Linked Questions

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1answer
40 views

how to tackle this partial differentiation problem [duplicate]

show that if $$f(x, y, z)=0$$ then $$\left ( \partial x \over \partial y \right )_{z}\left ( \partial y \over \partial z \right )_{x}\left ( \partial z \over \partial x \right )_{y}=-1$$ I don't know ...
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0answers
40 views

Applying the Implicit Function Theorem from R3 to R [duplicate]

Suppose $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ is such that the Implicit Function Theorem applies to $F(x,y,z) = 0$ to determine $z = f(x,y)$, $x=g(y,z)$ and $y=h(x,z)$ in a neighborhood of a point $...
4
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1answer
11k views

Proof on showing if F(x,y,z)=0 then product of partial derivatives (evaluated at an assigned coordinate) is -1

The task is as follows: Given: $$F(x,y,z) = 0$$ Goal: Show $\frac{\partial z}{\partial y}|_x \frac{\partial y}{\partial x}|_z \frac{\partial x}{\partial z} |_y = -1$ Here is my work so ...
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2answers
221 views

What is meant by $\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=-1$ ? How to interpret it?

Let $F(x,y,z)=0$. So $x,y,z$ are defined implicitly in function of the other variable, i.e. $x=x(y,z)$, $y=y(x,z)$ and $z=z(x,y)$. Now $$dx=\frac{\partial x}{\partial y}dy+\frac{\partial x}{\partial z}...
3
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2answers
194 views

What exactly is meant by $\frac {dv}{dt} = \frac {dv}{dx}\frac {dx}{dt}$?

I'm a bit embarrased to have to ask this, as I guess I'm missing something completely basic: I've seen various physics problems solved using $\frac {dv}{dt} ``=" \frac {dv}{dx}\frac {dx}{dt}$, but I'm ...
1
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1answer
363 views

Partial derivatives in circular permutation

So, we know from thermodynamics that (dy/dx)(dx/dz)(dz/dy), where the d's represent partial derivatives, is equal to -1, provided that z is a function of x and y. There are several proofs of that. My ...
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2answers
98 views

Un-proving $1=-1$ in the development of the implicit mapping theorem. Edwards: Advanced Calculus of Several Variables.

The follow conundrum arose while attempting to translate to tensor notation the development of the implicit mapping theorem in C.H. Edwards, Jr.'s Advanced Calculus of Several Variables. I refer to ...
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1answer
168 views

Show $\bigg(\frac{\partial x}{\partial y}\bigg) \bigg(\frac{\partial y}{\partial z}\bigg)\bigg(\frac{\partial z}{\partial x}\bigg) = - 1,$ [duplicate]

Question statement: Let $F(x,y,z)$ is a continuously differentiable function with nonvanishing partials at $(0,0,0).$ Define $x = x(y,z), \; y = y(x,z), \; z = z(x,y)$ as the solutions of the ...
2
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1answer
81 views

Why cant you cancel out partial derivatives like fractions here?

So we were asked that if $F(x,y,z)=0$, show $\frac{\partial z}{\partial y}|_x \frac{\partial y}{\partial x}|_z \frac{\partial x}{\partial z} |_y = -1$. Now I had to go about and show that $\frac{\...
2
votes
1answer
80 views

Given $F(x,y,z)=0$, $\frac{\partial x}{\partial y} \frac{\partial y}{\partial z}\frac{\partial z}{\partial x}$ should be -1, but I got 1

What went wrong in my calculation? Given $F(x,y,z)=0$ , $F_x\neq 0 , F_y\neq 0, F_z\neq 0$, and $z=f(x,y), y=g(x,z), x=h(y,z)$ $F_x\quad\,+F_y\frac{\partial y}{\partial x} + F_z \frac{\partial z}{\...
1
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0answers
115 views

Confusion about Partial Derivatives

I am doing some self-study in math. The problem below is from a Calculus text book. Problem: Establish the fact, widely used in hydrodynamics, that if $f(x,y,z) = 0$, then \begin{eqnarray*} \Big(\...
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2answers
40 views

Differentials and their meaning

For the equation $$PV=nRT$$ where $nR$ is constant $$\frac{\partial P}{\partial V}\cdot\frac{\partial V}{\partial T}\cdot\frac{\partial T}{\partial P} = -1$$ If we were to simplify this, it implies ...
0
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1answer
48 views

Cancelling differential terms

Can we cancel two differential terms while they are in a ratio. For example if we have (dx/dt) / (dy/dt), can we just directly cancel dt by dt and write it as dx/dy. I mean is is this step allowed?