Chain rule / Taylor expansion / functional derivative

In summary, the conversation discusses how to prove that when ##\rho(p',s)>\rho(p',s')##, then ##(\frac{\partial\rho}{\partial s})_p\frac{ds}{dz}<0##. The suggested approach is to use a Taylor expansion of ## \rho(p',s') ## in just the s variable about s, keeping in mind that s and s' differ by a differential amount. Furthermore, it is noted that the inequality may need to be multiplied by a "dz" term, which does not necessarily need to be positive.
  • #1
binbagsss
1,259
11

Homework Statement



To show that ##\rho(p',s)>\rho(p',s') => (\frac{\partial\rho}{\partial s})_p\frac{ds}{dz}<0##
where ##p=p(z)##, ##p'=p(z+dz)##, ##s'=s(z+dz)##, ##s=s(z)##

Homework Equations



I have no idea how to approach this. I'm thinking functional derivatives, taylor expansions, but if someone could please give me a clue where to start.

The Attempt at a Solution


as above.

many thanks in advance.
 
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  • #2
Suggest a Taylor expansion of ## \rho(p',s') ## in just the s variable about s. One comment is since s and s' differ by a differential amount, the the terms of the first inequality differ by only a differential amount. I think the inequality you are trying to prove actually needs a "dz" tacked on to (multiplying) the product of the derivatives. I would be interested in seeing if other readers concur. "dz" does not need to be positive.
 

Related to Chain rule / Taylor expansion / functional derivative

What is the chain rule in calculus?

The chain rule is a fundamental rule in calculus that allows you to find the derivative of a composite function. It states that the derivative of a composite function is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

What is Taylor expansion?

Taylor expansion is a mathematical technique used to approximate a function with a polynomial. It is based on the idea that any smooth function can be represented by a polynomial of infinite degree, and by taking more and more terms of the polynomial, we can get a better approximation of the original function.

What is a functional derivative?

A functional derivative is a mathematical concept used in functional analysis and calculus of variations. It is the analogue of the derivative in ordinary calculus and is defined as the rate of change of a functional with respect to a function. It is used to optimize functionals, which are functionals that take functions as inputs and output a real number.

Why is the chain rule important?

The chain rule is important because it allows us to find the derivatives of composite functions, which are functions that are composed of multiple functions. This is essential in many areas of mathematics and science, such as physics, engineering, and economics, where functions are often composed of smaller functions.

How is Taylor expansion used in science?

Taylor expansion is used in science to approximate complex functions and make them easier to analyze and understand. It is commonly used in physics, chemistry, and engineering to model and solve problems that involve non-linear functions. It is also used in statistics to estimate the values of unknown parameters in a model.

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