF=i(^) + ^ + ^}

r \ dr 89 dz

(c) Verify that, in spherical coordinates,

oiv r = —z—;^1 (d{r I 1 — h^2 smyFx) d{rF^2 ) , d(rsin^F^3 )

r^2 siii(p \ dr 09 dtp

Recall that the Laplacian V^2 / of a scalar field / is defined in every

coordinate system as V^2 / = div(grad/). Let / = 7(91,92,93) be a scalar

field defined in an orthogonal coordinate system Q.

EXERCISE 3.1.31. (a)B Using (3.142) and (3.143) verify that

v2/= 1 / d (h 2 h 3 df_ \ + _9_/Mi3 df\ + _d_fhih2 3/\\

hih 2 h 3 \dqi\ hi dqi) dq 2 \ h 2 dq 2 ) dq 3 \ h 3 dq 3 ))'