484 CHAPTER 25. CURVILINEAR COORDINATES

= εrst (x)

∂yi

∂xr∂y j

∂xs∂yk

∂xt Fp (x)∂xp

∂y j Gq (x)∂xq

∂yk el (x)∂xl

∂yi

= εrst (x)δ

ps δ

qt δ

lrFp (x)Gq (x)el (x) = ε

l pq (x)Fp (x)Gq (x)el (x) . (25.50)

We summarize these results in the following theorem.

Theorem 25.9.2 Suppose x is a system of curvilinear coordinates in R3 such that

det(

∂yi

∂x j

)> 0.

Letε

i jk (x)≡ εi jk 1√

g(x).

Then the following formulas for curl and cross product hold in this system of coordinates.

curl(F ) = εmqp (x)

∂Fp (x)

∂xq em (x) ,

andF ×G= ε

l pq (x)Fp (x)Gq (x)el (x) .