Proving differentiability in two dimensions

[quietrain]

Homework Statement

proof at 0,0 g(x,y) is differentiable

Homework Equations

notes says i have to write in the form
f_x(0,0)$$\Delta$$x + f_y(0,0)$$\Delta$$y + E₁$$\Delta$$x + E₂$$\Delta$$y

The Attempt at a Solution

i compute f_x(0,0) = 0
and f_y(0,0) = 0

but what's the E talking about?

what am i trying to do when i express the function in that form? and how does that show that it is differentiable? thanks!

 

[HallsofIvy]

The "E" is the non-linear partm, with one "x" or "y" factored out. For example, if the problem were $f(x,y)= x^2+ y^2+ x+ y$ then $f_x(0, 0)= 1$ and $f_y(0, 0)= 1$ so the "$f_x\Delta x+ f_y\Delta y$" part is just $\Delta x+ \Delta y$.

Now $F(\Delta x, \Delta y)= (\Delta x)^2+ (\Delta y)^2+ \Delta x+ \Delta y$ so that the "$E_1\Delta x+ E_2\Delta x$" is $(\Delta x)^2+ (\Delta y)^2$ which means that $E_1= \Delta x$ and $E_2= \Delta y$.

 

[quietrain]

erm ok but how does all those show that the function is differentiable at a point? you mean as long as i can manipulate the function g(x,y) into the form of
f_x(0,0)$$\Delta$$x + f_y(0,0)$$\Delta$$y + E₁$$\Delta$$x + E₂$$\Delta$$y
then i have shown it is differentiable at 0,0?

but the form of g(x,y) doesn't look kind :(