Partial differentiation as structured differentiation
Patrick Reany
presents the subject of "partial differentiation" in ways distinct from
the others. Why is it full of slippery phrases such as "if we regard" or
"if we interpret this in a such-and-such a way"? Why is it that the
so-called partial derivative is defined on primitive functions
(functions that have no functional dependencies among the arguments of
the function), but used on non-primitive functions where it has NOT been
defined. (The "unstructured" approach.) Why is it that the partial
derivative seems to bear little resemblance to the total derivative of
ordinary differentiation?
In the pages below I offer my own solution to this problem:
http://www.ajnpx.com/html/SD.html
Patrick
Severian
>
> Each of my advanced calculus books, physics books, and engineering books
Have you no analysis books?
> presents the subject of "partial differentiation" in ways distinct from
> the others. Why is it full of slippery phrases such as "if we regard" or
> "if we interpret this in a such-and-such a way"? Why is it that the
> so-called partial derivative is defined on primitive functions
> (functions that have no functional dependencies among the arguments of
> the function),
That's a new one on me: "primitive functions". Definition, please?
> but used on non-primitive functions where it has NOT been
> defined. (The "unstructured" approach.) Why is it that the partial
> derivative seems to bear little resemblance to the total derivative of
> ordinary differentiation?
Come again?
--
Severian
---------------------------------------------------------------------
"There is no limit to stupidity. Space itself is said to be bounded
by its own curvature, but stupidity continues beyond infinity."
Gene Wolfe, _The Citadel of the Autarch_
andre dajd
"Patrick Reany" <re...@firstworld.net> wrote in message
news:3B06A5DB...@firstworld.net...
> Each of my advanced calculus books, physics books, and engineering books
> presents the subject of "partial differentiation" in ways distinct from
> the others. Why is it full of slippery phrases such as "if we regard" or
> "if we interpret this in a such-and-such a way"?
Because the definitions are usually made without the differential geometric
context. So, they require some intuition. Not to defend it, but calculus
was developed before diff gem, so your advanced books just follow the old
tradition.
>Why is it that the
> so-called partial derivative is defined on primitive functions
> (functions that have no functional dependencies among the arguments of
> the function), but used on non-primitive functions where it has NOT been
> defined. (The "unstructured" approach.) Why is it that the partial
> derivative seems to bear little resemblance to the total derivative of
> ordinary differentiation?
Because the concepts are actually different. Differentiation (full) is
local approximation of a mapping with the linear one. A partial derivative
is simply the correspondent vector (matrix) element.
'functional dependencies among the arguments' is addressed by the linearity
of the first differential.
>
> In the pages below I offer my own solution to this problem:
Too late. You simply reinvent what is clearly presented in any normal book
on diff.gem.
>
> http://www.ajnpx.com/html/SD.html
>
> Patrick
>
Patrick Reany
Severian wrote:
> Patrick Reany wrote:
> [snip] Why is it that the
> > so-called partial derivative is defined on primitive functions
> > (functions that have no functional dependencies among the arguments of
> > the function),
>
> That's a new one on me: "primitive functions". Definition, please?
Yes, the term is my own. A function of two or more arguments is said to be
primitive if each argument is functionally independent of the others. Thus
primitive functions on n arguments are defined on n-space.
Patrick
David C. Ullrich
<seve...@matachin.fsnet.co.uk> wrote:
>Patrick Reany wrote:
>>
>> Each of my advanced calculus books, physics books, and engineering books
>
>Have you no analysis books?
>
>> presents the subject of "partial differentiation" in ways distinct from
>> the others. Why is it full of slippery phrases such as "if we regard" or
>> "if we interpret this in a such-and-such a way"? Why is it that the
>> so-called partial derivative is defined on primitive functions
>> (functions that have no functional dependencies among the arguments of
>> the function),
>
>That's a new one on me: "primitive functions". Definition, please?
If you look at his stuff you see that a "primitive function of n
variables" is simply a function of n variables.
The reason he needs the term is that he thinks for some
reason that an expression like f(x, y, z(x,y)) is a function
of three variables. If it _is_ a function of three variables
then things are as confusing as he suggests, and we
need the term "primitive" so we can say that this one
is not a "primitive" function of three variables.
(But of course in fact f(x,y, z(x,y)) is actually a function
of two variables so there's no problem to be solved.)
>> but used on non-primitive functions where it has NOT been
>> defined. (The "unstructured" approach.) Why is it that the partial
>> derivative seems to bear little resemblance to the total derivative of
>> ordinary differentiation?
>
>Come again?
>
>--
>Severian
>---------------------------------------------------------------------
>"There is no limit to stupidity. Space itself is said to be bounded
>by its own curvature, but stupidity continues beyond infinity."
