SOME OTHER EXPRESSIONS INVOLVING ∇

7. SOME OTHER EXPRESSIONS INVOLVING ∇

If we write ∇φ as [i(∂/∂x) + j(∂/∂y) + k(∂/∂z)]φ, we can then call the bracket ∇. By itself ∇ has no meaning (just as d/dx alone has no meaning; we must put some function after it to be differentiated). However, it is useful to use ∇ much as we use d/dx to indicate a certain operation.

We call ∇ a vector operator and write

It is more complicated than d/dx (which is a scalar operator ) because ∇ has vector properties too.

So far we have considered ∇φ where φ is a scalar; we next want to consider whether ∇ can operate on a vector.

Suppose V(x, y, z) is a vector function, that is, the three components V x ,V y ,V z of V are functions of x, y, z:

V(x, y, z) = iV x (x, y, z) + jV y (x, y, z) + kV z (x, y, z). (The subscripts mean components, not partial derivatives.) Physically, V represents

a vector field (for example, the electric field about a point charge). At each point of space there is a vector V, but the magnitude and direction of V may vary from point to point. We can form two useful combinations of ∇ and V. We define the divergence of V, abbreviated div V or ∇ · V, by (7.2):

We define the curl of V, written ∇ × V, by (7.3):

V = curl V

You should study these expressions to see how we are using ∇ as “almost” a vec- tor. The definitions of divergence and curl are the partial derivative expressions, of course. However, the similarity of the formulas (7.2) and (7.3) to those for A · B and A × B helps us to remember ∇ · V and ∇ × V. But you must remember to put the partial derivative “components” of ∇ before the components of V in each

Section 7 Some Other Expressions Involving ∇ 297

term [for example, in evaluating the determinant in (7.3)]. Note that ∇ · V is a scalar and ∇ × V is a vector (compare A · B and A × B). We shall discuss later the meaning and some of the applications of the divergence and the curl of a vector function.

The quantity ∇φ in (6.3) is a vector function; we can then let V = ∇φ in (7.2) and find ∇ · ∇φ = div grad φ. This is a very important expression called the Laplacian of φ; it is usually written as ∇ 2 φ. From (6.3) and (7.2), we have

∇ φ = ∇ · ∇φ = div grad φ =

∂x ∂x

∂y ∂y ∂z ∂z

(the Laplacian).

The Laplacian is part of several important equations in mathematical physics:

Laplace’s equation.

wave equation.

∂t 2

diffusion, heat conduction, Schr¨ odinger equation.

a 2 ∂t These equations arise in numerous problems in heat, hydrodynamics, electricity

and magnetism, aerodynamics, elasticity, optics, etc.; we shall discuss solving such equations in Chapter 13.

There are many other more complicated expressions involving ∇ and one or more scalar or vector functions, which arise in various applications of vector analysis. For reference we list a table of such expressions at the end of the chapter (page 339). Notice that these are of two kinds: (1) expressions involving two applications of ∇

such as ∇ · ∇φ = ∇ 2 φ; (2) combinations of ∇ with two functions (vectors or scalars) such as ∇ × (φV). We can verify these expressions simply by writing out

components. However, it is usually simpler to use the same formulas we would use if ∇ were an ordinary vector, being careful to remember that ∇ is also a differential operator.

Example 1. Evaluate ∇ × (∇ × V). We use (3.8) for A × (B × C) being careful to write both ∇’s before the vector function V which they must differentiate. Then we get

∇× (∇ × V) = ∇(∇ · V) − (∇ · ∇)V

= ∇(∇ · V) − ∇ 2 V.

This is a vector as it should be; the Laplacian of a vector, ∇ 2

V, simply means a

vector whose components are ∇ 2 V x

,∇ 2 V y

,∇ 2 V z .

Example 2. Find ∇ · (φV), where φ is a scalar function and V is a vector function. Here we must differentiate a product, so our result will contain two terms. We could write these as

∇· (φV) = ∇ φ · (φV) + ∇ V · (φV),

298 Vector Analysis Chapter 6

where the subscripts on ∇ indicate which function is to be differentiated. Since φ is a scalar, it can be moved past the dot. Then

∇ φ · (φV) = (∇ φ φ) · V = V · (∇φ),

where we have removed the subscript in the last step since V no longer appears after ∇. Actually you may see in books (∇φ) · V meaning that only the φ is to be differentiated, but it is clearer to write it as V · (∇φ). [Be careful with (∇φ) × V, however; assuming that this means that only φ is to be differentiated, the clear way to write it is −V × (∇φ); note the minus sign.] In the second term of (7.5), φ is a scalar and is not differentiated; thus it is just like a constant and we can write this term as φ(∇ · V). Collecting our results, we have

∇· (φV) = V · ∇φ + φ(∇ · V).

In Chapter 10, Section 9, we will derive the formulas for div V = ∇ · V and ∇ 2 f in cylindrical and spherical coordinates. However, it is useful to have the results for reference, so we state them here. Actually, these can be done as partial differentiation problems (see Chapter 4, Section 11), but the algebra is messy.

In cylindrical coordinates (or polar by omitting the z term):

1 ∂ 2 f ∂ 2 f (7.8)

∂f

∇ f=

r 2 ∂θ 2 ∂z 2 In spherical coordinates:

PROBLEMS, SECTION 7

The purpose in doing the following simple problems is to become familiar with the formulas we have discussed. So a good study method is to do them by hand and then check your results by computer. Compute the divergence and the curl of each of the following vector fields.

6. V=x 2 yi + y 2 xj + xyzk 7. V = x sin y i + cos y j + xyk

5. V=x 2 i+y 2 j+z 2 k

8. V = sinh z i + 2y j + x cosh z k

Section 8 Line Integrals 299

Calculate the Laplacian ∇ 2 of each of the following scalar fields. 9. x 3 2 +y − 3xy 3

10. ln(x 2 +y 2 )

11. − px 2 −y 2 12. (x + y) 1

13. xy(x 2 +y 2 2 − 5z − ) 14. (x 2 +y 2 +z 2 ) 1/2

16. ln(x 2 +y 2 +z 2 ) 17. Verify formulas (b), (c), (d), (g), (h), (i), (j), (k) of the table of vector identities

15. xyz(x 2 − 2y 2 +z 2 )

at the end of the chapter. Hint for (j): Start by expanding the two triple vector products on the right.

For r = xi + y j + zk, evaluate