what is the use of euler's theorem for partial derivatives and maxima ,minima of function of two variables

Question

please tel me the use in nature and science

Answer 1

Get a book from your library on applied math.

Internate must be full of such applictions.

Answer 2

No, once you've 2 maxima, there must be factors on a minimum of one area of that maxima that are below that optimal. In a nutshell, the definition of a max is in case you move to both area of it, the cost of the function is decrease. f(x) = Max, then f(x ± a) < Max the position a is a few infinitessimally small cost. If the function is non-stop, there must be some cost below both max, so there must be a min. this question does spoil down the position you've a horizontal line, yet the following, you absolutely have an unlimited quantity of max and minutes as each and every element is the max and min. i desire this helps

Answer 3

Euler's Theorem for LINEAR homogeneous systems lies at the basis of many thermodynamic identities, much used in chemical engineering, etc.

A basic differential relation between reversible changes in U (internal energy of a system), S (entropy), T (temperature), p (pressure), V (volume) for a particulate-CLOSED system (# of particles not changed) to which heat (TdS) can be added is:

TdS = dU + pdV (1); meaning

"Added heat increases internal energy and/or does work by expansion."

If one now adds in the possiblity of the numbers of particles being allowed to change, also, there are additional terms generally denoted by N_i (#'s of the ith component) and (Greek letter) mu_i, the "chemical potential" for the ith species, a kind of analogue (for changes in N_i) of pressure (for changes in V), related to the work needed to increase or extract particles under certain conditions. (For historical reasons, there's a sign convention such that the mu_i are more the analogue of "-p.")

The augmented relation allowing for #-changes is:

TdS = dU + pdV - [Sum over i] (mu_i)dN_i (2).

All the above is for SMALL changes.

Now to set up how Euler enters. Thermodynamic quantities can be divided into "intensive" (they hold throughout the system, like T, p, ...) and "extensive" (proportional to "amount of stuff," like S, U, N_i). Since the other terms of eqn. (2) involving S, U, V (with or without an appropriate multiplier) are extensive (proportional to "stuff"), the term involving the N_i must also be extensive. But since N_i are extensive by definition, that means that the mu_i must be intensive. (Just in case you were wondering.) That's why I could declare that they were a kind of analogue of p.

It might seem that I've forgotten about Euler, but it was essential to establish that all the (single or double variable) terms separated by "=", "+", and "-" in eqn. (2) are extensive, i.e. LINEARLY dependent on the amount of stuff.

Then by Euler's Theorem, eqn. (2) can be integrated on sight --- we simply erase all the "d's"! (Is that not audacious or what?!):

TS = U + pV - [Sum over i] (mu_i)N_i (3)

This INTEGRATED form of the small-changes relationship is basic to many more thermodynamic identities like the Maxwell Identities, in shifting back and forth between the most convenient variables to use under a variety of physical conditions, and in defining all the many other variables (enthalpy, free energy, ...) so beloved in practice by chemical engineers.

Eqn. (3) also helps one understand the miasma of errors into which people (including quite distinguished authors) have fallen in discussing the so-called Gibbs Paradox --- but that's a whole other story.

As to the second part of your question, let's simply say that the various stationary states possible (under an assumption of different basic thermodynamic variables) can tell one what state a system will come to with controls of a given kind. This is a very important thing to know so that, for example, your refinery or other chemical plant won't blow up!

Live long and prosper.