>Gene Wolfe, _The Citadel of the Autarch_
David C. Ullrich
*********************
"Sometimes you can have access violations all the
time and the program still works." (Michael Caracena,
comp.lang.pascal.delphi.misc 5/1/01)
Dave L. Renfro
[sci.math Sat, 19 May 2001 09:56:59 -0700]
<http://forum.swarthmore.edu/epigone/sci.math/zheldzermsna>
wrote
Your web page wasn't available when I tried it (7:50 A.M. Central
Standard Time, May 20).
Here's an interesting puzzle for you. Have fun with it!
##################################################################
Consider rectangular coordinates x and y, and polar
coordinates, r and %. [I'm using '%' for '\theta'.]
One relation between these is x = r*cos(%).
Hence, (partial x)/(partial r) = cos(%). (1)
Another relation is r = (x^2 + y^2)^(1/2).
Hence, (partial r)/(partial x) = (1/2)*[(x^2 + y^2)^(-1/2)]*(2x).
Substituting 1/r for (x^2 + y^2)^(-1/2) gives
(partial r)/(partial x) = x/r.
Finally, making use of x = r*cos(%) gives
(partial r)/(partial x) = cos(%). (2)
But (partial r)/(partial x) = 1 / [(partial x)/(partial r)],
and so (1) and (2) imply that
cos(%) = 1 / cos(%). (3)
Letting % = Pi/3 in (3) gives 1/2 = 2, or 1 = 4 (multiply by 2).
Now take the positive square root of both sides to get
1 = 2. (4)
##################################################################
The derivation up to (3) was discovered by a (now former) student
of mine, who stumbled upon it while working a problem I had
assigned --- find the Laplacian operator ("del squared") in
polar coordinates. [She was a very bright -- 1600 SAT -- high
school Junior at the time.] A friend of mine wrote this up and
submitted it for the two of us to the College Mathematics
Journal's "Fallacies, Flaws, and Flimflam" column. However, we
later learned that a similar partial derivative paradox had already
appeared, FFF #64 in volume 24 (1993), page 344.
Dave L. Renfro
Severian
>
> Severian wrote:
>
> > Patrick Reany wrote:
> > [snip] Why is it that the
> > > so-called partial derivative is defined on primitive functions
> > > (functions that have no functional dependencies among the arguments of
> > > the function),
> >
> > That's a new one on me: "primitive functions". Definition, please?
>
> Yes, the term is my own.
Nice of you to admit it.
> A function of two or more arguments is said to be
> primitive if each argument is functionally independent of the others. Thus
"Functionally independent"? Please define.
> primitive functions on n arguments are defined on n-space.
--
Patrick Reany
"David C. Ullrich" wrote:
> On Sat, 19 May 2001 19:33:58 +0100, Severian
> <seve...@matachin.fsnet.co.uk> wrote:
>
> >Patrick Reany wrote:
> >>
> >> Each of my advanced calculus books, physics books, and engineering books
> >
> >Have you no analysis books?
> >
> >> presents the subject of "partial differentiation" in ways distinct from
> >> the others. Why is it full of slippery phrases such as "if we regard" or
> >> "if we interpret this in a such-and-such a way"? Why is it that the
> >> so-called partial derivative is defined on primitive functions
> >> (functions that have no functional dependencies among the arguments of
> >> the function),
> >
> >That's a new one on me: "primitive functions". Definition, please?
>
> If you look at his stuff you see that a "primitive function of n
> variables" is simply a function of n variables.
>
> The reason he needs the term is that he thinks for some
> reason that an expression like f(x, y, z(x,y)) is a function
> of three variables. If it _is_ a function of three variables
> then things are as confusing as he suggests, and we
> need the term "primitive" so we can say that this one
> is not a "primitive" function of three variables.
>
> (But of course in fact f(x,y, z(x,y)) is actually a function
> of two variables so there's no problem to be solved.)
Here we have one of the many ambiguities found in partial differentiation; for
even in that notation we can supposedly take the "partial derivative" of f by
either x, y, or z, thus requiring f to be a "function" in some sense of all
three variables. It seems to me that we have two possible solutions here:
Either forbid the use of forms such as "f(x,y, z(x,y))" or else distinguish
between explicit and "fundamental" functional dependence. The former has
phrases like "is a function *of* three variables" and the latter has phrases
such as "is a function *on* R^2."
