The Derivative in Several Variables .... Hubbard and Hubbard, Section 1.7 ....

[Math Amateur]

I am reading the book: "Vector Calculus, Linear Algebra and Differential Forms" (Fourth Edition) by John H Hubbard and Barbara Burke Hubbard.

I am currently focused on Section 1.7: Derivatives in Several Variables as Linear Transformations ...

I need some help in order to understand some remarks by Hubbard and Hubbard on page 124 under the heading "The derivative in several variables ... ...

The relevant text reads as follows:
View attachment 8720
Referring to equation 1.7.10 H&H say the following:

" ... ... But this wouldn't work even in dimension \(\displaystyle 1\), because the limit changes sign as \(\displaystyle h\) approaches \(\displaystyle 0\) from the left and from the right. ... ... "Could someone please explain exactly how/why the limit changes sign as \(\displaystyle h\) approaches \(\displaystyle 0\) from the left and from the right. ... ... ?

I am puzzled because \(\displaystyle \mid h \mid\) doesn't change sign and \(\displaystyle f( a + h ) - f ( a)\) doesn't necessarily change sign as \(\displaystyle h\) approaches \(\displaystyle 0\) from the left and from the right. ... ...
Hope someone can help ...

Peter

 

[HOI]

Yes, |h| doesn't change sign but h does and so using |h| is a mistake. As the author says "this won't work even in dimension 1". Take the very simple case of $$f(x)= 2x+ 3$$ at x= 1. Since this is linear, its derivative at any x is the slope, 2. The usual definition of the derivative gives $$\lim_{h\to 0}\frac{f(1+ h)- f(1)}{h}= \lim_{h\to 0}\frac{(2(1+ h)+ 3)- 5}{h}= \lim_{h\to 0}\frac{2h}{h}= 2$$. But if we use "|h|" in the denominator instead of "h" that limit does not exist!

If h> 0 then $$\lim_{h\to 0}\frac{2h}{|h|}=\lim_{h\to 0^+}\frac{2h}{h}= 2$$ but if h< 0, |h|= -h so $$\lim_{h\to 0^-}\frac{2h}{|h|}= \lim_{h\to 0}\frac{2h}{-h}= -2$$. The two onesided limits are not the same so the limit itself does not exist.

 

[math771]

Hi Peter,

The reason why the limit changes sign as h approaches 0 from the left and from the right is because of the definition of a limit. In this case, we are looking at the limit of a function as h approaches 0, which means we are looking at the behavior of the function as h gets closer and closer to 0.

In the case of equation 1.7.10, we are looking at the limit of the function f(a+h) - f(a) as h approaches 0. This limit is not just dependent on the values of f(a+h) and f(a), but also on the direction from which h is approaching 0. When h approaches 0 from the left, the values of f(a+h) and f(a) will be different than when h approaches 0 from the right. This is because the behavior of the function may be different on the left and right sides of the point a.

To understand this concept, it may be helpful to think about a simple example in one dimension. Consider the function f(x) = x^2. When x is positive, the function is increasing, but when x is negative, the function is decreasing. So, when we take the limit of f(x) as x approaches 0, the value of the limit will be different depending on whether x is approaching 0 from the left (negative values) or from the right (positive values).

In the case of the function in equation 1.7.10, the limit will also be different depending on the direction from which h is approaching 0. This is because the function may have different behavior on the left and right sides of the point a, which will affect the values of f(a+h) and f(a).

I hope this helps to clarify the concept for you. Let me know if you have any further questions. Happy studying!