Post History
#2: Post edited
by
TextKit
·
2021-03-09T05:48:35Z (almost 4 years ago)
- [Hagen von Eitzen answered](https://math.stackexchange.com/a/738461/53259) that $z_1, z_2$
- >depend on only one variable - there's no comma between the parentheses.
- [John Doe commented](https://math.stackexchange.com/questions/2780026/why-arent-z-1-fxy-and-z-2-fx-y-multivariable#comment5733224_2780026)
- >the function $f(xy)=e^{xy}sin(xy)+(xy)^3$ may look like a multivariable function in x and y, but it can be written more simply as a univariate function, $f(t)=e^tsint+t^3$.
Yes, I agree that defining $t_1=xy$ transforms $f(xy)$ into $f(t_1)$, and $t_2= \dfrac xy$ transforms $f(x/y)$ into $f(t_2)$. Yes, $f(t_1)$ and $f(t_2)$ are single-variable. I agree that $z_1, z_2$ can be re-defined to be functions of ONE variable.- But I'm still befuddled why $z_1, z_2$ can't be construed as TWO variables. I feel that redefining with $t$ misrepresents the original 2 independent variables x, y. These definitions don't change the original fact that f did depend on 2 independent variables!
- [Hagen von Eitzen answered](https://math.stackexchange.com/a/738461/53259) that $z_1, z_2$
- >depend on only one variable - there's no comma between the parentheses.
- [John Doe commented](https://math.stackexchange.com/questions/2780026/why-arent-z-1-fxy-and-z-2-fx-y-multivariable#comment5733224_2780026)
- >the function $f(xy)=e^{xy}sin(xy)+(xy)^3$ may look like a multivariable function in x and y, but it can be written more simply as a univariate function, $f(t)=e^tsint+t^3$.
- Yes, I agree that defining $t_1=xy$ transforms $f(xy)$ into $f(t_1)$, and $t_2= \dfrac xy$ transforms $f(\dfrac xy)$ into $f(t_2)$. Yes, $f(t_1)$ and $f(t_2)$ are single-variable. I agree that $z_1, z_2$ can be re-defined to be functions of ONE variable.
- But I'm still befuddled why $z_1, z_2$ can't be construed as TWO variables. I feel that redefining with $t$ misrepresents the original 2 independent variables x, y. These definitions don't change the original fact that f did depend on 2 independent variables!
#1: Initial revision
by
TextKit
·
2021-03-09T05:48:14Z (almost 4 years ago)
Why aren't $z_1=f(xy)$ and $z_2=f(x/y)$ functions of 2 variables?
[Hagen von Eitzen answered](https://math.stackexchange.com/a/738461/53259) that $z_1, z_2$
>depend on only one variable - there's no comma between the parentheses.
[John Doe commented](https://math.stackexchange.com/questions/2780026/why-arent-z-1-fxy-and-z-2-fx-y-multivariable#comment5733224_2780026)
>the function $f(xy)=e^{xy}sin(xy)+(xy)^3$ may look like a multivariable function in x and y, but it can be written more simply as a univariate function, $f(t)=e^tsint+t^3$.
Yes, I agree that defining $t_1=xy$ transforms $f(xy)$ into $f(t_1)$, and $t_2= \dfrac xy$ transforms $f(x/y)$ into $f(t_2)$. Yes, $f(t_1)$ and $f(t_2)$ are single-variable. I agree that $z_1, z_2$ can be re-defined to be functions of ONE variable.
But I'm still befuddled why $z_1, z_2$ can't be construed as TWO variables. I feel that redefining with $t$ misrepresents the original 2 independent variables x, y. These definitions don't change the original fact that f did depend on 2 independent variables!