- #1

- 735

- 188

- Summary:
- When we restrict the domain a function then why it is no longer a function of ##x## and ##y##?

Good Morning!

If I have function of two variables ##f(x,y)## and if we write it like this $$ z = f(x,y)$$ then it means that for every point in the ##xy## plane there is a point above/below it and is related to it by ##f##. In simple words, every point in ##xy## plane has a point associated with it. We can say that ##f## in this case is a function of both ##x ~\textrm{and}~y##, because ##f## is varying as ##x ~\textrm{and}~y## are changing, and consequently it's derivative with respect to ##x## or ##y## is not zero.

Now, if we imagine that ##z=f(x,y)## but only those ##x~\textrm{and}~ y## are allowed which lie in the defined rectangular region:

let's assume that the corners of the rectangle have coordinates: ##(2,2) ; (4,2) ; (4,4) ; (2,4)##. And let's say any point lying in that rectangle is represented as ##(a, b)##. So, the function now looks like this $$ z= f(a, b)$$. My problem is: why ##f## is no longer a function of ##x ~\textrm{and}~y## . After all, ##a~\textrm{and}~y## are just the different symbols, we can even write our function as $$ z = f(x,y) ~~~~~~~~~~~~\bigg\{ 2\lt x,y \lt 4 $$ So, why ##f## is no longer a function of ##x## and ##y##, and why is the derivative of ##f## (I mean after the restriction is defined) is zero?

If I have function of two variables ##f(x,y)## and if we write it like this $$ z = f(x,y)$$ then it means that for every point in the ##xy## plane there is a point above/below it and is related to it by ##f##. In simple words, every point in ##xy## plane has a point associated with it. We can say that ##f## in this case is a function of both ##x ~\textrm{and}~y##, because ##f## is varying as ##x ~\textrm{and}~y## are changing, and consequently it's derivative with respect to ##x## or ##y## is not zero.

Now, if we imagine that ##z=f(x,y)## but only those ##x~\textrm{and}~ y## are allowed which lie in the defined rectangular region:

let's assume that the corners of the rectangle have coordinates: ##(2,2) ; (4,2) ; (4,4) ; (2,4)##. And let's say any point lying in that rectangle is represented as ##(a, b)##. So, the function now looks like this $$ z= f(a, b)$$. My problem is: why ##f## is no longer a function of ##x ~\textrm{and}~y## . After all, ##a~\textrm{and}~y## are just the different symbols, we can even write our function as $$ z = f(x,y) ~~~~~~~~~~~~\bigg\{ 2\lt x,y \lt 4 $$ So, why ##f## is no longer a function of ##x## and ##y##, and why is the derivative of ##f## (I mean after the restriction is defined) is zero?