400 CHAPTER 22. CALCULUS OF VECTOR FIELDS

This last symbol is important enough that it is given a name, the Laplacian.It is also de-noted by ∆. Thus ∇

2 f = ∆ f . In addition for f a vector field, the symbol f ·∇ is defined asa “differential operator” in the following way.

f ·∇(g)≡ f1 (x)∂g (x)

∂x1+ f2 (x)

∂g (x)

∂x2+ · · ·+ fp (x)

∂g (x)

∂xp.

Thus f ·∇ takes vector fields and makes them into new vector fields.

This definition is in terms of a given coordinate system but later coordinate free defini-tions of the curl and div are presented. For now, everything is defined in terms of a givenCartesian coordinate system. The divergence and curl have profound physical significanceand this will be discussed later. For now it is important to understand their definition interms of coordinates. Be sure you understand that for f a vector field, divf is a scalar fieldmeaning it is a scalar valued function of three variables. For a scalar field f , ∇ f is a vectorfield described earlier. For f a vector field having values in R3,curlf is another vectorfield.

Example 22.1.2 Let f (x) = xyi+(z− y)j+(sin(x)+ z)k. Find divf and curlf .

First the divergence of f is

∂ (xy)∂x

+∂ (z− y)

∂y+

∂ (sin(x)+ z)∂ z

= y+(−1)+1 = y.

Now curlf is obtained by evaluating∣∣∣∣∣∣∣i j k∂

∂x∂

∂y∂

∂ z

xy z− y sin(x)+ z

∣∣∣∣∣∣∣=i

(∂

∂y(sin(x)+ z)− ∂

∂ z(z− y)

)−j

(∂

∂x(sin(x)+ z)− ∂

∂ z(xy)

)+

k

(∂

∂x(z− y)− ∂

∂y(xy)

)=−i− cos(x)j− xk.

22.1.1 Vector IdentitiesThere are many interesting identities which relate the gradient, divergence and curl.

Theorem 22.1.3 Assuming f,g are a C2 vector fields whenever necessary, the followingidentities are valid.

1. ∇ · (∇×f) = 0

2. ∇×∇φ = 0

3. ∇× (∇×f) = ∇(∇ ·f)−∇2f where ∇

2f is a vector field whose ith component is∇

2 fi.

4. ∇ · (f ×g) = g·(∇×f)−f ·(∇×g)