Begin2.DVI

Chapter 8

I8-1.

I8-2. (i) gradφ= 4ˆe 1 − 3 ˆe 2

(iii)gradφ= 2xˆe 1 + 2yˆe 2 (ii)

I8-3. (iii)z=x^2 +y^2 paraboloid N~ =− 2 xˆe 1 − 2 yˆe 2 +ˆe 3
3 , 4 , 25

=− 6 ˆe 1 − 8 ˆe 2 +ˆe 3

(iv) z−xy= 0 hyperbolic paraboloidN~=−yˆe 1 −xˆe 2 +ˆe 3

2 , 3 , 6

=− 3 ˆe 1 − 2 ˆe 2 +ˆe 3

I8-4. ∂z
∂x

=y= 0and∂z∂y=x= 0simultaneously at the origin.

I8-5. Normal to sphereN~ 1 = 2xˆe 1 + 2yˆe 2 + 2zˆe 3 and normal to plane is

N~ 2 =ˆe 1 +ˆe 2 +ˆe 3

At the point(3, 2 ,6)one findsN~ 1 = 6ˆe 1 + 4ˆe 2 + 12ˆe 3 andN~ 2 =eˆ 1 +eˆ 2 +ˆe 3

A tangent vector to the curve of intersection is

T~=N~ 1 ×N~ 2 =

∣∣

∣∣

∣∣

ˆe 1 ˆe 2 ˆe 3

6 4 12

1 1 1

∣∣

∣∣

∣∣=−^8 ˆe^1 + 6ˆe^2 + 2ˆe^3 or −T~

Solutions Chapter 8