Calculus: Analytic Geometry and Calculus, with Vectors

11.1 Elementary partial derivatives 555

books and on blackboards and in notebooks and on scratch pads. Wher-
ever we find aw/aq, we automatically know that w is a function of q
and some other variables, we know that fixed values have been assigned
to all variables except q, and we know that aw/aq stands for the result
of differentiating the resulting function of q with respect to q. If w
is measured in gees and q is measured in haws, then aw/aq is measured
in gees per haw. When we see the formula

(11.13) u = x2 + y2 + e°x sin by or u(x,y) = x2 + y2 + e°S sin by,

we can compute partial derivatives with respect to x by supposing that

all variables except x are assigned fixed values, so that they are to be

regarded as constants when we differentiate with respect to x to obtain

(11.131) ax = 2x + ae°x sin by or u,(x,y) = 2x + ae°y sin by.

The second one of these formulas involves the standard subscript nota-

tion for partial derivatives. Similarly,

(11.132) e = 2y + be' cos by or 2y + be°= cos by.

It should now be apparent that calculating these partial derivatives is

equivalent to evaluating the limits in

(11.14) uz(x,Y) = e

o

u(x + Ox, Y) - u(x,Y)l

Lx

uv(x'Y) = lim

u(x, y + AY) - u(x,Y)

AY-0 Ay

Taking the partial derivatives with respect to x of the members of

(11.131) and (11.132) gives

(11.141)

azu

= 2 + a2e°S sin by or u=z(x,y) = 2 + a"e°Z sin by

and

(11.142)

axzay

= abe- cos by or ud=(x,y) = abet cos by.

Taking partial derivatives with respect to y of themembers of (11.131)

and (11.132) gives

z

(11.143) as ax =abe6z cos by or abe°= cos by

and

492U z z

(e°= sin by.11.144) aye = 2 - b ea sin by or u (x,y) =2 - b2

In the above formulas, we have used the formulas

z

(11.145) ay (ax =ay ax (uz),(x,Y)