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## Main Question or Discussion Point

Is [tex]\frac{∂y}{∂x}×\frac{∂x}{∂z}=-\frac{∂y}{∂z}[/tex]?

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- #1

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Is [tex]\frac{∂y}{∂x}×\frac{∂x}{∂z}=-\frac{∂y}{∂z}[/tex]?

- #2

CompuChip

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No, the chain rule does not involve such a minus sign.

Why are you asking?

Why are you asking?

- #3

arildno

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In these cases, it is true that we have the counter-intuitive result:

[tex]\frac{\partial{Y}}{\partial{x}}\frac{\partial{X}}{\partial{z}}\frac{\partial{Z}}{\partial{y}}=-1[/tex]

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arildno

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Consider the function G(x,y,z)=x+y+z

Now, the condition G(x,y,z)=0 gives rise to the equation x+y+z=0

We may now form three separate function definitions:

X(y,z)=-y-z

Y(x,z)=-x-z

Z(x,y)=-x-y

We have now that:

G(X(y,z),y,z)=-y-z+y+z=0, i.e, the condition G=0 is satisfied IDENTICALLY, for all choices of y and z.

Similarly with the other two substitutions.

We see that in this case, that we have:

[tex]\frac{\partial{Y}}{\partial{x}}=\frac{\partial{X}}{\partial{z}}=\frac{\partial{Z}}{\partial{y}}=-1[/tex]

and therefore,

[tex]\frac{\partial{Y}}{\partial{x}}\frac{\partial{X}}{\partial{z}}\frac{\partial{Z}}{\partial{y}}-1[/tex]

- #5

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Try using the simple example z = x + yNo, the chain rule does not involve such a minus sign.

Why are you asking?

Isn't there a minus sign?

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HallsofIvy

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- #7

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Is there a general formula for partial derivatives or is it a collection of several formulas based on different conditions?

Anyway, consider a function w ( x, y, z ). We can then derive z ( x, y, w ).

In this case, is

[tex]\frac{∂w}{∂z}×\frac{∂z}{∂x}=-\frac{∂w}{∂x}?[/tex]

- #8

arildno

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"w ( x, y, z ). We can then derive z ( x, y, w )."

This is meaningless.

This is meaningless.

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