Mixing
Reciprocal relation derivation
$begingroup$
I will use a snippet of the work done on Andrew Rex's textbook Finn's Thermal Physics to illustrate my confusion.
What I find confusing is this:
First, we are to suppose what happens we consider x is a function of y and z and isn't itself a dependent variable - that is to say that setting y and z will evaluate x, which then will evaluate f. B.3 is fine.
However, we then consider y is ALSO not a dependent variable, and it is then plugged in for B.3.
However, in order for B.4 to be valid, it has to be a function of $x$ and $z$.
So, to me, it seems that for B.5 to be a valid equation, BOTH $x$ and $y$ must be DEPENDENT variables on $z$. However, it says that we are meant to consider only two of the variables being independent, despite considering what would be implied if $x$ and $y$ became dependent. Does this make sense? I'm wondering how we can assume this equation still implies two independent variables despite considering what would happen if two of the variables $F$ is a function of were given dependencies, and it looks like a contradiction.
multivariable-calculus
asked Jan 29 at 20:18
sangstarsangstar
853215
$endgroup$
add a comment |
$begingroup$
I will use a snippet of the work done on Andrew Rex's textbook Finn's Thermal Physics to illustrate my confusion.
What I find confusing is this:
First, we are to suppose what happens we consider x is a function of y and z and isn't itself a dependent variable - that is to say that setting y and z will evaluate x, which then will evaluate f. B.3 is fine.
However, we then consider y is ALSO not a dependent variable, and it is then plugged in for B.3.
However, in order for B.4 to be valid, it has to be a function of $x$ and $z$.
So, to me, it seems that for B.5 to be a valid equation, BOTH $x$ and $y$ must be DEPENDENT variables on $z$. However, it says that we are meant to consider only two of the variables being independent, despite considering what would be implied if $x$ and $y$ became dependent. Does this make sense? I'm wondering how we can assume this equation still implies two independent variables despite considering what would happen if two of the variables $F$ is a function of were given dependencies, and it looks like a contradiction.
multivariable-calculus
asked Jan 29 at 20:18
sangstarsangstar
853215
$endgroup$
add a comment |
$begingroup$
I will use a snippet of the work done on Andrew Rex's textbook Finn's Thermal Physics to illustrate my confusion.
What I find confusing is this:
First, we are to suppose what happens we consider x is a function of y and z and isn't itself a dependent variable - that is to say that setting y and z will evaluate x, which then will evaluate f. B.3 is fine.
However, we then consider y is ALSO not a dependent variable, and it is then plugged in for B.3.
However, in order for B.4 to be valid, it has to be a function of $x$ and $z$.
So, to me, it seems that for B.5 to be a valid equation, BOTH $x$ and $y$ must be DEPENDENT variables on $z$. However, it says that we are meant to consider only two of the variables being independent, despite considering what would be implied if $x$ and $y$ became dependent. Does this make sense? I'm wondering how we can assume this equation still implies two independent variables despite considering what would happen if two of the variables $F$ is a function of were given dependencies, and it looks like a contradiction.
multivariable-calculus
asked Jan 29 at 20:18
sangstarsangstar
853215
$endgroup$
I will use a snippet of the work done on Andrew Rex's textbook Finn's Thermal Physics to illustrate my confusion.
What I find confusing is this:
First, we are to suppose what happens we consider x is a function of y and z and isn't itself a dependent variable - that is to say that setting y and z will evaluate x, which then will evaluate f. B.3 is fine.
However, we then consider y is ALSO not a dependent variable, and it is then plugged in for B.3.
However, in order for B.4 to be valid, it has to be a function of $x$ and $z$.
So, to me, it seems that for B.5 to be a valid equation, BOTH $x$ and $y$ must be DEPENDENT variables on $z$. However, it says that we are meant to consider only two of the variables being independent, despite considering what would be implied if $x$ and $y$ became dependent. Does this make sense? I'm wondering how we can assume this equation still implies two independent variables despite considering what would happen if two of the variables $F$ is a function of were given dependencies, and it looks like a contradiction.
multivariable-calculus
multivariable-calculus
asked Jan 29 at 20:18
sangstarsangstar
853215
asked Jan 29 at 20:18
sangstarsangstar
853215
asked Jan 29 at 20:18
sangstarsangstar
853215
asked Jan 29 at 20:18
sangstarsangstar
853215
asked Jan 29 at 20:18
sangstarsangstar
853215
853215
add a comment |
add a comment |
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