Or, we can treat the form "f(x,y, z(x,y))" as nasty and forbidden. In which
case we may use a proxy valid function such as g(x,y) = f(x,y, z(x,y)) to do
our differentiations on. (But does it make any sense to say that the rigorously
proper function g is equal to the rigorously *improper* function f?) In which
case, we can say that "f(x,y, z(x,y)) has been reduced to the primitive form
g(x,y)." This IS exactly what gets done in the advanced calculus books! They
just don't use the term "primitive." However, those same advanced calculus
books write stuff like
\partial f
----------
\partial x
which is not valid according to how the "partial derivative" is defined: being
defined only on primitive functions.
Patrick
Patrick Reany
Severian wrote:
> [snip]
>
> "Functionally independent"? Please define.
I will for two variables.
Let x \in D_1 and y \in D_2. Then x and y are said to be "functionally
independent" if whenever there exists a function F of x and y such that F(x,y) =
0 then the equality is an identity for all x \in D_1 and for all y \in D_2.
Patrick
G.E. Ivey
derivatives while NOT being "differentiable". One can define THE
derivative of a function from R^n to R^m (essentially, it is the
linear function that "best" approximates the function) (assuming, of
course, that it is differentable). In the case of R^1 to R^1 (real
valued function of a real variable) it is the linear function
corresponding to multiplication by a number- that number being what we
consider the derivative in Calculus I. In the case of R^n to R^1
(real valued function of several real variables) it is a the vector
whose components (in a given coord system) are the partial
derivatives- in other words, THE derivative is the gradient vector
(and, so, independent of the coord system). In the general case R^n
to R^m, the derivative is the linear transformation which can be
represented, in any given coord system, as a matrix whose components
are the second derivatives- a_ij= d^2 f/ dx^i dx^j (the "d"s are, of
course, partial derivatives.
This is, by the way, where that strange "second derivative"
test for max and min in two variables (if d^2f/dx^2*d^2f/dy^2-
(d^f/dxdy)^2 is positive, either max of min-check d^2f/dx^2 to see
which, if negative, saddle point comes from) comes from- it is the
determinant of the second derivative matrix- which is independent of
coord. system. If that determinant is positive, both eigenvalues have
the same sign, if negative, different signs. For functions of more
that two variables, its harder to find the derivative, and the
derivative gives less information (if positive, it might be that all
eigenvalues are the same but also might be that an even number of them
are negative) but the same principle is true- if all eigenvalues are
positive, minimum, if all negative, maximum, if some are positive and
others are negative, the nth dimension generalization of "saddle
point".
Severian
>
> Severian wrote:
>
> > [snip]
> >
> > "Functionally independent"? Please define.
>
> I will for two variables.
>
> Let x \in D_1 and y \in D_2. Then x and y are said to be "functionally
D_1? D_2? Please define.
> independent" if whenever there exists a function F of x and y such that F(x,y) =
> 0 then the equality is an identity for all x \in D_1 and for all y \in D_2.
Patrick Reany
Severian wrote:
> Patrick Reany wrote:
> >
> > Severian wrote:
> >
> > > [snip]
> > >
> > > "Functionally independent"? Please define.
> >
> > I will for two variables.
> >
> > Let x \in D_1 and y \in D_2. Then x and y are said to be "functionally
>
> D_1? D_2? Please define.
>
Let's keep it simple: They are open sets.
Patrick
Kevin Foltinek
> "David C. Ullrich" wrote:
>
> > (But of course in fact f(x,y, z(x,y)) is actually a function
> > of two variables so there's no problem to be solved.)
>
> Here we have one of the many ambiguities found in partial
> differentiation; for even in that notation we can supposedly take
> the "partial derivative" of f by either x, y, or z, thus requiring f
> to be a "function" in some sense of all three variables.
Sigh.
There is no "ambiguity ... in partial differntiation" here. There is
a confusion about what "function" means. The "ambiguity" you describe
is present only when you (or the authors of a textbook) fail to
understand or use correctly the notion of "function". Given the
expression
f(x,y,z(x,y)) ,
you refer to partial differentiation "of f by either x, y, or z", and
you seem surprised that f is a function "in some sense" of "all three
variables".
There are at least four errors here, which are obvious to anyone who
understands functions, and none of which have anything to do with
differentiation.
Error 1: Refering to "x" or "y" or "z" in the context of "the function
f". "f" is the name of the function, which (see Error 2) is a
function of three things. You can refer to the first, second, or
third argument of f, but by referring to "z", you introduce confusion.
(One could very well write f(z(x,y),x,y), or even f(a,b,c).) (More on
this below.)
Error 2: f is obviously a function of three things. The expression
f(x,y,z(x,y))
depends on two things (x and y); the dependency is via the functions z
and f. z is a function of two, f is a function of three.
Error 3: There is no "in some sense" about it. The use of this phrase
reflects a conditioned response to confuse "f" with "f(x,...)".
