Problem 45 - An Introduction to Thermal Physics

Question

Problem 1.45. As an illustration of why it matters which variables you hold fixed when taking partial derivatives, consider the following mathematical example. Let \(w=x y\) and \(x=y z\) (a) Write \(w\) purely in terms of \(x\) and \(z,\) and then purely in terms of \(y\) and \(z\) (b) Compute the partial derivatives $$ \left(\frac{\partial w}{\partial x}\right)_{y} \quad \text { and } \quad\left(\frac{\partial w}{\partial x}\right)_{z} $$ and show that they are not equal. (Hint: To compute \((\partial w / \partial x)_{y},\) use a formula for \(w\) in terms of \(x\) and \(y,\) not \(z .\) Similarly, compute \((\partial w / \partial x)_{z}\) from a formula for \(w\) in terms of only \(x\) and \(z .\) ) (c) Compute the other four partial derivatives of \(w\) (two each with respect to \(y \text { and } z),\) and show that it matters which variable is held fixed.

Step-by-Step Solution

Answer

The different partial derivatives confirm that which variable is held constant affects the result, illustrating the significance of selected held variables.

Step 1: Express w in Terms of x and z

Given the relations \( w = xy \) and \( x = yz \), we can substitute \( x \) in the expression for \( w \): \[ w = (yz)y = y^2z \]This expresses \( w \) purely in terms of \( y \) and \( z \).

Step 2: Express w in Terms of y and z

Based on the substitution from step one, \( w = y^2 z \) is already in terms of \( y \) and \( z \). There's no need to express it again, as it's computed in Step 1.

Step 3: Compute \( \left(\frac{\partial w}{\partial x}\right)_{y} \)

Using the formula for \( w = xy \), the derivative of \( w \) with respect to \( x \) while holding \( y \) constant is:\[ \left(\frac{\partial w}{\partial x}\right)_{y} = \frac{\partial (xy)}{\partial x} = y \]

Step 4: Compute \( \left(\frac{\partial w}{\partial x}\right)_{z} \)

Using the formula \( w = \frac{x^2}{z} \) based on the relation \( x = yz \), express \( y \) in terms of \( x \) and \( z \) (\( y = \frac{x}{z} \)) and substitute in \( w \):\( w = x\left(\frac{x}{z}\right) = \frac{x^2}{z} \)Then, the partial derivative is:\[ \left(\frac{\partial w}{\partial x}\right)_{z} = \frac{\partial}{\partial x}\left(\frac{x^2}{z}\right) = \frac{2x}{z} \]Thus, \( \left(\frac{\partial w}{\partial x}\right)_{y} eq \left(\frac{\partial w}{\partial x}\right)_{z} \).

Step 5: Compute Partial Derivatives with Respect to y

For \( w = xy \), \( \left(\frac{\partial w}{\partial y}\right)_{x} = x \). For \( w = y^2z \), \( \left(\frac{\partial w}{\partial y}\right)_{z} = 2yz \).

Step 6: Compute Partial Derivatives with Respect to z

From \( w = \frac{x^2}{z} \), the partial derivative of \( w \) with respect to \( z \) is:\[ \left(\frac{\partial w}{\partial z}\right)_{x} = -\frac{x^2}{z^2} \]From \( w = y^2z \), the partial derivative with respect to \( z \) is:\[ \left(\frac{\partial w}{\partial z}\right)_{y} = y^2 \] This again shows how the fixed variable impacts the value of the partial derivative.

Key Concepts

Mathematical Relations

Mathematical relations are fundamental in understanding how different quantities interact with each other. In this exercise, we're dealing with the equations \(w = xy\) and \(x = yz\). These relations imply a connection between the three variables \(x\), \(y\), and \(z\), which in turn defines \(w\) in terms of these variables.

When working with such equations, your first task is often to express one variable in terms of others to simplify the calculations. In our problem, we first express \(w\) in terms of \(x\) and \(z\), then in terms of \(y\) and \(z\). This exercise illustrates how flexibility in choice of expression can later aid in derivative computation.

Variables

Variables, in mathematics, are symbols used to represent numbers or values. They are essential for representing functions and expressing relationships between quantities. In this exercise, \(w\), \(x\), \(y\), and \(z\) are our variables. Each represents a specific measurement or value in the context of the relations given.

Understanding the role each variable plays is crucial. For instance, when computing partial derivatives, knowing which variables should be held constant and which can change is critical. In \(\left(\frac{\partial w}{\partial x}\right)_y\), \(y\) is held constant, whereas for \(\left(\frac{\partial w}{\partial x}\right)_z\), \(z\) is the constant variable. This distinction is what gives the two partial derivatives different values.

Derivative Computation

Derivative computation involves finding how a function changes as its input changes. In simpler terms, it's about calculating rates of change. A partial derivative, such as \(\left(\frac{\partial w}{\partial x}\right)_y\), focuses on the rate of change of a function with respect to one variable while keeping others constant.

The process in this exercise shows two different methods based on two different expressions for \(w\). Holding \(y\) constant utilizes \(w = xy\), which gives \(\left(\frac{\partial w}{\partial x}\right)_y = y\). On the other hand, holding \(z\) constant uses \(w = \frac{x^2}{z}\), resulting in \(\left(\frac{\partial w}{\partial x}\right)_z = \frac{2x}{z}\). The different expressions of \(w\) lead to differing results for the partial derivatives, highlighting the importance of the chosen relation for computation.

Function Representation

Function representation allows one to identify the dependency between variables. By choosing which variables to express a function with, you gain insights into how changes in one affect another. Representing \(w\) solely in terms of \(y\) and \(z\) as \(w = y^2z\) or in terms of \(x\) and \(z\) as \(w = \frac{x^2}{z}\) gives detailed perspectives of the relationship between those variables.

How a function is represented significantly influences the derivative outcome. Each representation provides a unique lens on the problem, which can drastically change the solution process and its results. This is a powerful tool in multivariable calculus, offering deeper understanding and flexibility in tackling differential problems.