$$

Stoichiometry is indeed a method where we can find the various amounts of ingredients in a chemical reaction, using the conservation of matter.

]]>Let . Then one needs

I messed up with building up with this last step.

What is needed is

But

.

What is the mistake here and how can I correct it?

]]>If is dimensionless, then its scaling behaviour with respect to the dimensional parameter (playing the role of in the post; one can also use if one prefers) is given by . From the chain rule one then has , and so has dimension .

]]>in (3) ]]>

*If is dimensionless, then has the units of *

I guess this follows from the definition in the post:

I feel that this seems to relate to some calculation in the one-dimensional case like

.

But I can’t match the notation/language in definition (2) to (1) and (3).

Also, I’m confused that why you say “ has units of ” instead of has units of .

]]>In terms of the language of the post, the dimensional parameter is , and varies in according to the formula , thus is the dilation of a fixed region by .

]]>

… thus, by (3) (now using parameters {L,A} instead of {M,L,T}), we have

In (3), only the abstract exponents are given. Would you elaborate how to get the formula above by (3)?

]]>*We say that a function of a bounded open set , , is scale invariant if for all obtained from by a rigid transformation and a dilation .
*

Is the bounded domain in the beginning of the quoted text above a “dimensionful set” and the scalar invariant function a dimensionful function?

]]>We say that a function of a bounded open set , , is scale invariant if for all obtained from by a rigid transformation and a dilation .

Denote by the operation mapping functions defined on to function defined on :

When we refer to the way something scales we mean under the dilations and operations . One can modify the definition of the norms of in such a way that they scale as the pure -th order derivatives do. We shall denote by

and by

the linear size of .

*Define with
With this definition the quantity scales like , i.e.,
is scale invariant.
*

I’m wondering if one can use the “dimensional analysis” in this post to answer the following questions:

Is there a heuristic way to cook up the exponent in the term

A direct simplification gives

Why is this scale invariant?