482 CHAPTER 25. CURVILINEAR COORDINATES

25.9 Curl and Cross ProductsIn this section is the curl and cross product in general curvilinear coordinates in R3. Wewill always assume that for x a set of curvilinear coordinates,

det(

∂yi

∂x j

)> 0 (25.45)

Where the yi are the usual coordinates in which ek (y) = ik.

Theorem 25.9.1 Let 25.45 hold. Then

det(

∂yi

∂x j

)=√

g(x) (25.46)

and

det(

∂xi

∂y j

)=

1√g(x)

. (25.47)

Proof:

ei (x) =∂yk

∂xi ik

and so

gi j (x) =∂yk

∂xi ik ·∂yl

∂x j il =∂yk

∂xi∂yk

∂x j .

Therefore, g = det(gi j (x)) =(

det(

∂yk

∂xi

))2. By 25.45,

√g = det

(∂yk

∂xi

)as claimed. Now

∂yk

∂xi∂xi

∂yr = δkr

and so

det(

∂xi

∂yr

)=

1√g(x)

.

This proves the theorem.To get the curl and cross product in curvilinear coordinates, let ε i jk be the usual permu-

tation symbol. Thus,ε

123 = 1

and when any two indices in ε i jk are switched, the sign changes. Thus

ε132 =−1,ε312 = 1, etc.

Now defineε

i jk (x)≡ εi jk 1√

g(x).

Then for x and z satisfying 25.45,

εi jk (x)

∂ zr

∂xi∂ zs

∂x j∂ zt

∂xk = εi jk det

(∂xp

∂yq

)∂ zr

∂xi∂ zs

∂x j∂ zt

∂xk