Calculus: Analytic Geometry and Calculus, with Vectors

11.1 Elementary partial derivatives 561

is called the divergence of V. Finally, the vector function V X V defined by

i j k

(4)

V X V =

a a

ax ay az

P Q R

is called the curl of V. The expanded form of (4) is

(5)

aR_aQl aP ORll (Q_aP

VX V =Cay az/1+Caz ^ax/1+\ax ay)k.

The inverted delta is called "del," and we shall hear more about the formula

(6) V=a i+a,j+a k.

Meanwhile, use the definitions to show that

(7) F}

82F

V- (VF) =

a2 a2F

axe aye az2

i

a

j

a

k

a

(8) VX (VF) = ax ay az = 0.

OF

ax

OF

ay

a1

az

10 Start with the formulas

V=ad ayj+k

F(x,y,z) = x2 + y2 + z2

V(x,y,z) = xi + yj + zk

and calculate all of the following things that are meaningful:

(a) VF (b) V X F

(d) VV (e) V V

X

V (V XV) (1)

(^11) Letting u be the thoroughly reasonable function having the value yp sin 2¢
or p sin ¢ cos 0 at the point having polar coordinates p and ¢, show that, in
rectangular coordinates,
() (^1) U( 0 , 0 ) = 0 , u(x,Y) 22 + 2 ,40)
(x2 + y2)3s
(x y^5 -.
Show that
au y$ au x2
(2) '

TX (x2 + y.)% ay (x 2 + y 2 ) 74 (x2 + Y2 0 0)
Show that
(3) lira uz(O,y) = 1, lim u:(O,y) _ -1-
V-O+ u-+0-
Show that u=(x,0) = 0 for each x and that uy(O,y) = 0 for each y.