Calculus: Analytic Geometry and Calculus, with Vectors

568 Partial derivatives

can we estimate the error resulting from using f(xo,yo,zo) instead of f(x,y,z)?

11ns.: Put

x=xo, Y=Yo, z=zo, Ax=x - xo, AY=Y - Yo, Az=z - zo

in formulas from (11.21) to (11.223). Remark: Very often we do not want to

hunt up books and copy formulas from them. See the next problem.

2 Remember the following modus operandi because it is useful when properly

used. As an approximation to the number Du defined by

(1) Au = u(x + dx, y + dy, z + dz) - u(x,y,z)

use the number du defined by

au au au

(2)

du=axdx+aydy+azdz.

Remember that (2) can, in appropriate circumstances, be obtained by differ-

entiating with respect tot by the chain rule (Theorem 11.24) and multiplying

by dt. Note the similarity between this modus operandi and the one involving

(3.96)

3 Supposing that y = p sin 0, derive the formula

Idyl =< Ip cos 01 Id4)I + Isin 4)I Idpl

which gives information about the error in y resulting from use of erroneous values

of p and 4).

4 Supposing that y = p cos 4), where p and 0 are functions of t, find a

formula for dy/dt in two different ways. First use partial derivatives and the

chain rule. Then use ordinary (not partial) derivatives and the rule for differ-

entiating products of functions of t. Make the results agree.

5 Formulate and solve another problem more or less like the preceding one.

(^6) Supposing that .1, B, C are constants for which .42 + B2 + C2 = 1, find
the directional derivative of the function (or potential function)
(1) Y =
1
'V (x - x1)2 + (y - Y1)2 + (z - zj)2
at the point (xo,yo,zo) in the direction of the vector
(2) D = 1i + Bj + Ck
by two different (or superficially different) methods. In the first place, put
(3) x = xo + -41, y = yo + Bt, z = zo + Ct
in (1) and find dY/dt by differentiating with respect to t without use of partial
derivatives. Then put t = 0. In the second place, show that
OP= - (x-x1)i.+/(-y-yl)j+(z-z1)k
[(x - xl)" + (y - Y02 + (z - z1)2],>
and calculate the scalar product Then put x = xo, y = Yo, z = zo.
Make the results agree. Remark: It is worthwhile to observe that, by introducing