Problem 17 - Calculus: Early Transcendental Functions

Question

Find the gradient field corresponding to \(f\) Use a CAS to graph it. $$f(x, y)=x e^{-y}$$

Step-by-Step Solution

Answer

The gradient field corresponding to the function \(f(x, y)=x e^{-y}\), \(\nabla f=(e^{-y}, -x e^{-y})\), which indicates the rate and direction of change of the function at each point in the plane. Visualization of this gradient field can be done using a suitable Computer Algebra System (CAS).

Step 1: Calculate the partial derivatives

The first step is to find the partial derivatives of the given function \(f(x, y)=x e^{-y}\). The partial derivative of \(f\) with respect to \(x\) is \(f_x = e^{-y}\) and the partial derivative with respect to \(y\) is \(f_y = -x e^{-y}\). So, \(\nabla f = (e^{-y}, -x e^{-y})\).

Step 2: Present the Gradient Field

The gradient field corresponding to the function \(f\), denoted by \(\nabla f\), is now given by \(\nabla f = (e^{-y}, -x e^{-y})\). It describes the rate and direction of change of the function \(f\). Each point in this field represents the vector formed by the partial derivatives at that point.

Step 3: Graph the Gradient Field

The final step is to graph the gradient field on a suitably scaled Cartesian coordinate system using Computer Algebra Systems (CAS). Plot the function \(\nabla f = (e^{-y}, -x e^{-y})\) noting that every point in the plane has a vector associated with it. The direction of the vector represents the direction of maximum growth of the function at that point, and the magnitude of the vector indicates the rate of growth.

Key Concepts

Partial Derivatives

When studying functions of multiple variables, like our function \( f(x, y) = x e^{-y} \), partial derivatives are crucial to understanding how each variable impacts the function independently. In this context:

• A partial derivative with respect to \( x \) (\( f_x \)) looks at how the function changes as \( x \) varies while keeping \( y \) constant. For our function, this derivative is \( f_x = e^{-y} \).
• Conversely, the partial derivative with respect to \( y \) (\( f_y \)) measures the change in function as \( y \) changes, while \( x \) remains constant. This is \( f_y = -x e^{-y} \) for our function.

Understanding partial derivatives helps us determine how sensitive the function is to changes in individual variables. This insight is foundational for analyzing more complex phenomena such as the behavior of surfaces and optimizing functions dependent on several variables.
Each partial derivative contributes to forming the gradient vector, providing a vectorial representation of the function's changes.

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector fields and differentiable functions of several variables, as seen in our function \( f(x, y) = x e^{-y} \).
In this context, the gradient vector, \( abla f \), is a central concept:

• It provides a way to concisely represent both the direction and rate of change of a function.
• For our function, the gradient \( abla f = (e^{-y}, -x e^{-y}) \) symbolizes how the function's output changes in the \( x \) and \( y \) directions simultaneously.

This vector field places arrows at each point in the 2-dimensional plane, pointing in the direction of steepest ascent of the function.
Vector calculus not only deals with gradient vectors but also with operations like divergence and curl, which further explore the properties of vector fields. These concepts have practical applications in physics, engineering, and beyond, making vector calculus a powerful tool for mathematical analysis.

Rate and Direction of Change

The gradient field we calculated in vector calculus is all about understanding the rate and direction of change for the function \( f(x, y) = x e^{-y} \).
This gradient \( abla f = (e^{-y}, -x e^{-y}) \) tells us several things:

• The direction of this vector at any given point represents the direction in which the function increases most rapidly.
• The length or magnitude of the vector determines how quickly the value of the function is increasing or decreasing in that particular direction.

When plotted, these vectors illustrate not just where a function is experiencing its greatest changes but also the relative size of those changes; larger vectors indicate faster increases. Understanding both the direction and rate of change helps in maximizing or minimizing functions, vital in economics, structural engineering, and various scientific computations where optimal solutions are sought.