Solve Partial Derivatives: Chain Rule Explained

[jesuslovesu]

[SOLVED] Partial Derivatives

Whoops, never mind my calculus book explained it.

Homework Statement

F(x,y,z) = 0
$$(\frac{\partial x}{\partial y})\right)_{z} (\frac{\partial y}{\partial x})\right)_{z}$$ = 0

Show
$$(\frac{\partial x}{\partial y})\right)_{z} (\frac{\partial y}{\partial z})\right)_{x} (\frac{\partial z}{\partial x})\right)_{y}$$ = -1

The Attempt at a Solution

Well I drew out a diagram
F
dF/dx dF/dy dF/dz

dx/dy dx/dz dy/dx dy/dz dz/dx dz/dy

So I would assume that the reason the second expression is = -1 because of the chain rule, however, I really don't see why it would be -1...
If it were dx/dy dy/dx dz/dx would it be 1? Just because the middle term is dy/dz the sign will change?

 

[mighty2000]

Hello there,

Thank you for your post. It seems like you are on the right track with your understanding of the chain rule. Let me explain why the second expression is equal to -1.

First, let's rewrite the second expression using the chain rule:

(\frac{\partial x}{\partial y})\right)_{z} (\frac{\partial y}{\partial z})\right)_{x} (\frac{\partial z}{\partial x})\right)_{y} = (\frac{\partial x}{\partial y} \frac{\partial y}{\partial z} \frac{\partial z}{\partial x})\right)_{y}

Now, let's look at the partial derivatives in the parentheses. We have dx/dy, dy/dz, and dz/dx. Notice that when we multiply these three terms together, we get (-1)(-1)(1) = 1. This is because when we take the derivative of x with respect to y, we are essentially taking the derivative of x with respect to z and then taking the derivative of z with respect to y. This introduces a negative sign in the chain rule.

So, why is the final expression equal to -1? This is because we are taking the partial derivatives with respect to x, y, and z in a specific order - z, x, y. This means that when we take the derivative of x with respect to y, we are essentially taking the derivative of x with respect to z and then taking the derivative of z with respect to y. This introduces a negative sign in the chain rule. Similarly, when we take the derivative of y with respect to z, we are essentially taking the derivative of y with respect to x and then taking the derivative of x with respect to z. This also introduces a negative sign in the chain rule. Finally, when we take the derivative of z with respect to x, we are essentially taking the derivative of z with respect to y and then taking the derivative of y with respect to x. This does not introduce a negative sign in the chain rule.

So, when we multiply all these partial derivatives together, we get (-1)(-1)(1) = 1. However, since we are taking the partial derivatives in a specific order - z, x, y - the final expression is equal to -1.

I hope this explanation helps. Let me know if you have any further questions.