Error 4: The phrase "all three variables" reflects confusion over what
"z" is. z is a function. (See Error 1.)
Regarding the point in Error 1 about "z" vs. the third argument: It is
unfortunately common practice to use a notation such as "df/dx", even
in single-variable calculus. I discourage this notation, for many
reasons. A preferable notation (in my opinion) is " f' " or "Df" to
represent the derivative, and "D_3 f" to represent the partial
derivative with respect to the third argument. For explicit
expressions, something like D_x(x^2+y^2) or (x^2+y^2)_x is necessary
and unambiguous.
I recall having a rather lengthy discussion about this a while back.
Patrick does have some valid points, that textbook authors are often
rather sloppy (e.g., "f=f(x,y,z(x,y))") in ways which can be confusing
to the student. His "solution" to this problem is to address it at
the level of differentiation (rather than at the level of function,
which obviously I think is where it should be addressed).
His "solution" is (as far as I can tell) not incorrect (if you ignore
the "primitive function" stuff); in fact, it seems to be nothing more
than the notion of "differential" (as in the 1-form from differential
geometry) combined with new notation involving the "primite function"
idea, to indicate that some things are functions of others (and to
hide the technicalities of the 1-form).
Kevin.
Severian
Open sets? Open subsets of what? Presumably the real numbers or
something similar? So we have a notion of "functional dependence"
of pairs of real numbers?
> independent" if whenever there exists a function F of x and y such that
> F(x,y) = 0 then the equality is an identity for all x \in D_1 and for all
> y \in D_2.
Now the plot thickens. The quantification has become somewhat obscure:
Does
"Let x \in D_1 and y \in D_2 ... there exists a function F of x and y
such that F(x,y) = 0"
mean
"for all x in D_1, for all y in D_2 there exists F ..."
or
"There exists F ... for all x in D_1, for all in D_2" ....
The latter sounds more convincing, but then the thing veers towards
the impenentrable:
"then the equality is an identity for all x \in D_1 and for all y \in
D_2."
What do "equality" and "identity" mean here?
David C. Ullrich
There is a third solution. The standard one. Works perfectly well:
Yes, of course f is a function of three variables. Not in some sense,
f _is_ a function of three variables. I didn't say anything to the
contrary.
What is not a function of three variables is f(x, y, z(x,y)). That
is a function of two variables. If you think that f and f(x,y,z(x,y))
are the same thing that explains the confusion - they're not
the same thing at all.
(Yes, there is a certain amount of stuff out there using notation
that would seem to indicate that f and f(x,y,z(x,y)) are the
same thing. But that's just bad use of traditional notation and
concepts - doesn't require any revolutionary new notation or
concepts to fix, just requires that the traditional notation be
used correctly. And it's not something that any revolutionary
new notation _can_ fix - whatever the new notation is if it's
used incorrectly it will lead to problems.)
>Or, we can treat the form "f(x,y, z(x,y))" as nasty and forbidden. In which
>case we may use a proxy valid function such as g(x,y) = f(x,y, z(x,y)) to do
>our differentiations on. (But does it make any sense to say that the rigorously
>proper function g is equal to the rigorously *improper* function f?
No, and if I'd ever seen anyone say that I might think you had a
point. But I didn't say _anything_ about f, I said something about
f(x,y,z(x,y)), and the fact that so much of your reply sounds as
though you think I said something about f indicates to me that
the location of the confusion that SD fixes is exactly where
I suspected it was.
>) In which
>case, we can say that "f(x,y, z(x,y)) has been reduced to the primitive form
>g(x,y)." This IS exactly what gets done in the advanced calculus books! They
>just don't use the term "primitive." However, those same advanced calculus
>books write stuff like
>
>\partial f
>----------
>\partial x
Which advanced calculus book thinks that f and f(x,y,z(x,y)) are
the same thing? Writing df/dx where (d/dx) f(x,y,z(x,y)) is meant?
>which is not valid according to how the "partial derivative" is defined: being
>defined only on primitive functions.
>
>Patrick
>
David C. Ullrich
***********************
Daryl McCullough
>There is no "ambiguity ... in partial differntiation" here. There is
>a confusion about what "function" means. The "ambiguity" you describe
>is present only when you (or the authors of a textbook) fail to
>understand or use correctly the notion of "function". Given the
>expression
> f(x,y,z(x,y)) ,
>you refer to partial differentiation "of f by either x, y, or z", and
>you seem surprised that f is a function "in some sense" of "all three
>variables".
I certainly agree with all that you, and David Ullrich, (and whoever
else has participated in this thread) have said. However, if you read
a typical undergraduate textbook on, for example, thermodynamics, you
will see things written such as
1. The total energy of a system is a function of the pressure,
volume, temperature, and number of particles: U(P,V,N,T).
