function and variable...some thoughts
Aitken
I have some thoughts about variables and functions that have changed
over time. When I was first learning calculus, I thought the variable
(e.g., the x in f(x) = x^2) was essential in identifying the function
throughout the context where the function is used. But over time I
came to think of the variable as something that was used only to show
interdependencies between functions and to show the "rule" that the
function takes.
So taking some ideas from set theory, I think I can define a function
like this:
f : R -> R , where f maps each element of R to it's square in R
so,
f ' : R - > R , where f' maps each element of R to two times that
element in R
So there is no need for a variable at all here (similar to defining set
A = { set of all positive integers } )
so now f(y) is just as important as f(x). They both denote the
function's value at x or y.
But when someone says,
f : R -> R, where f(x) = x^2
then goes onto say...
df/dx = 2x this suggents that x is simehow importat to the function.
And if someone were to do this in the same context, people would be
shocked:
df/dy = 2y WRONG! f is a function of x!!!!
Of course, the PRIME NOTATION is much better:
f '(x) = 2x
f '(y) = 2y <=== see how it allows for function f to be independent of
variable
or don't need variables at all:
f '(0) = 0, f'(1) = 2, etc.
So it seems as some notations tie the function closely to the variable
it is defined with originally, while others make the function an
independent thing (where derivative is with "respect to the argument"
whatever that maybe be).
This is most evident in the multivariable case:
f : R^2 -> R, where f(x,y) = x^2 + y^2
f_x (a,b) = 2a
f_y (a,b) = 2b
The very notation of partial derivatives suggest that the x,y are
intrinsically tied to the function (which is why I prefer the D_1, D_2
operation notations - which pisses off the markers).
Also the deltaf/deltax notation also suggets that "x" is somehow
improtant to f.
So my question is. Are these variable important? Or are they dummy?
Anyone ever experience the same confusion I've had (where I've shifted
slowly into thinking that all notation should suggest independent of
variables from functions).
My post might be confusing, but that's because I am myself confused. I
can not poinpoint the difficulty I'm having.
lite.o...@gmail.com
T = f(x,y) , x = x(t), y = y(t)
by chain rule
d/dt of f(x,y) = f_x(x, y) * x'(t) + f_y(x, y) * y'(t)
In this context, the variable x and y ARE VERY IMPORTANT, because not
only do the partial derivative notations suggesting this, but x,y are
being assigned to.
I don't mind the assigned to part, because the variable act as a way to
show interdependencies between the function. Nevertheless, the partial
derivative notation does suggest that some mathematicians must view the
darn x, y as important to f.
To make all of the above variable-independent I would do this...
define T = f(x,y)
define other function g(t), h(t)
define a new function
G(t) = f(g(t), h(t))
now,
G'(t) = D_1 * f(g(t), h(t)) * g'(t) + D_2 * f(g(t), h(t)) * h'(t)
how I can simply change the variable:
G'(u) = D_1 * f(g(u), h(u)) * g'(u) + D_2 * f(g(u), h(u)) * h'(u)
Which makes the variable truly dummy.
Definiing the new composite function removes the dependence on x,y to
show how the 2 functions of t were being input into f. And the use of
the D_n notation removes the need to suggest that the partials of f
were being taken with respect to any variable.
Actually this can be done without defining a new function G!! But it
looks better this way.
2 ways of thinking? My calclus book shows first way, but I usually
translate it into the second way in my head (got into habit of doing
this, not needed, but it makes me feel more comfortable).
Hero
> Hi,
>
> I have some thoughts about variables and functions that have changed
> over time. When I was first learning calculus, I thought the variable
> (e.g., the x in f(x) = x^2) was essential in identifying the function
> throughout the context where the function is used. But over time I
> came to think of the variable as something that was used only to show
> interdependencies between functions and to show the "rule" that the
> function takes.
>
> So taking some ideas from set theory, I think I can define a function
> like this:
>
> f : R -> R , where f maps each element of R to it's square in R
>
> so,
>
> f ' : R - > R , where f' maps each element of R to two times that
> element in R
>
> So there is no need for a variable at all here (similar to defining set
> A = { set of all positive integers } )
>
Regarding a function as how a dependent variable varies with the
variation of an independent variable has it's merits.
f: R -----> R+ ( the positive reals): x ----> x²
With set theory one first looks at the set of all ordered pairs (a, b )
with a Element R and b Element R+, thus RxR+. Every subset of this is
called a relation.
