Triple Product Rule Equivalency

In summary, the equations ##(\frac {\partial p} {\partial v})_T (\frac {\partial v} {\partial T})_p (\frac {\partial T} {\partial p})_v = -1## and ##(\frac {\partial p} {\partial T})_v (\frac {\partial v} {\partial p})_T (\frac {\partial T} {\partial v})_p = -1## are equivalent. The choice of which variable to take derivatives with respect to does not affect the result, as long as the cyclic rule is followed.
  • #1
WhiteWolf98
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Homework Statement
I've been trying to apply the cyclic rule to the ideal gas law, and got a different answer to the one given. I was wondering if the two solutions are equivalent; knowing this would be a great help for me, as the derivation of the Maxwell equations are next ~
I'm rather new to partial derivatives, so apologies if this is a silly question
Relevant Equations
##(\frac {\partial x} {\partial y})_z (\frac {\partial y} {\partial z})_x (\frac {\partial z} {\partial x})_y = -1##
##p=\frac {RT} v;~p=p(T,v)~...1##

##v=\frac {RT} p;~v=v(T,p)~...2##

##T=\frac {pv} R;~T=T(p,v)~...3##

##Considering~eq.~1:##

##p=\frac {RT} v \Rightarrow (\frac {\partial p} {\partial v})_T=-\frac {RT} {v^2}##

##Considering~eq.~2:##

##v=\frac {RT} p \Rightarrow (\frac {\partial v} {\partial t})_p=\frac R p##

##Considering~eq.~3:##

##T=\frac {pv} R \Rightarrow (\frac {\partial T} {\partial p})_v=\frac v R##

##(\frac {\partial p} {\partial v})_T (\frac {\partial v} {\partial T})_p (\frac {\partial T} {\partial p})_v =-\frac {RT} {v^2} \cdot \frac R p \cdot \frac v R= -\frac {RT} {pv} = -1 ##

That's the first, given solution. I was able to get to this on my own, but there's also another I got to in a similar way:

##(\frac {\partial p} {\partial T})_v (\frac {\partial v} {\partial p})_T (\frac {\partial T} {\partial v})_p = -\frac {RT} {pv} = -1 ##

I'd like to know if these two final equations are the same; and if not, why aren't they? I'm finding it a little difficult to find a pattern with the triple product rule. I understand that there has to be a partial derivative of each variable, but is the second part (what the partial derivative is with respect to), random? With three variables, the partial derivative of a variable can be with respect to two other variables; how do you choose these? If one wants to solve for a general solution, is there any convention/ procedure to follow?
 
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  • #2
The two equations are the same. In the first case you considered the partial derivative ##(\frac {\partial p} {\partial v})_T##, that is you took the derivative with respect to the second variable in ##p=p(T,v)## keeping the first variable constant. You could just as well have written ##p=p(v,T)## and following the same procedure would have given you ##(\frac{\partial p} {\partial T})_v## which is what you have in the second case. Note that once you write the first derivative, the other two follow from the cyclic rule, so whether you write ##v=v(T,p)## or ##v=v(p,T)## makes no difference to the product. Which variable you choose to take derivatives with respect to depends on what you are trying to show or derive.
 
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  • #3
I think that answers my question. So the first derivative, in this case, will decide what the other two are with respect to. Once you've decided the first, there is only one choice for what the second and third variable could be with respect to; and as you add variables, you can have more and more, 'combinations' as such, but which are still all equivalent.
 

FAQ: Triple Product Rule Equivalency

What is the Triple Product Rule Equivalency?

The Triple Product Rule Equivalency is a mathematical rule used in vector calculus to simplify the calculation of the derivative of a vector triple product. It states that the derivative of a vector triple product is equal to the sum of the individual triple products of the derivatives of the original vectors.

How is the Triple Product Rule Equivalency used in physics?

In physics, the Triple Product Rule Equivalency is used to simplify the calculation of momentum and angular momentum in rotational motion problems. It allows for a more efficient and concise way of calculating these quantities, which are important in understanding the dynamics of objects in motion.

What are the applications of the Triple Product Rule Equivalency?

The Triple Product Rule Equivalency has various applications in mathematics, physics, and engineering. It is commonly used in vector calculus, mechanics, and electromagnetism to simplify complex calculations involving vector quantities. It can also be applied in computer graphics and robotics for motion planning and control.

Is the Triple Product Rule Equivalency always valid?

Yes, the Triple Product Rule Equivalency is a fundamental rule in vector calculus and is always valid. It is based on the properties of vector operations, such as the distributive and associative laws, and can be mathematically proven to be true.

What are some common mistakes when applying the Triple Product Rule Equivalency?

One common mistake when using the Triple Product Rule Equivalency is forgetting to take into account the order of the vectors. The order in which the vectors are multiplied can affect the final result, so it is important to follow the correct order when applying the rule. Another mistake is confusing the Triple Product Rule Equivalency with the Product Rule, which is used for calculating the derivative of a product of two functions.

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