2. P, V, N and T satisfy PV = NRT where R is Boltzmann's constant.
3. The entropy S and the energy U are related by dS/dU = 1/T
The thermodynamic quantities U, S, T, N, P, V, etc. are used
ambiguously as variables, and as functions of each other.
Roughly, what's going on is this: the state of a system of
particles can be characterized by three quantities: the volume V,
the number of particles N, and the total energy U. In terms of
these quantities, we can define the entropy, temperature, and
pressure:
S(V,N,U)
T(V,N,U)
P(V,N,U)
But these functions can also be inverted to get V, N, and U
as a function of S,T, and P. Or you can get S,U, and P as a
function of V, N, and T. Basically, any three of them can be
determined from the remaining three. On top of that, you can
introduce yet more thermodynamic quantities, such as the
Gibbs free energy or the enthalpy, or the chemical potential,
etc.
Of course this can all be sorted out in terms of the usual mathematical
notions of variables and functions, but it's almost never done in
undergrad thermodynamics classes. Thus, it is not surprising that
non-math majors are confused about the meaning of partial
differentiation.
--
Daryl McCullough
CoGenTex, Inc.
Ithaca, NY
David C. Ullrich
<folt...@math.utexas.edu> wrote:
>Patrick Reany <re...@firstworld.net> writes:
>
>> "David C. Ullrich" wrote:
>>
>> > (But of course in fact f(x,y, z(x,y)) is actually a function
>> > of two variables so there's no problem to be solved.)
>>
>> Here we have one of the many ambiguities found in partial
>> differentiation; for even in that notation we can supposedly take
>> the "partial derivative" of f by either x, y, or z, thus requiring f
>> to be a "function" in some sense of all three variables.
>
>Sigh.
[...]
>
>I recall having a rather lengthy discussion about this a while back.
Ah, you must be the "Defender" he refers to on his web site?
I'm honored (wonder why he thought that you (or whoever)
didn't want to be identified?)
>Patrick does have some valid points, that textbook authors are often
>rather sloppy (e.g., "f=f(x,y,z(x,y))") in ways which can be confusing
>to the student. His "solution" to this problem is to address it at
>the level of differentiation (rather than at the level of function,
>which obviously I think is where it should be addressed).
Meaning it's obvious that you think this? I think it's obvious
that this is where the problem should be addressed.
>His "solution" is (as far as I can tell) not incorrect (if you ignore
>the "primitive function" stuff); in fact, it seems to be nothing more
>than the notion of "differential" (as in the 1-form from differential
>geometry) combined with new notation involving the "primite function"
>idea, to indicate that some things are functions of others (and to
>hide the technicalities of the 1-form).
>
>Kevin.
David C. Ullrich
David C. Ullrich
wrote:
>Kevin says...
>
>>There is no "ambiguity ... in partial differntiation" here. There is
>>a confusion about what "function" means. The "ambiguity" you describe
>>is present only when you (or the authors of a textbook) fail to
>>understand or use correctly the notion of "function". Given the
>>expression
>> f(x,y,z(x,y)) ,
>>you refer to partial differentiation "of f by either x, y, or z", and
>>you seem surprised that f is a function "in some sense" of "all three
>>variables".
>
>I certainly agree with all that you, and David Ullrich, (and whoever
>else has participated in this thread) have said. However, if you read
>a typical undergraduate textbook on, for example, thermodynamics, you
>will see things written such as
>
[details of traditional nonsensical notation in thermodynamics
snipped]
>
>Of course this can all be sorted out in terms of the usual mathematical
>notions of variables and functions, but it's almost never done in
>undergrad thermodynamics classes. Thus, it is not surprising that
>non-math majors are confused about the meaning of partial
>differentiation.
Indeed. But (as I suspect you agree) partial derivatives are just
where the student _realizes_ he's confused; the actual problems
with the notation have nothing specifically to do with partial
derivatives, the problems have to do with confusion about
functions and variables.
>Daryl McCullough
>CoGenTex, Inc.
>Ithaca, NY
>
David C. Ullrich
George Jones
I certainly agree. Surprising many university math and science students
don't properly understand the concept of function. A student will have
a fuzzy intuitive notion of a function that is often incomplete or wrong
(often something to do with a formula), but they manage to get along OK
until multivariable calculus, when notation and the multivariable chain
rule do them in. In this thread, Dave Renfro has a nice example of what
can be shown through incorrect application of the chain rule.
