Begin2.DVI

(c)

∇^2 A
=

∂^2

∂x^2 (x

(^2) ˆe 1 +xy ˆe 2 +y (^2) ˆe 3 ) + ∂^2
∂y^2 (x
(^2) eˆ 1 +xy ˆe 2 +y (^2) ˆe 3 ) + ∂^2
∂z^2 (x
(^2) ˆe 1 +xy ˆe 2 +y (^2) ˆe 3 )
= 2 ˆe 1 + 2 ˆe 3
(d)
(A
×∇ )φ= (xy ∂φ
∂z
−y^2 ∂φ
∂y
)ˆe 1 + (y^2 ∂φ
∂x
−x^2 ∂φ
∂z
)ˆe 2 + (x^2 ∂φ
∂y
−xy ∂φ
∂x
)ˆe 3 ,

where ∂φ

∂x

= 2xy^2 +yz^2 , ∂φ

∂y

= 2yx^2 +xz^2 , ∂φ

∂z

= 2xyz

and (A
×∇ )φ= (2 x^2 y^2 z− 2 x^2 y^3 −xy^2 z^2 )ˆe 1

+ (2xy^4 +y^3 z^2 − 2 x^3 yz)ˆe 2

+ (2x^4 y+x^3 z^2 − 2 x^2 y^3 −xy^2 z^2 )ˆe 3

Relations Involving the Del Operator

In summary, the following table illustrates a variety of relations involving the

del operator. In these tables the functions f, g are assumed to be differentiable scalar

functions of position and A,
B
are vector functions of position, which are continuous

and differentiable.

The ∇operator and differentiation

1. ∇(f+g) = ∇f+∇g or grad (f+g) = grad f+ grad g

2. ∇·(A
+B
) = ∇·A
+∇·B
or div (A
+B
) = div A
+ div B

3. ∇× (A
+B
) = ∇× A
+∇× B
or curl (A
+B
) = curlA
+ curlB

4. ∇(fA
) = (∇f)·A
+f(∇·A
)

5. ∇× (fA
) = (∇f)×A
+f(∇× A
)

6. ∇· (A
×B
) = B
(∇× A
)−A
(∇× B
)

7. (A
×∇ )f=A
×∇ f

8. For f=f(u)and u=u(x, y, z ), then ∇f=dudf ∇u

9. For f=f(u 1 , u 2 ,... , u n)and ui=ui(x, y, z )for i= 1, 2 ,.. ., n, then

∇f= ∂f

∂u 1

∇u 1 + ∂f

∂u 2

∇u 2 +···+ ∂f

∂u n

∇un

10. ∇× (A
×B
) = ∇(∇·A
)−B
(∇·A
)

11. ∇(A
·B
) = A
×(∇× B
) + B
×(∇× A
) + (B
∇)A
+ (A
∇)B

12. ∇× (∇× A
) = ∇(∇·A
)−∇^2 A

13. ∇·(∇f) = ∇^2 f=

∂^2 f

∂x^2 +

∂^2 f

∂y^2 +

∂^2 f

∂z^2

14. ∇×∇ f) =
0 The curl of a gradient is the zero vector.

15. ∇·(∇× A
) = 0 The divergence of a curl is zero.