Question

partial differentiation f(x,y)=xy(x^2-y^2)/x^2+y^2), when x^2+y^2 is not equal to 0 and f(0,0)=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