Writing f(x,y) = f(x, y, z(x,y)) or g(r,theta) = g(x(r,theta),y(r,theta))
can cause all kinds of problems for someone who doesn't understand that
the symbol f is doing dual duty as mappings with different domains. I try,
even in physics classes, to use different symbols, maybe put a tilde over
one f to show that the f's are different but related.
Sometimes I dispair that horrible thermodynamics notation is so ingrained
that the situation is beyond salvation.
While we're on the subject of functional notation, consider the problems
that sin^n x for n a positive integer and sin^(-1) x can cause (even when
the differences are explained over and over again) for the typical
freshman calculus student.
Regards,
George
Patrick Reany
"David C. Ullrich" wrote:
> [snip]
> >Patrick does have some valid points, that textbook authors are often
> >rather sloppy (e.g., "f=f(x,y,z(x,y))") in ways which can be confusing
> >to the student. His "solution" to this problem is to address it at
> >the level of differentiation (rather than at the level of function,
> >which obviously I think is where it should be addressed).
>
> Meaning it's obvious that you think this? I think it's obvious
> that this is where the problem should be addressed.
>
I had self-designed constraints on the way I tried to revamp the subject of
partial differentiation. The most important thing I can say is that SD was
never intended to be a PERFECT replacement for partial differentiation
conventions. It's obvious that no one else in this thread likes my
solutions to the problem, and most don't even admit to a problem existing
at all, except to say that students often don't get functions very well,
which is true too, of course.
Anyway, these are the constraints I placed on my revamping effort:
1. Leave ordinary differentiation notation alone
2. Leave the conventions of partial differentiation as minimally changed as
possible.
3. Try to make so-called partial differentiation appear as a simple
generalization
of ordinary differentiation.
4. Attempt a notation unambiguous when used by either mathematicians or
physicists,
minimally interfering with their customs
I believe that these constraints are more restrictive than is at first
apparent and certainly not in line with goals of mathematical "purists." If
in the end, whatever can be produced under these constraints is little more
than a band aid to the problem, then so be it. I have the thanks of some
and the condemnation of others for what SD does to "clear up" and/or
obfuscate the subject.
It was never my intention to fix or deal with all conceptual problems
regarding the function concept as used in partial differentiation. Posters
in this thread maintain that f and f(x,y,z(x,y)) are really functionally
different objects. I don't deny that this is a perfectly good way to avoid
confusions, but physicists won't accept it and I don't think any
mathematician is really so pedantic as to adhere to this distinction *all*
the time. Say we write g(x,y) = f(x,y,z(x,y)), where x and y are the
"independent" variables that both g and f depend on. What's the partial
derivative of g by x in terms of partial derivatives of f?
\partial_x g = \partial_x f + \partial_z f \partial_x z
I have no problem with the partial derivative of g on the left. What I have
a problem with is the partial derivative of f on the right. Conventional
partial derivatives are defined only on primitive functions, such as g is.
For the partial of f by x to be legit, it must be taken on a primitive
function, but which one? Is f primitive? Just what is f? Is it
f(x,y,z(x,y))? Or is it f(x,y,z), where x,y,z are mutually independent of
each other? In SD the partial derivative is an explicit derivative and it
doesn't care how you interpret f, primitive or not. You get the same result
as in partial differentiation because defining the partial as explicit in
SD has the same effect as taking the conventional partial of a primitive
function.
My weakness is that when I see the equation g(x,y) = f(x,y,z(x,y)), I
regard g (the function) as defined on R^2, and I regard f (the function) as
defined on a 2-dimensional surface in R^3, thus f is a non-primitive
function. The equality says to me that the value of g at a given x,y is the
same as the value of f for the same x,y. I do not regard g and f as being
the same function because I do not regard them as being defined on the same
set of points in R^3. The interpretation I have seems quite natural to me,
which is why an expression like f(x,y,z(x,y)) seems like an "unnatural"
expression in partial differentiation. I accept that partial
differentiation wants me to regard f(x,y,z(x,y)) as being "defined" on a
set of points in the x-y plane, but this seems unnatural to me, though I
can accept it formally -- it amounts to a rule of interpretation. It seems
quite reasonable, however, to want to pull back the points on the surface
to the points on the plane to do differentiation as conventional partial
differentiation wants us to do. This is really NOT the problem I've had. I
just can't seem to get used to the expression f(x,y,z(x,y)) as meaning
this, and so I am left without any writable expression that makes a good
candidate for this for me. This issue just never arises in SD.
It also seems natural to me to think of f(x,y,z(x,y)) as being a "function"
of z, and more generally, that f is a "function" of each of the variables
it is explicitly dependent on, regardless of their possible
interdependencies. I allow for both explicit and fundamental functional
dependence notions.