Now a subset with the property, that
(a, b) = ( a, c) ====> b = c
is a function.
So f = { ( a, b ) | ( a, b ) Element of RxR+, and a² = b }
Impulsive one might say, that is the picture or the graph of the
function - and i think, that is how they got this definition.
Now a graph is showing all ( okay nearly all) values at the same time,
with one look one can see them, but it's no longer a movement, its the
trace, the movement has left on the paper.
Return to the variables, the independent x has this uniform change
which R allows, between 3 and 5 it changes as much as between pi/2 +
0.1 and pi/2 + 2.1. This can be looked at as two hours or as 2 cm.From
time and space the number space is developed.
So far and with friendly greetings
Hero
eKo1
> So taking some ideas from set theory, I think I can define a function
> like this:
>
> f : R -> R , where f maps each element of R to it's square in R
>
> so,
>
> f ' : R - > R , where f' maps each element of R to two times that
> element in R
>
> So there is no need for a variable at all here (similar to defining set
> A = { set of all positive integers } )
Err, you are still using variables. What do you think f, f', R and A
are?
> so now f(y) is just as important as f(x). They both denote the
> function's value at x or y.
>
> But when someone says,
>
> f : R -> R, where f(x) = x^2
>
> then goes onto say...
>
> df/dx = 2x this suggents that x is simehow importat to the function.
>
> And if someone were to do this in the same context, people would be
> shocked:
>
> df/dy = 2y WRONG! f is a function of x!!!!
Why are you substituting dx with dy? You're not being consistent with
your variables. 2y is what you get after evaluating df/dx at x = y.
> Of course, the PRIME NOTATION is much better:
>
> f '(x) = 2x
> f '(y) = 2y <=== see how it allows for function f to be independent of
> variable
As long as you're consistent, you can replace the variables.
> So it seems as some notations tie the function closely to the variable
> it is defined with originally, while others make the function an
> independent thing (where derivative is with "respect to the argument"
> whatever that maybe be).
>
> This is most evident in the multivariable case:
>
> f : R^2 -> R, where f(x,y) = x^2 + y^2
>
> f_x (a,b) = 2a
>
> f_y (a,b) = 2b
>
> The very notation of partial derivatives suggest that the x,y are
> intrinsically tied to the function (which is why I prefer the D_1, D_2
> operation notations - which pisses off the markers).
Use whatever notation you like as long as it you explain what it means,
how it is used and are consistent with its use.
> So my question is. Are these variable important? Or are they dummy?
> Anyone ever experience the same confusion I've had (where I've shifted
> slowly into thinking that all notation should suggest independent of
> variables from functions).
Variables are place-holders for values. Nothing more and nothing less.
lite.o...@gmail.com
What I don't get is the point behind the f_x (a,b) notation.
This seems to mean "first take the partial derivative of the expression
you get when you plug x,y into f - namely f(x,y) - with respect to x,
then find value of this expression at x=a, y=b"
[Of course y can be replaced by something else since I make no mention
of f_y ]
But f doesn't require any variable x to have meaning, so what's this
f_x all about?
Seems kind of odd, doesn't it?
David Marcus
Dummy. Leibniz notation dates from the dawn of the calculus. Back in
those days, people thought of functions as expressions, not as mappings.
The more logical notation is much more recent. I generally find I'm less
likely to get confused if I use the logical notation.
> Anyone ever experience the same confusion I've had (where I've shifted
> slowly into thinking that all notation should suggest independent of
> variables from functions).
>
> My post might be confusing, but that's because I am myself confused. I
> can not poinpoint the difficulty I'm having.
You apparently learned the material using the less logical notation.
That accounts for your difficulty.
--
David Marcus
Mike Terry
news:1162155507....@b28g2000cwb.googlegroups.com...
> Hi,
>
> I have some thoughts about variables and functions that have changed
> over time. When I was first learning calculus, I thought the variable
> (e.g., the x in f(x) = x^2) was essential in identifying the function
> throughout the context where the function is used. But over time I
> came to think of the variable as something that was used only to show
> interdependencies between functions and to show the "rule" that the
> function takes.
>
> So taking some ideas from set theory, I think I can define a function
> like this:
>
> f : R -> R , where f maps each element of R to it's square in R
>
> so,
>
> f ' : R - > R , where f' maps each element of R to two times that
> element in R
>
> So there is no need for a variable at all here (similar to defining set
> A = { set of all positive integers } )
>
Sure - you can view f(x) = x^2 as just another way of saying this. The x
itself is not important - what is important is the mapping itself. In set
theory we can rigourously define a function as a particular set using the
underlying concept of ordered pairs to tie all the from and to elements
together. (No variables involved here, either...)
