478 CHAPTER 25. CURVILINEAR COORDINATES

25.7 Gradients and DivergenceThe purpose of this section is to express the gradient and the divergence of a vector field ingeneral curvilinear coordinates. As before,

(y1, ...,yn

)will denote the standard coordinates

with respect to the usual basis vectors. Thus

y ≡ ykik, ek (y) = ik = ek (y) .

Let φ : U → R be a differentiable scalar function, sometimes called a “scalar field” inthis subject. Write φ (x) to denote the value of φ at the point whose coordinates are x. Thesame convention is used for a vector field. Thus F (x) is the value of a vector field at thepoint of U determined by the coordinates x. In the standard rectangular coordinates, thegradient is well understood from earlier.

∇φ (y) =∂φ (y)

∂yk ek (y) =∂φ (y)

∂yk ik.

However, the idea is to express the gradient in arbitrary coordinates. Therefore, using thechain rule, if the coordinates of the point of U are given as x,

∇φ (x) = ∇φ (y) =∂φ (x)

∂xr∂xr

∂yk ek (y) =

∂φ (x)

∂xr∂xr

∂yk∂yk

∂xs es (x) =

∂φ (x)

∂xr δrse

s (x) =∂φ (x)

∂xr er (x) .

This shows the covariant components of ∇φ (x) are

(∇φ (x))r =∂φ (x)

∂xr , (25.35)

Formally the same as in rectangular coordinates. To find the contravariant components,“raise the index” in the usual way. Thus

(∇φ (x))r = grk (x)(∇φ (x))k = grk (x)∂φ (x)

∂xk . (25.36)

What about the divergence of a vector field? The divergence of a vector field F definedon U is a scalar field, div(F ) which from calculus is

∂Fk

∂yk (y) = Fk,k (y)

in terms of the usual rectangular coordinates y. The reason the above equation holds inthis case is that ek (y) is a constant and so the Christoffel symbols are zero. We want anexpression for the divergence in arbitrary coordinates. From Theorem 25.6.1,

F i, j (y) = Fr

,s (x)∂xs

∂y j∂yi

∂xr

From 25.27,

=

(∂Fr (x)

∂xs +Fk (x)

{rks

}(x)

)∂xs

∂y j∂yi

∂xr .