Question

If u = r = √(x² + y² + z²); then find uxx at (1,0,0)

Answer 1

Answer:

In single-variable calculus we were concerned with functions that map the real numbers R

to R, sometimes called “real functions of one variable”, meaning the “input” is a single real

number and the “output” is likewise a single real number. In the last chapter we considered

functions taking a real number to a vector, which may also be viewed as functions f: R →

R

3

, that is, for each input value we get a position in space. Now we turn to functions

of several variables, meaning several input variables, functions f: R

n → R. We will deal

primarily with n = 2 and to a lesser extent n = 3; in fact many of the techniques we

discuss can be applied to larger values of n as well.

A function f: R

2 → R maps a pair of values (x, y) to a single real number. The three-

dimensional coordinate system we have already used is a convenient way to visualize such

functions: above each point (x, y) in the x-y plane we graph the point (x, y, z), where of

course z = f(x, y).

EXAMPLE 14.1.1 Consider f(x, y) = 3x+ 4y −5. Writing this as z = 3x+ 4y −5 and

then 3x+4y−z = 5 we recognize the equation of a plane. In the form f(x, y) = 3x+4y−5

the emphasis has shifted: we now think of x and y as independent variables and z as a

variable dependent on them, but the geometry is unchanged.

EXAMPLE 14.1.2 We have seen that x

2 + y

2 + z

2 = 4 represents a sphere of radius 2.

We cannot write this in the form f(x, y), since for each x and y in the disk x

2+y

2 < 4 there

are two corresponding points on the sphere. As with the equation of a circle, we can resolve

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