# Relationship Between Total Derivatives and Directional Derivatives .... ....

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In summary, Peter is struggling to understand the concept of a function having a finite directional derivative but not being continuous at a particular point. He is looking for help understanding this concept.
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MHB
I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.4: The Total Derivative ... ...I need help in order to fully understand Theorem 12.3, Section 12.4 ...Theorem 12.3 (including its proof) reads as follows:
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Regarding the above Theorem, I am finding it difficult to understand how, when $$\displaystyle T_c(u) = f'(c;u)$$, that a function can have a finite directional derivative $$\displaystyle f'(c;u)$$ for every $$\displaystyle u$$ but may fail to be continuous at $$\displaystyle c$$ ... whereas! ... for a total derivative ... when a function has a total derivative it is continuous ... YET! ... $$\displaystyle T_c(u) = f'(c;u)$$ ...Can someone please explain what is going on ...
Help will be appreciated ...

Peter
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It may help MHB readers of the above post to have access to Section 12.4 on the total derivative ... so I am providing access to the same ... as follows...
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It may also help MHB readers of the above post to have access to Section 12.2 on the directional derivative ... so I am providing access to the same ... as follows...

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Hope that helps ...

Peter

#### Attachments

• Apostol - Theorem 12.3 .png
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• Apostol - 1 - Section 12.4 ... PART 1 .png
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• Apostol - 2 - Section 12.4 ... PART 2 ... .png
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• Apostol - 1 - Section 12.2 ... PART 1 ... .png
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• Apostol - 2 - Section 12.2 ... PART 2 .png
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Peter said:
I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.4: The Total Derivative ... ...I need help in order to fully understand Theorem 12.3, Section 12.4 ...Theorem 12.3 (including its proof) reads as follows:Regarding the above Theorem, I am finding it difficult to understand how, when $$\displaystyle T_c(u) = f'(c;u)$$, that a function can have a finite directional derivative $$\displaystyle f'(c;u)$$ for every $$\displaystyle u$$ but may fail to be continuous at $$\displaystyle c$$ ... whereas! ... for a total derivative ... when a function has a total derivative it is continuous ... YET! ... $$\displaystyle T_c(u) = f'(c;u)$$ ...Can someone please explain what is going on …
Consider the function f(x, y)= 0 if xy= 0, f(x,y)= 1 otherwise. xy= 0 if and only if x= 0 or y= 0. Think of this as the plane z= f(x,y)= 1 with (x, 0, 1) and (0, y, 1), above x and y axes, lowered down to z= 0. At any point on the x-axis the derivative in the x-direction exists (and is 0). At any point on the y-axis the derivative in the y-direction exists and is 0. At the origin, the derivative in both x and y directions exist. But the function is not continuous on the axes.

Peter said:
I am finding it difficult to understand how ... a function can have a finite directional derivative $$\displaystyle f'(c;u)$$ for every $$\displaystyle u$$ but may fail to be continuous at $$\displaystyle c$$.
An example of this is given by the function $f(x,y) = \dfrac{x^2y}{x^4+y^2}$ for all $(x,y)\ne(0,0)$, with $f(0,0) = 0.$ This function has a directional derivative in every direction at the origin. In fact, in the direction of the vector $(u,v)$, $\frac{f(hu,hv) - f(0,0)}h = \frac{h^3u^2v}{h(h^4u^4 + h^2v^2)} = \frac{u^2v}{h^2u^4 + v^2}.$ Taking the limit as $h\to0$, $\lim_{h\to0}\frac{u^2v}{h^2u^4 + v^2} = \begin{cases}0&\text{if }v=0,\\ \frac{u^2}v&\text{if }v\ne0. \end{cases}$ So that limit exists for every direction $(u,v).$

On the other hand, $f(x,x^2) = \dfrac{x^4}{2x^4} \to \frac12$ as $x\to0$. So there are points arbitrarily close to $(0,0)$ at which the function takes the value $\frac12$. Therefore $f$ is not continuous at the origin. Geometrically, what is happening here is that the function tends to $0$ as you approach the origin from any direction in a straight line. But if you approach the origin along the curve $y=x^2$, the function tends to $\frac12$ rather than $0$.

Of course, this function cannot have a total derivative at the origin, because that would imply continuity there.

Last edited:
Opalg said:
An example of this is given by the function $f(x,y) = \dfrac{x^2y}{x^4+y^2}$ for all $(x,y)\ne(0,0)$, with $f(0,0) = 0.$ This function has a directional derivative in every direction at the origin. In fact, in the direction of the vector $(u,v)$, $\frac{f(hu,hv) - f(0,0)}h = \frac{h^3u^2v}{h(h^4u^4 + h^2v^2)} = \frac{u^2v}{h^2u^4 + v^2}.$ Taking the limit as $h\to0$, $\lim_{h\to0}\frac{u^2v}{h^2u^4 + v^2} = \begin{cases}0&\text{if }v=0,\\ \frac{u^2}v&\text{if }v\ne0. \end{cases}$ So that limit exists for every direction $(u,v).$

On the other hand, $f(x,x^2) = \dfrac{x^4}{2x^4} \to \frac12$ as $x\to0$. So there are points arbitrarily close to $(0,0)$ at which the function takes the value $\frac12$. Therefore $f$ is not continuous at the origin. Geometrically, what is happening here is that the function tends to $0$ as you approach the origin from any direction in a straight line. But if you approach the origin along the curve $y=x^2$, the function tends to $\frac12$ rather than $0$.

Of course, this function cannot have a total derivative at the origin, because that would imply continuity there.
Thanks Country Boy and Opalg

Peter

## What is the difference between total derivatives and directional derivatives?

Total derivatives represent the overall change in a function as its input variables change, while directional derivatives represent the rate of change in a particular direction.

## How are total derivatives and directional derivatives related?

Total derivatives can be calculated using directional derivatives in all possible directions.

## Why is it important to understand the relationship between total derivatives and directional derivatives?

Understanding this relationship allows us to better analyze functions and their behavior in different directions, which can be useful in fields such as physics, engineering, and economics.

## Can directional derivatives be negative?

Yes, directional derivatives can be negative if the function is decreasing in that particular direction.

## What is the geometric interpretation of total derivatives and directional derivatives?

Total derivatives represent the slope of the tangent plane to a function's graph at a specific point, while directional derivatives represent the slope of a line tangent to the function's graph in a particular direction at that point.

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