Gradient of a Simple Surface at a point (without knowledge of derivative)

[mathnovice1]

Homework Statement

The statement that the simple surface F has gradient at the ordered pair ((x,y),F(x,y)) of F means that there exists only one ordered number pair (p,q) such that if c is a positive number then there exists a rectangular segment , S, containing (x,y) such that, if (u,v) is in S and in the domain of F then
F(u,v) - F(x,y) = (u-x)p + (v-y)q + ε|(u,v)-(x,y)|
where ε is a number between -c and c

Homework Equations

none

The Attempt at a Solution

I realize that this is not a 'problem' per se, however I am simply trying to make sense of the definition of gradient without resorting to derivatives. I would like to prove that if F has gradient at a point then it is continuous at the point, again without using any notion of derivative, just the definition provided. But I would like to attempt to prove the implication of continuity myself.

I am simply looking for anyone to assist me to break down the definition of gradient given and understand it a little better. I have been staring at it for a few days and I am just not getting anywhere. I don't understand where the positive number c comes into play and therefore the ε as well.

Any assistance/insight/guidance would be appreciated, thanks!

 

[Ray Vickson]

Homework Statement

The statement that the simple surface F has gradient at the ordered pair ((x,y),F(x,y)) of F means that there exists only one ordered number pair (p,q) such that if c is a positive number then there exists a rectangular segment , S, containing (x,y) such that, if (u,v) is in S and in the domain of F then
F(u,v) - F(x,y) = (u-x)p + (v-y)q + ε|(u,v)-(x,y)|
where ε is a number between -c and c

Homework Equations

none

The Attempt at a Solution

I realize that this is not a 'problem' per se, however I am simply trying to make sense of the definition of gradient without resorting to derivatives. I would like to prove that if F has gradient at a point then it is continuous at the point, again without using any notion of derivative, just the definition provided. But I would like to attempt to prove the implication of continuity myself.

I am simply looking for anyone to assist me to break down the definition of gradient given and understand it a little better. I have been staring at it for a few days and I am just not getting anywhere. I don't understand where the positive number c comes into play and therefore the ε as well.

Any assistance/insight/guidance would be appreciated, thanks!

P.S. I am getting the definition from CreativeMathematics by HS Wall on page 94

Essentially, it claims that the pair (p,q) is the gradient. Note that if you take y=v and let u → x you get F(u,y)-F(x,y) = p(u-x) + ε|u-x|. Presumably, the 'c' referred to can be any chosen positive number (and after choosing c, we get to specify the 'rectangle'). If that is what the author meant, then, of course, p will just be the partial derivative ∂F(x,y)/∂x, because \frac{F(u,y)-F(x,y)}{u-x} = p + \epsilon \frac{|u-x|}{u-x} \rightarrow p as u \rightarrow x. This holds because for any positive number c we have
\left|\frac{F(u,y)-F(x,y)}{u-x} -p\right| \leq |\epsilon| \leq c for any u sufficiently near x.

RGV

 

[voko]

It basically means that for any desired degree of "accuracy", which is denoted by c, the difference from any point close enough to (x, y) can be approximated by a linear function of the displacement. Which is very similar to the differential concept, except that the differential deal with infinitesimal displacements.

 

[mathnovice1]

RGV - thanks for the quick reply. Your response is very helpful to me.

Can you elaborate a bit on the rationale behind letting y=v as well as how we know that the following is true:

This holds because for any positive number c we have
\left|\frac{F(u,y)-F(x,y)}{u-x} -p\right| \leq |\epsilon| \leq c for any u sufficiently near x.

 

[Ray Vickson]

RGV - thanks for the quick reply. Your response is very helpful to me.

Can you elaborate a bit on the rationale behind letting y=v as well as how we know that the following is true:

I want to know if p is ∂F(x,y)/∂x, which is the x-derivative with y held fixed. So I don't want to change y to v ≠ y; I just want to change x by itself. Certainly the horizontal line segment from (x,y) to (u,y) is in the rectangle S, so I am allowed to write the first equation.

The second equation is obtained from the first one by dividing through by u-x (assuming u ≠ x, which I neglected to say).

RGV