Multivariable Derivative Error

[Kitty Kat]

Homework Statement

f(x,y) =
(xy) / (x² + y⁴), when (x, y) ≠ (0,0)
0, when (x,y) = (0,0)

Homework Equations

Explicitly show that f(x,y) does not satisfy
lim h -> 0 [ E(v,h) / ||h|| ] = 0 when v = 0
(h, v, and 0 are all vectors; I'm not sure how to put a hat on them)

The Attempt at a Solution

I have no idea how to actually tackle this problem using multivariable calculus methods so I just compared the slopes instead (not sure if this is a valid approach either).
Further more I don't really get the concept of E(v,h). All my professor said is that it should approach 0 if the function is differentiable. I'd really appreciate an explanation of E(v,h) please (:

f(h) = f(0,0) + (df/dx(0,0) * x) + (df/dy(0,0) * y) + E(h)
f(h) = 0 + 0x + 0y + E(h)
f(h) = E(h)

Let f(x,y) = f(h,h)
f(h,h) = h / (1+h²)
f'(h) = (-h² + 1) / (1+h²)²

||h|| = 2h²
f'(||h||) = 4h

lim h-> 0 [ f'(h) / f'(||h||) ] = ∞

 

[SammyS]

Homework Statement

f(x,y) =
(xy) / (x² + y⁴), when (x, y) ≠ (0,0)
0, when (x,y) = (0,0)

Homework Equations

Explicitly show that f(x,y) does not satisfy
lim h -> 0 [ E(v,h) / ||h|| ] = 0 when v = 0
(h, v, and 0 are all vectors; I'm not sure how to put a hat on them)

The Attempt at a Solution

I have no idea how to actually tackle this problem using multivariable calculus methods so I just compared the slopes instead (not sure if this is a valid approach either).
Further more I don't really get the concept of E(v,h). All my professor said is that it should approach 0 if the function is differentiable. I'd really appreciate an explanation of E(v,h) please (:

f(h) = f(0,0) + (df/dx(0,0) * x) + (df/dy(0,0) * y) + E(h)
f(h) = 0 + 0x + 0y + E(h)
f(h) = E(h)

Let f(x,y) = f(h,h)
f(h,h) = h / (1+h²)
f'(h) = (-h² + 1) / (1+h²)²

||h|| = 2h²
f'(||h||) = 4h

lim h-> 0 [ f'(h) / f'(||h||) ] = ∞

Do you have a definition of E(v,h)?

 

[Kitty Kat]

Do you have a definition of E(v,h)?

Not quite exactly. Here's a picture of my notes since I don't know how to type it here with proper notation.
Link: https://dl.pushbulletusercontent.com/wwTIGu5Cjgt9OlhFWP6geOc1nuuUgJJX/20151024_164749.jpg

 

[HallsofIvy]

I get a virus warning when I try to open that.

 

[Kitty Kat]

I get a virus warning when I try to open that.

umm that's weird, let me mirror it..
Try https://gyazo.com/875b7ec82e66e0333e383a73e63a077c

 

[Ray Vickson]

Homework Statement

f(x,y) =
(xy) / (x² + y⁴), when (x, y) ≠ (0,0)
0, when (x,y) = (0,0)

Homework Equations

Explicitly show that f(x,y) does not satisfy
lim h -> 0 [ E(v,h) / ||h|| ] = 0 when v = 0
(h, v, and 0 are all vectors; I'm not sure how to put a hat on them)

The Attempt at a Solution

I have no idea how to actually tackle this problem using multivariable calculus methods so I just compared the slopes instead (not sure if this is a valid approach either).
Further more I don't really get the concept of E(v,h). All my professor said is that it should approach 0 if the function is differentiable. I'd really appreciate an explanation of E(v,h) please (:

f(h) = f(0,0) + (df/dx(0,0) * x) + (df/dy(0,0) * y) + E(h)
f(h) = 0 + 0x + 0y + E(h)
f(h) = E(h)

Let f(x,y) = f(h,h)
f(h,h) = h / (1+h²)
f'(h) = (-h² + 1) / (1+h²)²

||h|| = 2h²
f'(||h||) = 4h

lim h-> 0 [ f'(h) / f'(||h||) ] = ∞

Some of your equations make no sense, or else are not true for your current function. For example, your "formula" for $f(h)$ has only $h$ on one side and $x,y$ on the other; that does not mean anything. Besides that, even if we were to correct the formula, it might be false for your function, because it makes some assumptions about the nature of the function in a neighborhood of $(0,0)$ that may not hold for your $f(x,y)$.

Basically, if we define the directional derivative of f at the point $(x_0,y_0)$ in the direction $(p,q)$ as
$$D_{(p,q)} f(x_0,y_0) = \lim_{t \to 0} \frac{f(x_0 + t p, y_0 + t q) - f(x_0,y_0)}{t},$$
then, in order for your result about $E(h)$ to hold, it is necessary (but perhaps not sufficient) that $D_{(p,q)} f(x_0,y_0)$ be linear in $(p,q)$; that is, it would be necessary to have $D_{(p,q)} f(x_0,y_0) = a p + b q$ for some real numbers $a$ and $b$. Is that true at $(x_0,y_0) = (0,0)$ in your current example?

 

[vela]

Furthermore, I don't really get the concept of E(v,h). All my professor said is that it should approach 0 if the function is differentiable. I'd really appreciate an explanation of E(v,h) please (:

Read this page: http://www.math.ku.edu/~lerner/m291F08/Chapter9.pdf