>
> so now f(y) is just as important as f(x). They both denote the
> function's value at x or y.
>
> But when someone says,
>
> f : R -> R, where f(x) = x^2
>
> then goes onto say...
>
> df/dx = 2x this suggents that x is simehow importat to the function.
Your earlier view of this is better - best to view the above as nothing more
than an alternative way of stating what you said before.
>
> And if someone were to do this in the same context, people would be
> shocked:
>
> df/dy = 2y WRONG! f is a function of x!!!!
>
> Of course, the PRIME NOTATION is much better:
>
> f '(x) = 2x
> f '(y) = 2y <=== see how it allows for function f to be independent of
> variable
>
> or don't need variables at all:
>
> f '(0) = 0, f'(1) = 2, etc.
>
Agreed.
>
> So it seems as some notations tie the function closely to the variable
> it is defined with originally, while others make the function an
> independent thing (where derivative is with "respect to the argument"
> whatever that maybe be).
>
> This is most evident in the multivariable case:
>
> f : R^2 -> R, where f(x,y) = x^2 + y^2
>
> f_x (a,b) = 2a
>
> f_y (a,b) = 2b
>
> The very notation of partial derivatives suggest that the x,y are
> intrinsically tied to the function (which is why I prefer the D_1, D_2
> operation notations - which pisses off the markers).
>
> Also the deltaf/deltax notation also suggets that "x" is somehow
> improtant to f.
>
> So my question is. Are these variable important? Or are they dummy?
> Anyone ever experience the same confusion I've had (where I've shifted
> slowly into thinking that all notation should suggest independent of
> variables from functions).
Definitely best to view them as dummy. And the x in f_x(a,b) should be
taken as meaning nothing more than "partial derivative with respect to the
1st argument". I.e. the x isn't really there either.
I have been confused myself in the past, e.g. I think when I was dealing
with Lagrangians L(q_i, q_i') in mechanics I became confused for a while,
because we deal with partial derivatives with respect to q_i, and q_i'
(q_i' being d/dt(q_i), the rate of change of generalised coordinate q_i).
How could we differentiate wrt q_i', holding q_i constant etc.? Potentially
quite confusing until you step back and view L as a formal function in the
sense you note above, rather than as an expression made up of particular
variables... Anyway, I can sympathise with your confusions, but the fact
that you've seen the confusion suggests you'll also see your way through it
clearly in the end, and be a better maths person for it! (Worse to be one
of the many who never sees the distinctions and potential confusions in the
first place...)
Regards,
Mike.
David Bernier
Perhaps you'd prefer using D_1 (f) instead...
For 2-dimensional calculus, there are cartesian coordinates, but
also polar coordinates (rho, theta). In 3-dimensional calculus,
there are cartesian coordinates, cylindrical coordinates
and spherical coordinates (r, theta, phi).
In vector calculus, one has the "differential operators"
div, grad and curl.
If h: R^3 -> R, grad(h) is a vector field, a vector-valued
function. div operates on vector-valued functions to give
real-valued functions. div(grad(h)) is then a real-valued,
or scalar, function. The div (grad( [something])) operation, or
laplacian([something]), is used frequently in mathematical
physics. In applications, it's frequent to change coordinates,
say from cartesian to spherical. Then one talks of "transforming"
a function from cartesian to spherical coordinates. Strictly
speaking, one has a new function, but it's related to
the other one since spherical coordinates depend on
cartesian coordinates. The laplacian also gets
"transformed". It turns out that expressions for the laplacian in
cartesian, cylindrical and spherical coordinates are
quite dissimilar.
In the study and solution of the Schrodinger equation
for the hydrogen atom, people commonly use
spherical coordinates.
cf.:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/sch3d.html
[equation 2].
It would be possible to use D_1, D_2, D_3 in place
of del/(del r), del/(del theta) and del/(del phi).
However, I don't see any advantage.
One would have to constantly recall the
association
1 <-> r
2 <-> theta
3 <-> phi.
David Bernier
Dave L. Renfro
> I have some thoughts about variables and functions that
> have changed over time. When I was first learning calculus,
> I thought the variable (e.g., the x in f(x) = x^2) was
> essential in identifying the function throughout the
> context where the function is used.
