ato said:
Fredrik said:
You don't take partial derivatives of variables. You take partial derivatives of functions.
i don't see the difference .
A
variable is a symbol that's used to represent a member of some set. A
constant is a variable that always represents the same thing. A
function should be thought of as a "rule" that associates exactly one member of a set (called the
codomain) with each member of a set (called the
domain). (This is
not a definition of "function". It's just an explanation of the concept. The actual definition is a bit tricky and not very relevant here. If you're curious, see
this post. Definition 2 is probably the most useful one).
Most functions can be defined by specifying a relationship between variables. For example, the specification x-y=1 implicitly defines two functions ##f,g:\mathbb R\to\mathbb R## that can also be defined by f(t)=1+t for all ##t\in\mathbb R##, and g(t)=1-t for all ##t\in\mathbb R##. (Note that it never matters what variable is used in a "for all" statement).
These functions are differentiable. For all ##t\in\mathbb R##, we have f'(t)=1 and ##g'(t)=-1##. Note that this makes f' and g'
constant functions, not to be confused with
constants as defined above.
The notation dy/dx doesn't refer to a derivative of y. It can't, because y isn't a function. The y in the numerator and the x in the denominator let's us know that we're supposed to compute the derivative of the function that "takes x to y" and then plug x into the result (which is another function) to get the final result. If x-y=1, then the function that "takes x to y" is the one I called f.
There's no such thing as a partial derivative of a variable. Typically, a math book will say something like this: Let n be a positive integer. Let E be a subset of ##\mathbb R^n##. Let x be an interior point of ##E##. Let ##\{e_1,e_2,\dots,e_n\}## be the standard basis for ##\mathbb R^n##. Let ##k\in\{1,2,\dots,n\}## be arbitrary. If there's a number A such that the limit
$$\lim_{t\to 0}\frac{f(x+te_k)-f(x)}{t}$$
exists, then this number is called the
kth partial derivative of f at x, or the "partial derivative of f at x, with respect to the kth variable". There are many different notations for it, for example ##D_kf(x)##, ##\partial_kf(x)##, ##\partial f(x)/\partial x_k##, ##f^{(k)}(x)## and ##f_{,\,k}(x)##.
Now, if n=2, we will usually write f(x,y) instead of f(x) (with ##x\in\mathbb R^2##) or ##f(x_1,x_2)##. Because x is traditionally put into the first variable "slot" of f, the notation ##\partial f(x,y)/\partial x## is an alternative to the five notations mentioned above (with k=1 and (x,y) replacing x).
ato said:
assuming for a function,all parameters are independent.
then you can't say
z(x,y)=x+y
because we don't know if x,y are independent of each other.
The thing you said that he can't say is "z(x, y) = x + y. It is a function of two parameters." Tac-tics was talking about the ##z:\mathbb R^2\to\mathbb R## defined by ##z(x,y)=x+y## for all ##x,y\in\mathbb R##. It never matters what variables are used in a "for all" statement, so we could have said "...defined by ##z(s,t)=s+t## for all ##s,t\in\mathbb R##." The definition of z doesn't in any way depend on some variables x and y, so there's nothing you can say about x and y that will change the function z.
On the other hand, if you say that x,y,z are variables representing real numbers, and then say that z=x+y, then this (i.e. the string of text "z=x+y") is a constraint that prevents us from assigning arbitrary values to all three variables, and also ensures that if we assign values to any two of them, the value of the third is fixed. This means that the constraint implicitly defines at least three functions, and we can compute the partial derivatives of those.
If you also specify that x=y, then this further reduces our ability to assign values to the three variables. Now if we assign a value to any
one of them, the values of the other two are fixed. So the pair of constraints (z=x+y, x=y) defines at least six functions implicitly, and we can compute the derivatives of those.
ato said:
so i don't understand why its false
$$\frac{\partial}{\partial t}f(x_{i},x_{j},x_{k},...) = \frac{\partial}{\partial t}f(x_{i})=\frac{d}{dt}f(x_{i})$$
The partial derivative notation doesn't really make sense here. Since t isn't one of the variables in the "slots" of f, there's no way to interpret ##\partial f/\partial t## as ##D_kf## for some k.
ato said:
as Tac-Tics suggested, i AM confused about function and variable.
That's OK. A lot of people are. I think the reason is that math books and math teachers don't explain the stuff that I said at the start of this post.
ato said:
1. in an equation, if a side is variable then the other side is also a variable.
I wouldn't say that. ##x^2=\pi## is an equation. Here x is a variable because it represents a real number, and ##\pi## is a constant because it represents a real number
and it represents that same real number when it appears in
other equations.
ato said:
2. f(x1,x2,x3...) is a short notation which says f is equal to some expression involving x1,x2,x3... . eg,
$$x+y=f(x,y)=f(x)=f(y)$$
more importantly f(x1,x2,x3...) does not assumes no relation between x1,x2,x3... . moer than one function notation can be formed out of a single expression.
f is a function. (x
1,x
2,x
3...) is a member of the domain of f. f(x
1,x
2,x
3...) is a member the codomain of f, called "the value of f at (x
1,x
2,x
3...)".
For example, if ##f:\mathbb R^2\to\mathbb R## is defined by ##f(x,y)=5x+y^2## for all ##x,y\in\mathbb R##, then f is a function, x and y are real numbers, (x,y) is an ordered pair of real numbers, i.e. a member of ##\mathbb R^2##, f(x,y) is a real number (but we only know which one if we know the values of x and y), f(7,3) is a real number (and we know that it's 44).
It is never OK to write f(x,y)=f(x), because the left-hand side only makes sense if the domain of f is a subset of ##\mathbb R^2## and the right-hand side only makes sense if the domain of f is a subset of ##\mathbb R##.
Note that the definition of a function is always a "for all" statement, even if the words "for all" have been omitted. For example, the words "the function ##x^2##" is strictly speaking nonsense. The correct way to say it is "the function ##f:\mathbb R\to\mathbb R## defined by ##f(x)=x^2## for all ##x\in\mathbb R##". Even the phrase "the function ##f(x)=x^2##" is flawed in
four(!) different ways: 1. Neither of the strings of text "f(x)=x
2" or "f(x)" represents a function. The function is denoted by f. 2. There's no specification of the domain. 3. There's no specification of the codomain. 4. The absence of the words "for all" hides the fact that x is a dummy variable (one that can be replaced by any other symbol without changing the meaning of the statement).