MHB Understanding Differential to Calculus Basics

[Yankel]

Hello,

I have a theoretical question, I am struggling to understand the meaning of differential. I know the formula, I read it all in the calculus textbook. I don't understand what is the meaning, maybe geometric interpretation and use.

In addition, I don't understand the difference between dz and

\[\Delta z\]

or between dx and

\[\Delta x\]

and same goes for y.

Oh, and I am talking about functions with two variables z=f(x,y)

(not that in 1 variable I do understand it)

if you can explain it in an understandable way I would appreciate it...

thanks !

 

[caffeinemachine]

Hello,

I have a theoretical question, I am struggling to understand the meaning of differential. I know the formula, I read it all in the calculus textbook. I don't understand what is the meaning, maybe geometric interpretation and use.

In addition, I don't understand the difference between dz and

\[\Delta z\]

or between dx and

\[\Delta x\]

and same goes for y.

Oh, and I am talking about functions with two variables z=f(x,y)

(not that in 1 variable I do understand it)

if you can explain it in an understandable way I would appreciate it...

thanks !

You are currently following the non-rigorous approach to calculus and it is good that you are having such questions.

To differentiate between $\Delta z$ and $dz$ first you should know the definitions of both of these.

Say $z:\mathbb R\to\mathbb R$ be a real function.
Then $\Delta z$ is quantity which represents the difference between the values $z$ achieves at two different points of its domain.
It is a quantity of the type $z(b)-z(a)$.
Not much of a definition really.

As for $dz$, I don't think anybody has ever defined it.
It appeals to our intuition. You may have seen the use of the word 'infinitesimal' when somebody tries to explain the meaning of $dz$. But what is an 'infinitesimal'? An infinitesimal has a precise definition, but on the real number line infinitesimals do not exist. And that is why infinitesimals are no longer talked about in modern texts (which develop calculus on real numbers).

So what is the solution to your problem?

I think you should start reading calculus with a rigorous approach.
Any real analysis book will help you understand single-variable calculus and books like 'Calculus on Manifolds' by Spivak and 'Analysis on Manifolds' by Munkres will help you understand multivariable-calculus rigorously.