Calculus: Analytic Geometry and Calculus, with Vectors

11.2 Increments, chain rule, and gradients 569

summation signs, we can easily extend all of these calculations to cover more
complex situations in which it is a positive integer and
n
Y /,mk
k=1 (x - xk)2 + (y - yk)2 + (Z - Zk)2

There are times when it is not unreasonable to start with a special situation in
which n > 3 and make quite extensive calculations to learn about directional
derivatives, gradients, and equipotential surfaces. An equipotential surface is a
surface consisting of points (x,y,z) such that, for some constant c, Y(x,y,z) = c.
7 Let the temperature it at the point (x,y) in an xy plane be defined by

u=r- sin v.

Modify the procedure of Problem 6 to obtain, by two methods, the directional

derivative of u at (xo,yo) in the direction of the unit vector

(cos a)i + (sin a)j.

Make the results agree.

8 Supposing that

show that

u = Ix + By + Cz,

Vu = .4i + Bj + Ck.

Use this result to show that the equation of the plane tangent to the graphof

the equation

/Ix+By+Cz=D

at a point Po(xo,yo,zo) on the graph is

11(x-xo)+B(y-yo)+C(z-zo) =0

or

9 Supposing that

,4x + By + Cz = D.

x2 Lv2

--

z2

"= a2+b2 `'

Vu = axi+ b',j +zk.

Use this result to show that the equation of the plane tangent tothe graph of the

equation

(3) x,2,+1y 2+Z2a- b2 c

at the point (xo,yo,zo) on the graph is

(4) ZQ°°(x-xo)+o°(y-yo)+ -°(z-zo)=0

or

xa2+

b

+`oti =1.

Remark: In case a = b = c, the graph of (3) is a sphere. Otherwise, the graph

is an ellipsoid; see Figure 6.631 and the accompanying discussion.