5. Vector Calculus

The GRADIENT and RATES of CHANGE lS/s.'Hfil-r- ~Gl' C~ ,.;;k ~r3) let (x, y, z)and its first partial derivatives be continuous in some sphere about Po and suppose that V (Po) *O. Then

i) At Po, (x, y, z) has its maximum rate of change in the direction of V (Po), This maximum rate of change is II V ( Po) II .

ii) At Po, (x, y, z) has its minimum rate of change in the direction of - V( Po). This minimum rate of change is -II V ( Po) II .

ExampleS: let (x,y,z)=2xz+eYz2. Find the direction from (2, 1, 1) in which (x, y, z)has its greatest and least rates of change, and find the values of these rates of change.

Then

V(2,1,1)= 2i +ej+(4+ 2e)k

This vector points in the direction in which (x, y, z) has its greatestrate of change at (2,1,1). This maximum rate of change is the magnitude of this gradient vector, or k0ky

I.A',[ ~ = ,j20+16e+5e2 ~ p\llVI

The vector - 2i - ej - (4 +2e)k is in the direction in which (x, y, z) has its least rate of change at

(2,1, I), and this minimum rate ofcbange is .j20+16e + 5e' .~1V ~ k G~

Advanced Engineering Mathematics [ENG4200] Page 7 of 31

1 . r 5. Vector Calculus

Recall, the VECTOR DIFFERENTIAL OPERATOR, written V , called "del" :

n d. d. d d. d. d v = -1+-J+-k = -1+-J+-k dx dy dZ dx dy dZ

In formally, del times= V~ =gradientof~= (~i+ d~ j+ d~k)dx dy dz

. (dV dV dV) t:tdel dot = V V = divergence of V = ~+_2+_3 V f.f vevfcr:' (IX dy dz

-tV'LA-fl (l\-\) del cross = VxV = curl of V =(dV3 _ dV2)i+ (d~ _dV;)j + (dV2 _ d~)k

dy dz dz dx dx dy

Definition 5.4: Curl of a Gradient

Vx(V~) = curl (V~)=O. Thecurlofthegradientof~iszero.

Definition 5.5: Divergence of a Curl

V (V x V) = O. The divergence of the curl of V is zero.

( cJ 1

1 r . 5. Vector Calculus

Laplace's Equation

Laplace's equation in three dimensions it is

The symbol V 2u is call the laplacian of u, and is defined by

Using this notation, laplace's equation is

In two variable x and y, we simply omit the z dependence. In the plane, then the Laplacian of u(x, y) is

Advanced Engineering Mathematics [ENG4200] Page 11 of 31