Calculus: Analytic Geometry and Calculus, with Vectors

11.1 Elementary partial derivatives 559

us to make any change we please in order of differentiation when u is

differentiated more than once with respect to variablesupon which it

depends.

Problems 11.19

I Let

f(x,y) = x2 + y2

and observe that f.(x,y) = 2x. Then perform every single step required to

show in the most tedious possible way that

limf(x + AX, y) - f(x,y)- 2x.

AX-0 Ax

Then think about the whole business.

2 If

show that

u(x,y,z) = x2 + 2y2 + 3z2,

u.(x,y,z) = 2x uy(x,y,z) = 4y * u.(x,y,z) = 6z

us(1,1,1) = 2, uy(1,1,1) = 4, u,(1,1,1) = 6.

(^3) Supposing that p > 0 and
P2 = x2 + y2, = tan-' yX
find the first partial derivatives of p and 45 with respect to x and y and then use
the formulas x = p cos q5, y = p sin 0 to put the results in the form
ap ap ao sin ¢ a4, cos 0
TX = cos4, ay = sin 0,
ax
= -
P
, ay = P
(^4) Obtain the simplest possible expression for
a2u a2u
axe+ ays
when
4u=x2-y2 (b) u=3x2y-y=
(c) u = log (x2 + y2) (d) u = ex cos y

(e) u=xsiny (f) u=tan-, y

x

(g) u= log (x - a)2 + (y - b)2

_ns.: With one exception, each answer is 0. Remark: The equation

191U^2

axe + aye - 0

and those that appear in the following problems are called partial differential
equations. It is never too early to start learning that the above equation is the
Laplace equation. A function u is said to be harmonic over a region if it satisfies
the Laplace equation and is continuous and has continuous partial derivatives
at each point of the region. Problem 12 gives an example of a function which