I admit that SD is not exactly the content of conventional partial
differentiation. Yet, it seems to me to relax an overly restrictive
definition of what a function means. Of course, as has been stated
elsewhere, SD can be interpreted more conventionally, in which case
expressions like f(x,y,z(x,y)) are always interpreted as primitive (in this
case a function over R^2), and then perhaps all that's left of it is some
notational distinctions which we have not gone into at all in this thread
(and probably should not). Whether SD is itself revamped into a more
conventional interpretation or not, it gets the same results as
conventional partial differentiation does in solving problems. I just find
SD a much more efficient tool in problem solving than conventional partial
differentiation.
It is my custom every few months to post a notice to sci.math announcing SD
on my website. I have noticed that if I say nothing provocative in the post
then nobody replies to the post, though a few readers take the link. But if
I include any provocative statements in the announcement at all, then the
post can lead to a thread, which wasn't my intention this time, though it
has been instructive for me and hopefully for other readers as well. Next
time I think I'll say nothing provocative.
Patrick
David C. Ullrich
wrote:
>
>
>"David C. Ullrich" wrote:
>
>> [snip]
>> >Patrick does have some valid points, that textbook authors are often
>> >rather sloppy (e.g., "f=f(x,y,z(x,y))") in ways which can be confusing
>> >to the student. His "solution" to this problem is to address it at
>> >the level of differentiation (rather than at the level of function,
>> >which obviously I think is where it should be addressed).
>>
>> Meaning it's obvious that you think this? I think it's obvious
>> that this is where the problem should be addressed.
>>
>
>I had self-designed constraints on the way I tried to revamp the subject of
>partial differentiation. The most important thing I can say is that SD was
>never intended to be a PERFECT replacement for partial differentiation
>conventions. It's obvious that no one else in this thread likes my
>solutions to the problem, and most don't even admit to a problem existing
>at all, except to say that students often don't get functions very well,
>which is true too, of course.
When I say that f(x,y,z(x,y)) is a function of two variables and
in your reply you state that no, f is a function of three variables
it seems very clear that _you_ are one of the people who
doesn't get functions very well, thinking that f and f(x,y,z(x,y))
are the same thing!
There _is_ a certain amount of badly-written sloppy stuff out there.
But "revamping" things is not a solution to that - whatever the
new system is, if people use the new notation incorrectly it will
lead to confusion or worse.
On the other hand there's also plenty of stuff out there that's
written precisely, with no confusion at all. A few posts ago
you said something about seeing various things in advanced
calculus books, I asked what ac books contained such statements,
and you never replied.
As long as you think that f and f(x,y,z(x,y)) are the same thing,
as you evidently do, then you cannot possibly understand the
stuff that's written correctly - hence your opinion that there's
a big problem that requires that we redefine partial
differentiation. The problem is caused by your confusion
about what the standard notation means - it's a problem
about what functional notation means, _not_ a problem
about derivatives.
>Anyway, these are the constraints I placed on my revamping effort:
>
>1. Leave ordinary differentiation notation alone
>2. Leave the conventions of partial differentiation as minimally changed as
>possible.
>3. Try to make so-called partial differentiation appear as a simple
>generalization
> of ordinary differentiation.
>4. Attempt a notation unambiguous when used by either mathematicians or
>physicists,
> minimally interfering with their customs
>
>I believe that these constraints are more restrictive than is at first
>apparent and certainly not in line with goals of mathematical "purists." If
>in the end, whatever can be produced under these constraints is little more
>than a band aid to the problem, then so be it. I have the thanks of some
>and the condemnation of others for what SD does to "clear up" and/or
>obfuscate the subject.
>
>It was never my intention to fix or deal with all conceptual problems
>regarding the function concept as used in partial differentiation. Posters
>in this thread maintain that f and f(x,y,z(x,y)) are really functionally
>different objects. I don't deny that this is a perfectly good way to avoid
>confusions, but physicists won't accept it and I don't think any
>mathematician is really so pedantic as to adhere to this distinction *all*
>the time.
See, the fact that you think it would be pedantic to adhere to this
just shows that you're confused about the notation. Plenty of
people _do_ adhere to this _all_ the time - the idea that f and
f(x,y,z(x,y)) are different things is not pedantic, the idea that
they're the same thing is simply ridiculous.
But some sorts of confusions are self-reinforcing. If you
don't realize that people _do_ regard f and f(x,y,z(x,y))
as different things then when you read things that are
carefully written, where the author never confuses the
two, you will not have any way to _realize_ that the
author is not confusing the two! He writes f when he
means f, he writes f(x,y,z(x,y)) when he means
f(x,y,z(x,y)), there is no confusion at all in what he
writes. But if you _think_ that everyone always
confuses the two then when you see him write
f you don't _realize_ that he meant f and not
f(x,y,z(x,y)), and then you come away with the
idea that he's using the notation sloppily, like
in those thermo texts.