[snip rest]
You might be interested in the text "Calculus: A Modern Approach"
by Karl Menger (Ginn, 1955, 354 pages). Menger was trying to
reform the notation used in elementary calculus texts in
the 1950's. I think his ideas began at least the decade
before this, and during these two decades he published several
papers in teaching/expository math journals about his ideas.
After coming across a few of these articles myself last winter,
I decided to look at his book. [I had to request it from
storage at the university library near me.] Menger's book
is certainly different, but it didn't really appeal to
me very much, although I don't know if I would have felt
differently when I was learning calculus. I guess I was
expecting that his text might have a number of novel topics,
like one finds in Spivak's elementary calculus book, but
Menger's text didn't have this.
Still, if you feel strongly about notation, you may find
it interesting to see what one fairly good mathematician,
one who had a strong interest in fixing these problems, did.
If Herman Rubin sees this, I'd be interested in what he
thinks of Menger's efforts.
http://scholar.google.com/scholar?q=Calculus-A-Modern-Approach+menger
http://books.google.com/books?q=Calculus-A-Modern-Approach+menger
http://www.google.com/search?q=Calculus-A-Modern-Approach+menger
Dave L. Renfro
David Marcus
> Aitken wrote:
>
> > I have some thoughts about variables and functions that
> > have changed over time. When I was first learning calculus,
> > I thought the variable (e.g., the x in f(x) = x^2) was
> > essential in identifying the function throughout the
> > context where the function is used.
>
> [snip rest]
>
> You might be interested in the text "Calculus: A Modern Approach"
> by Karl Menger (Ginn, 1955, 354 pages). Menger was trying to
> reform the notation used in elementary calculus texts in
> the 1950's. I think his ideas began at least the decade
> before this, and during these two decades he published several
> papers in teaching/expository math journals about his ideas.
> After coming across a few of these articles myself last winter,
> I decided to look at his book. [I had to request it from
> storage at the university library near me.] Menger's book
> is certainly different, but it didn't really appeal to
> me very much, although I don't know if I would have felt
> differently when I was learning calculus. I guess I was
> expecting that his text might have a number of novel topics,
> like one finds in Spivak's elementary calculus book, but
> Menger's text didn't have this.
How about Spivak's Calculus on Manifolds? I like the notation in it.
> Still, if you feel strongly about notation, you may find
> it interesting to see what one fairly good mathematician,
> one who had a strong interest in fixing these problems, did.
>
> If Herman Rubin sees this, I'd be interested in what he
> thinks of Menger's efforts.
>
> http://scholar.google.com/scholar?q=Calculus-A-Modern-Approach+menger
>
> http://books.google.com/books?q=Calculus-A-Modern-Approach+menger
>
> http://www.google.com/search?q=Calculus-A-Modern-Approach+menger
--
David Marcus
alainv...@yahoo.fr
David Marcus a écrit :
Bon Dimanche ,
in equation f(x^2) = f(x) + 1
what is dummy x , f , f(x) ?
in f(h(x)) ? ...
Alain
thero...@hotmail.com
> I have some thoughts about variables and functions that have changed
> over time. When I was first learning calculus, I thought the variable
> (e.g., the x in f(x) = x^2) was essential in identifying the function
> throughout the context where the function is used. But over time I
> came to think of the variable as something that was used only to show
> interdependencies between functions and to show the "rule" that the
> function takes.
The notation "f(x) = x^2" in this context can be thought as compressed
form of:
f is R -> R function such that for every x in R: f(x) = x^2.
There is only one such f.
Now, let's change the letters:
g is R -> R function such that for every y in R: g(y) = y^2.
There is only one such g, and g = f because f and g are the same
mapping.
The mappings do not contain symbolic information about the expression
with which they are defined as well as the numbers do not contain
symbolic information about the expression with witch they are defined.
> So taking some ideas from set theory, I think I can define a function
> like this:
>
> f : R -> R , where f maps each element of R to it's square in R
Here is another way to construct f without variables:
l is the identity function on R.
f = l * l
Theron
David Marcus
> Bon Dimanche ,
>
> in equation f(x^2) = f(x) + 1
> what is dummy x , f , f(x) ?
"f" is the name of the function. "x" is probably dummy, assuming you
haven't previously used "x".
> in f(h(x)) ? ...
That's not a full sentence.
--
David Marcus
David Marcus
In other words, without further context, "f", "h", and "x" should all be
assumed to be non-dummy.
--
David Marcus