Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 7-12, find \(\nabla f\). $$ f(x, y, z)=x e^{y} \cos z $$

Step-by-Step Solution

Answer

\(\nabla f = (e^y \cos z, \, x e^y \cos z, \, -x e^y \sin z)\).

Step 1: Understanding the Problem

The problem asks us to find the gradient of the function \(f(x, y, z) = x e^y \, \cos z\). The gradient of a function is a vector consisting of its partial derivatives with respect to each variable.

Step 2: Find Partial Derivative with Respect to \(x\)

To find the partial derivative of \(f\) with respect to \(x\), treat \(y\) and \(z\) as constants. Thus, the derivative of \(x e^y \cos z\) with respect to \(x\) is just \(e^y \cos z\).

Step 3: Find Partial Derivative with Respect to \(y\)

For the partial derivative with respect to \(y\), treat \(x\) and \(z\) as constants. The derivative of \(x e^y\) is \(x e^y\), so the partial derivative is \(x e^y \cos z\).

Step 4: Find Partial Derivative with Respect to \(z\)

To find the partial derivative with respect to \(z\), treat \(x\) and \(y\) as constants. The derivative of \(\cos z\) is \(-\sin z\), so the partial derivative is \(-x e^y \sin z\).

Step 5: Assemble the Gradient Vector

The gradient vector \(abla f\) is formed by combining the partial derivatives found in Steps 2, 3, and 4. Therefore, \(abla f = \left( e^y \cos z, \, x e^y \cos z, \, -x e^y \sin z \right)\).

Key Concepts

Partial Derivatives

Partial derivatives are essential in multivariable calculus. They help us understand how a function changes as each individual variable changes. When dealing with a function of several variables, like \( f(x, y, z) \), a partial derivative with respect to one variable is calculated while keeping the others constant. This allows us to isolate the effect of a single variable's change.

• Example: To compute the partial derivative of \( f(x, y, z) = x e^y \, \cos z \) with respect to \( x \), consider \( y \) and \( z \) as constants and differentiate.
• This results in \( \frac{\partial f}{\partial x} = e^y \, \cos z \).
• This step-by-step computation focuses uniquely on how \( x \) influences \( f \), offering a piece of the overall gradient vector.

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions with more than one variable. It builds the framework for understanding how changes in inputs affect outputs across complex systems. This branch of calculus includes techniques for finding maxima, minima, and optimization in multi-dimensional spaces.

• Functions can be in the form \( f(x, y, z) \), representing systems depending on multiple independent variables.
• Calculating partial derivatives helps develop a deeper understanding of how the function behaves as each variable changes independently.
• It emphasizes not just the magnitude of change but also in what direction in the multi-dimensional space these changes occur.

Vector Calculus

Vector calculus is a field that deals with vector fields and the functions operating on them. It allows mathematicians and engineers to describe physical phenomena involving vectors, such as force fields.

• The gradient vector is a fundamental concept within vector calculus, providing a direction and rate of maximum increase of a scalar field.
• For \( f(x, y, z) \), the gradient \( abla f \) indicates how quickly the output changes and in which direction, given a tiny variation in input variables.
• Composed of all partial derivatives, this vector offers critical insight into multi-dimensional changes.

Functions of Several Variables

Functions of several variables, like \( f(x, y, z) = x e^y \, \cos z \), involve multiple input variables impacting a single output. Such functions are vital in real-world applications, modeling various systems.

• Understanding: These functions represent scenarios where an outcome depends simultaneously on different factors or dimensions.
• They require considering how each variable contributes uniquely to the overall behavior of the function.
• Tools like partial derivatives and the gradient vector help break down these complex interactions into manageable, understandable parts.
• They are crucial in fields like physics, engineering, economics, and any discipline requiring the modeling of complex systems.