Mathematics of Physics and Engineering

148 Functions of Several Variables

determinant similar to (3.1.27):

curl F^1

hih 2 h 3

hiqx h 2 q 2 h 3 q 3

dqi dq 2 dqz

hxFx h 2 F 2 h 3 F 3

(3.1.46)

EXERCISE 3.1.32. A (a) Verify (3.1.46). (b) Use (3.1.46) in spherical
coordinates to show that curl(g(r) r) = 0 in R^3.
We conclude this section with a brief discussion of INTEGRATION
in curvilinear coordinates. As before, let X be the cartesian coordi-
nates x\,x 2 ,X3 and Q, some other coordinates qi,q 2 ,qz so that Xk =
Xk(qi,q2,qs), k — 1,2,3; we no longer assume that the Q coordinates are
orthogonal. At a non-special point P, consider the three vectors r'fc(0),
k = 1,2,3, defined in (3.1.35), and assume that

»-i(0)-(r' 2 (0)x^(0))^0.

EXERCISE 3.1.33.B Verify that

ri(0)-(r' 2 (0)xr^(0)) =

(3.1.47)

dx\

dqi

dxi

dq 2

dxi

dqz

dX2

dqi

dX2

dq 2

9X2

9qz

9xz

dqi

9X3

9q2

9X3

9q3

(3.1.48)

Hint: we have cartesian coordinates in (3.1.35).

The determinant in (3.1.48) is called the Jacobian, after the Ger-
man mathematician CARL GUSTAV JACOB JACOBI (1804-1851), who,
in 1841, published a paper containing a detailed study of such determi-
nants. The Jacobian (3.1.48) is denoted either by J — J(qi,q2,q3) or by
9{xi,x 2 ,xs)/d(qi,q2,q3); similarly, we either call it the Jacobian of the
system of functions Xi>X2iX3 or the Jacobian of the transformation from
the X coordinates to the Q coordinates. It was one of the results in the
original paper by Jacobi that the change of variables is one-to-one if J ^ 0.

EXERCISE 3.1.34. (a)A We know that, in orthogonal coordinates,