What are the limitations of directional derivatives at (0,0)?

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In summary, the conversation is discussing the directional derivatives of f(x,y) at (0,0) in directions a\mathbf{i} + b\mathbf{j}, where a and b are nonzero. The limit of the directional derivative is found to be ab/sqrt(a^2 + b^2), but it is mentioned that there might be an issue in showing that this limit exists. The idea of using polar coordinates is suggested as a possible solution.
  • #1
jdstokes
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Let [itex]f(x,y) = \begin{array}{cc}
\frac{xy}{\sqrt{x^2 + y^2}} &, (x,y) \neq(0,0) \\
0 & ,(x,y) = (0,0) \\
\end{array}[/itex]

Show that the directional derivatives at (0,0) in directions [itex]a\mathbf{i} + b\mathbf{j}[/itex] with [itex]a\neq 0[/itex] and [itex]b\neq 0[/itex], do not exist.

Let [itex]\mathbf{u} = a\mathbf{i} + b\mathbf{j}[/itex]

[itex]
\begin{align*}
D_{\mathbf{u}}f(0,0) & = \lim_{h\rightarrow 0}\frac{f(0+ ha,0 + hb) - f(0,0)}{h}
& = \lim_{h \rightarrow 0} \frac{\frac{h^2ab}{\sqrt{h^2a^2 + h^2b^2}}}{h}
= \frac{ab}{\sqrt{a^2 + b^2}}
\end{align*}
[/itex]

which if I'm not mistaken, exists. How do I show that this doesn't exist? Also, in order to show that f(x,y) is everywhere continuous, will it suffice to say that xy/sqrt(x^2 + y^2) is continuous when (x,y) != 0 and that the limit of xy/sqrt(x^2 + y^2) as (x,y) tends to (0,0) along the x-axis is 0?
 
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  • #2
Anyone have any ideas? I'm really confused about this.
 
  • #3
I'm not sure about this, but maybe they want you to take the directional derivative the usual way (dot the gradient into the direction vector), and then take the limit as (x,y)->0. I tried this, and it seemed like I got two different answers depending on how I approached 0. I think this would mean the derivative doesn't exist, even though both answers were finite. If you want to try this, I used polar coordinates and I think it made it easier.
 
  • #4
[itex]
\lim_{h \rightarrow 0} \frac{\frac{h^2ab}{\sqrt{h^2a^2 + h^2b^2}}}{h}
= \frac{ab}{\sqrt{a^2 + b^2}}
[/itex]

You sure about that?
 

What is a directional derivative?

A directional derivative is a measure of the instantaneous rate of change of a function in a particular direction. It represents the slope of the function in a specific direction at a given point.

How is a directional derivative calculated?

The directional derivative is calculated using the gradient of the function, which is a vector that points in the direction of the steepest increase of the function at a given point. The directional derivative is then found by taking the dot product of the gradient vector and a unit vector in the desired direction.

What is the significance of directional derivatives?

Directional derivatives are important in understanding the behavior and properties of a function. They can be used to determine the direction of steepest ascent or descent of a function, as well as the rate of change in a particular direction.

What is the relationship between directional derivatives and partial derivatives?

Directional derivatives are a generalization of partial derivatives. While partial derivatives measure the rate of change of a function with respect to one variable, directional derivatives can measure the rate of change in any direction. In fact, the partial derivatives in each direction can be combined to find the directional derivative in any given direction.

How are directional derivatives used in real-world applications?

Directional derivatives have many applications in fields such as physics, engineering, and economics. They can be used to optimize processes and systems by finding the direction of maximum increase or decrease, as well as in predicting the behavior of physical systems and financial markets.

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