Re: Partial derivatives don't give full picture


Kenneth Bull wrote:
Can someone explain to me how derivatives (as functions) for single
variable functions tell the whole story, while partials for
multivariable function do not?

The function

f(x,y) = xy / (x^2 + y^2), if (x,y) is not (0,0)
0, if (x,y) = (0,0)

has partial derivatives at 0 (f_x(0,0) = f_y(0,0) = 0), but the limit
of f(x,y) as (x,y) approaches (0,0) does not exist, so f is not even
continuous.

(If we approach along y = 0, we get a limit of 0; if we approach along
y = x, we get a limit of 1/2.)

For example, sometimes partial derivatives (as functions) don't give a
value for certain inputs, but when using the limit definition of the
partial derivative at the point (input), a value comes up.

Why is this the case with higher dimension functions, but with single
variable functions, the derivative as a function is always right?

I don't know what you mean by "always right", but it is certainly tied
up with the fact you can only approach 0 from two directions, but you
can approach (0,0,...,0) from an infinite number of directions.
(Provided you have at least 2 0's, that is.)

--- Christopher Heckman

.

Relevant Pages

  • Re: Partial derivatives dont give full picture
    ... variable functions tell the whole story, while partials for ... the derivative takes into account all possible ... directions in the domain; ... partial derivatives take into account only two perpendicular ...
    (sci.math)
  • Partial derivatives dont give full picture
    ... Can someone explain to me how derivatives for single ... variable functions tell the whole story, while partials for ... multivariable function do not? ... variable functions, the derivative as a function is always right? ...
    (sci.math)
Quantcast