Problem 22 - Calculus with Concepts in Calculus

Question

Determine \(d f\). $$ f(x, y, z)=\frac{x}{x^{2}+y^{2}+z^{2}} $$

Step-by-Step Solution

Answer

The differential \( df \) is \( (y^2 + z^2 - x^2) dx - 2xy \, dy - 2xz \, dz \) divided by \((x^2 + y^2 + z^2)^2\).

Step 1: Identify the Variables and Function

Given the function \( f(x, y, z) = \frac{x}{x^2 + y^2 + z^2} \), identify \( x, y, \) and \( z \) as the variables. We need to find the differential \( df \).

Step 2: Use the Differential Formula

The differential \( df \) of a function \( f(x, y, z) \) is given by \( df = f_x \, dx + f_y \, dy + f_z \, dz \), where \( f_x \), \( f_y \), and \( f_z \) are the partial derivatives of \( f \) with respect to \( x, y, \) and \( z \) respectively.

Step 3: Compute Partial Derivative with Respect to x

Use the quotient rule to differentiate \( f(x, y, z) = \frac{x}{x^2 + y^2 + z^2} \) with respect to \( x \). This gives \( f_x = \frac{(x^2 + y^2 + z^2) \cdot 1 - x \cdot 2x}{(x^2 + y^2 + z^2)^2} = \frac{y^2 + z^2 - x^2}{(x^2 + y^2 + z^2)^2} \).

Step 4: Compute Partial Derivative with Respect to y

Differentiate \( f(x, y, z) = \frac{x}{x^2 + y^2 + z^2} \) with respect to \( y \) using the quotient rule. This gives \( f_y = \frac{(x^2 + y^2 + z^2) \cdot 0 - x \cdot 2y}{(x^2 + y^2 + z^2)^2} = -\frac{2xy}{(x^2 + y^2 + z^2)^2} \).

Step 5: Compute Partial Derivative with Respect to z

Differentiate \( f(x, y, z) = \frac{x}{x^2 + y^2 + z^2} \) with respect to \( z \) using the quotient rule. This gives \( f_z = \frac{(x^2 + y^2 + z^2) \cdot 0 - x \cdot 2z}{(x^2 + y^2 + z^2)^2} = -\frac{2xz}{(x^2 + y^2 + z^2)^2} \).

Step 6: Combine to Find df

Combine the partial derivatives into the differential formula: \[ df = \left( \frac{y^2 + z^2 - x^2}{(x^2 + y^2 + z^2)^2} \right) dx - \left( \frac{2xy}{(x^2 + y^2 + z^2)^2} \right) dy - \left( \frac{2xz}{(x^2 + y^2 + z^2)^2} \right) dz \].

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental concept in calculus, especially when dealing with functions of multiple variables. They help us understand how a function changes as one variable changes while the others are held constant. In the context of the original exercise, we have the function \( f(x, y, z) = \frac{x}{x^2 + y^2 + z^2} \). This means the function is dependent on three variables: \( x \), \( y \), and \( z \).

To find partial derivatives, we differentiate the function with respect to one variable, treating the others as constants. For \( f(x, y, z) \):

• \( f_x \) is the partial derivative with respect to \( x \).
• \( f_y \) is the partial derivative with respect to \( y \).
• \( f_z \) is the partial derivative with respect to \( z \).

This process is crucial for understanding local changes in multivariable functions and has applications in fields ranging from physics to economics.

Differential Calculus

Differential calculus involves the study of how functions change, and it provides tools like derivatives to analyze this change. In our exercise, differential calculus is applied through finding the differential \( df \).

The differential of a function with several variables can provide information about the function's rate of change. The formula for the differential of a function of three variables, \( f(x, y, z) \), is \( df = f_x \, dx + f_y \, dy + f_z \, dz \).

Using the calculated partial derivatives, the differential becomes a linear approximation of how the function \( f \) changes based on small changes in \( x, y, \) and \( z \). This is particularly useful in optimization problems and other applications where understanding small changes can lead to significant insights.

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions with more than one variable. It is essential in analyzing systems where more than one factor is in play, as seen with \( f(x, y, z) = \frac{x}{x^2 + y^2 + z^2} \).

Key concepts in multivariable calculus include partial derivatives and differentials, both of which are used in our exercise.

• Partial derivatives allow us to see how each individual variable affects the function.
• The differential provides a way to model how small changes in variables influence the overall function.

This branch of calculus is widely used in physical sciences, engineering, and economics to model behaviors and solve complex problems involving multiple factors. Multivariable calculus is a powerful tool that helps to translate real-world issues into mathematical language, allowing for deeper analysis and understanding.