Hey everyone I have a question on the proof of Implicit Function Theorem. My book says, "Suppose that z is given implicitly as a function z = f(x,y) by an equation of the form F(x,y,z) = 0. This means that F(x,y,f(x,y)) = 0 for all (x,y) in the domain of f. If F and f are differentiable, then we can use the Chain Rule to differentiate the equation F(x,y,z) = 0 as follows:
But and
So the equation becomes
My question is, how does ? It doesn't make sense. Is it just saying that let/assume ? Confuseddd.
Thanks in advance!
Sorry I should have made it clearer. I think a main point of confusion here is that we're dealing with partial derivatives (i.e. dy/dx is a partial derivative, so is dz/dx)
What does it mean that z is defined implicitly as a function of x and y?
For your equation, it means z must be a function of x and y that satisfies 3x + y + z = 0. So
z = -(3x + y). That is the explicit equation of your F(x,y,f(x,y)) = 0
Now note that the derivative
So if we go with what you suggested, that we solve F(x,y,f(x,y)) = 0 for y
y = -3x - z.
differentiating with respect to x gives you
Do you see now why will always be zero?
:O! Yes it does!
This seems a bit recursive though. We're basically saying:
y = -3x - z, where z = -(3x+y) and by direct substitution, we get y = -3x + (3x+y) = y
The partial derivative of y = y with respect to x is 0.
In that case, wouldn't dx/dx equal to 0 as well?
It is a bit circular, but you would expect it to be.
After all, F(x,y,f(x,y)) = 3x + y + f(x,y) = 0
describes the exact same thing as f(x,y) = -3x - y
Interchanging the two would seem to be recursive.
As for dx/dx, dx/dx always equals one.
Lets try this again on your example
3x = -y -z
To see what I mean, I'll give a worded illustration. It is easy to get caught up with just the maths and forget why we study it in the first place.
Say we have a container filled with two miscible liquids (two liquids which mix evenly), Suppose z is the amount of liquid flowing out of a container, x be the amount of liquid A flowing out, y be the amount of liquid B flowing out.
Now dy/dz means for every amount dz of total liquid flowing out, a dy amount of liquid B flows out. Similarly, dx/dz is the proportion of liquid A compared to the combined liquid
What does it mean to have dz/dz, dx,dx or dy/dy? For every dz amount of total liquid flowing out, you'll have a dz amount of liquid flowing out. The proportion between total liquid out/ total liquid out is 1.