>Say we write g(x,y) = f(x,y,z(x,y)), where x and y are the
>"independent" variables that both g and f depend on. What's the partial
>derivative of g by x in terms of partial derivatives of f?
>
>\partial_x g = \partial_x f + \partial_z f \partial_x z
This is exactly why Epstein said that the notation D_1,
etc is better than D_x, etc. If g(x,y) = f(x,y,z(x,y))
then
D_1 g(x,y) = D_1 f (x,y,z(x,y)) + D_3 f(x,y,z(x,y)) * D_1 z(x,y)
>I have no problem with the partial derivative of g on the left. What I have
>a problem with is the partial derivative of f on the right. Conventional
>partial derivatives are defined only on primitive functions, such as g is.
>For the partial of f by x to be legit, it must be taken on a primitive
>function, but which one? Is f primitive? Just what is f? Is it
>f(x,y,z(x,y))? Or is it f(x,y,z), where x,y,z are mutually independent of
>each other?
f is f(x,y,z). That _is_ what f means. That's not the same
thing as f(x,y,z(x,y)), and there's simply no reason to
think that f(x,y,z(x,y)) is what is meant by "f". If you insist
this is not clear then it will not be clear to you and you
will then think that there is a problem.
> In SD the partial derivative is an explicit derivative and it
>doesn't care how you interpret f, primitive or not. You get the same result
>as in partial differentiation because defining the partial as explicit in
>SD has the same effect as taking the conventional partial of a primitive
>function.
>
>My weakness is that when I see the equation g(x,y) = f(x,y,z(x,y)), I
>regard g (the function) as defined on R^2, and I regard f (the function) as
>defined on a 2-dimensional surface in R^3, thus f is a non-primitive
>function.
"Your weakness" is _exactly_ right! When I see the equation 2 + 2 = 4
I regard "2" as 3, while I regard "4" as denoting 4; hence the
equation 2 + 2 = 4 looks wrong to me. But I have _no_ _reason_
to regard the symbol "2" as 3 - if I would simply realize that "2"
denotes 2 my confusion would vanish.
> The equality says to me that the value of g at a given x,y is the
>same as the value of f for the same x,y.
And it says to me that I am the king of France. But that's as
relevant as what it says to you. What it _actually_ _says_
is that the value of g at a given x,y is the same as the
value of f at x, y, z(x,y).
When you talk about the value of f at x, y that simply makes
no sense at all - the fact that you think it makes sense is
the source of your confusion.
>I do not regard g and f as being
>the same function because I do not regard them as being defined on the same
>set of points in R^3. The interpretation I have seems quite natural to me,
Of course f and g are not the same function!
>which is why an expression like f(x,y,z(x,y)) seems like an "unnatural"
>expression in partial differentiation. I accept that partial
>differentiation wants me to regard f(x,y,z(x,y)) as being "defined" on a
>set of points in the x-y plane, but this seems unnatural to me, though I
>can accept it formally -- it amounts to a rule of interpretation. It seems
>quite reasonable, however, to want to pull back the points on the surface
>to the points on the plane to do differentiation as conventional partial
>differentiation wants us to do. This is really NOT the problem I've had. I
>just can't seem to get used to the expression f(x,y,z(x,y)) as meaning
>this,
Just like I simply cannot get used to the idea that "2" denotes the
integer 2 - it seems to me that "2" really denotes 3.
So I need to set up a web site explaining that if we wrote
"3" for the successor of 1 things would be much simpler.
David C. Ullrich
Цэсдзн МЯгдзн
"David C. Ullrich" <ull...@math.okstate.edu> wrote in message
news:3b0bc5f4...@nntp.sprynet.com...
>
> f is f(x,y,z). That _is_ what f means. That's not the same
> >
>
> >same as the value of f for the same x,y.
>
> And it says to me that I am the king of France. But that's as
> relevant as what it says to you. What it _actually_ _says_
> is that the value of g at a given x,y is the same as the
> value of f at x, y, z(x,y).
>
Robin Chapman
Indeed it's the value. To be really pedantic one should insist
that (x,y,z) |--> f(x,y,z) is the function f, not f(x,y,z).
------------------------------------------------------------
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"His mind has been corrupted by colours, sounds and shapes."
The League of Gentlemen
David C. Ullrich
Officially f is a function and f(x,y,z) is not a function, it is the
value of f at the given x,y,z.
Informally people often say things like "the function x^2" where
officially they mean "the function f defined by f(x)=x^2".
Yes, I'm guilty of this informality here.