Problem 259 - Calculus 3

Question

Let \(u=u(x, y, z)\) where \(x=x(w, t), y=y(w, t), z=z(w, t), w=w(r, s),\) and \(t=t(r, s)\) Use a tree diagram and the chain rule to find an expression for \(\frac{\partial u}{\partial r}\)

Step-by-Step Solution

Answer

Use the chain rule to sum products of partial derivatives from \( u \) to \( r \) via the intermediate variables.

Step 1: Identify Dependencies

We begin by identifying how each variable is functionally related. We have:- \( u = u(x, y, z) \)- \( x = x(w, t) \)- \( y = y(w, t) \)- \( z = z(w, t) \)- \( w = w(r, s) \)- \( t = t(r, s) \)

Step 2: Draw the Tree Diagram

Create a tree diagram to visualize the dependencies:- The root node is \( u \).- Branch out to \( x \), \( y \), and \( z \) which connect to \( u \).- Each of \( x \), \( y \), and \( z \) branches to \( w \) and \( t \).- \( w \) and \( t \) both depend on \( r \) and \( s \).This diagram helps us track the chain of derivatives.

Step 3: Apply the Chain Rule

According to the chain rule, the partial derivative \( \frac{\partial u}{\partial r} \) is the sum of products of partial derivatives along each path from \( u \) to \( r \) through the interconnected variables:\[ \frac{\partial u}{\partial r} = \frac{\partial u}{\partial x} \cdot \frac{\partial x}{\partial w} \cdot \frac{\partial w}{\partial r} + \frac{\partial u}{\partial y} \cdot \frac{\partial y}{\partial w} \cdot \frac{\partial w}{\partial r} + \frac{\partial u}{\partial z} \cdot \frac{\partial z}{\partial w} \cdot \frac{\partial w}{\partial r} + \frac{\partial u}{\partial x} \cdot \frac{\partial x}{\partial t} \cdot \frac{\partial t}{\partial r} + \frac{\partial u}{\partial y} \cdot \frac{\partial y}{\partial t} \cdot \frac{\partial t}{\partial r} + \frac{\partial u}{\partial z} \cdot \frac{\partial z}{\partial t} \cdot \frac{\partial t}{\partial r} \]

Step 4: Write the Expression

Combine all terms from Step 3:\[ \frac{\partial u}{\partial r} = \sum_{cyc} \left(\frac{\partial u}{\partial x_i} \cdot \frac{\partial x_i}{\partial w} \cdot \frac{\partial w}{\partial r} + \frac{\partial u}{\partial x_i} \cdot \frac{\partial x_i}{\partial t} \cdot \frac{\partial t}{\partial r}\right)\]- The expression shows the derivative of \( u \) with respect to \( r \) as the sum of products of the derivatives, using the chain rule.

Key Concepts

Partial Derivatives

Partial derivatives are used in calculus when dealing with functions of two or more variables. They measure how a function changes as each variable changes individually, while keeping the other variables constant. This is crucial when we have a multivariable function like \( u = u(x, y, z) \), where each component \( x, y, \) and \( z \) impacts \( u \).

To find a partial derivative, such as \( \frac{\partial u}{\partial x} \), we assume \( y \) and \( z \) are constants, and only \( x \) varies. Similarly, for \( \frac{\partial u}{\partial y} \) and \( \frac{\partial u}{\partial z} \), only \( y \) and \( z \) vary respectively.

Using partial derivatives helps in understanding the rate of change of a function in relation to one of its variables, while the other variables remain unchanged. This concept is widely used in fields such as physics, engineering, and economics to model and understand complex systems.

Multivariable Functions

Multivariable functions, like \( u = u(x, y, z) \), depend on multiple variables. These functions are more complicated than single-variable functions but offer a more realistic representation of natural phenomena, where several factors often influence the outcome.

When dealing with functions of multiple variables, we explore how variations in each variable can affect the function. These functions often involve differentials with respect to each variable, like partial derivatives. They are crucial in describing physical systems and models because most real-world problems have several influencing variables.

Understanding how changes in one variable affect the entire function requires techniques like the chain rule, allowing us to derive complex derivative expressions from interrelated variables. Applications of multivariable functions range from predicting financial trends to solving scientific problems.

Tree Diagrams in Calculus

Tree diagrams in calculus are visual representations used to track the relationships between variables in complex functions. These diagrams help us visualize how different variables are interconnected, making it easier to apply the chain rule and calculate derivatives.

Consider a situation where we have a function \( u = u(x, y, z) \). To find the derivative of \( u \) with respect to another variable like \( r \), a tree diagram illustrates the indirect dependencies through other variables, such as \( w \) and \( t \).

• The root of the tree is the main function \( u \).
• From \( u \), branches lead to \( x, y, \text{ and } z \).
• Each of these then branches to \( w \) and \( t \), which depend further on \( r \) and \( s \).

By mapping out all paths from the output to the independent variables, tree diagrams simplify the task of tracking the "propagation" of changes through a function. This systematic approach is particularly useful in applying the chain rule, as it helps ensure all variable paths are considered.