Problem 20 - Calculus Early Transcendentals

Question

If \(\mathbf{F}(x, y)=a y \mathbf{i}+b x \mathbf{j}\) is a conservative vector field, then \(a=b\)

Step-by-Step Solution

Answer

For \( \mathbf{F} \) to be conservative, \( a = b \).

Step 1: Understand Conservative Vector Field

A vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} \) is conservative if there exists a potential function \( f(x, y) \) such that \( \mathbf{F} = abla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} \). It implies that the curl of \( \mathbf{F} \) must be zero.

Step 2: Identify Components of the Vector Field

Given \( \mathbf{F}(x, y) = ay\mathbf{i} + bx\mathbf{j} \), we identify \( P(x, y) = ay \) and \( Q(x, y) = bx \).

Step 3: Compute the Curl of the Vector Field

The curl of a two-dimensional vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} \) is \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \). Compute this for \( \mathbf{F}(x, y) = ay\mathbf{i} + bx\mathbf{j} \):\[\frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(bx) = b, \quad \frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(ay) = a.\]Thus, the curl is \( b - a \).

Step 4: Set the Curl to Zero

For \( \mathbf{F} \) to be conservative, set the curl to zero:\[ b - a = 0. \]

Step 5: Solve for Equality

From the equation \( b - a = 0 \), solve for \( a \) and \( b \): \[ b = a. \]Thus, for the vector field to be conservative, \( a \) must equal \( b \).

Key Concepts

Potential Function

In mathematics, a potential function is a scalar function whose gradient vector field is equal to a given vector field. If a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} \) can be expressed as the gradient of some potential function \( f(x, y) \), then \( \mathbf{F} \) is said to be conservative. This means there exists a function \( f(x, y) \) such that:

• \( \frac{\partial f}{\partial x} = P(x, y) \)
• \( \frac{\partial f}{\partial y} = Q(x, y) \)

Finding such a function \( f \) involves integration and ensuring the cross-partial derivatives satisfy a condition (often related to the curl of the vector field). It is also important to note that potential functions are not unique—any constant added to \( f(x, y) \) results in another valid potential function. Moreover, the concept of a potential function is foundational in physics, especially in fields like electrostatics and gravitational fields.

Curl of Vector Field

The curl of a vector field is an operation that describes the field's tendency to rotate around a point in two-dimensional or three-dimensional space. For a two-dimensional vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} \), the curl is calculated as:\[\text{curl}(\mathbf{F}) = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}.\]In the exercise provided, we start with \( \mathbf{F}(x, y) = ay\mathbf{i} + bx\mathbf{j} \). By computing the partial derivatives,

• \( \frac{\partial Q}{\partial x} = b \)
• \( \frac{\partial P}{\partial y} = a \)

Thus, \( \text{curl}(\mathbf{F}) = b - a. \)For the vector field to be conservative, the curl must be zero (\( b - a = 0 \)). This means there should be no rotation or circulation around any point in the field, and so it permits the existence of a potential function for the vector field.

Two-Dimensional Vector Field

A two-dimensional vector field assigns a two-dimensional vector to each point in a plane. It can be visualized by plotting arrows in a coordinate system, where each arrow represents the magnitude and direction of the vector at that point. The general form of a 2D vector field is \( \mathbf{F}(x, y) = P(x, y)\mathbf{i} + Q(x, y)\mathbf{j} \), where \( P(x, y) \) and \( Q(x, y) \) are functions of \( x \) and \( y \).In the given exercise, \( \mathbf{F}(x, y) = ay\mathbf{i} + bx\mathbf{j} \) represents such a field. Here, \( P(x, y) = ay \) and \( Q(x, y) = bx \), which define the field. Understanding these components helps determine field properties like curl and conservativeness.2D vector fields are frequently used in physics and engineering to describe various phenomena such as fluid flow, electromagnetic fields, and more. They provide a simplified model to understand the interaction between objects in a two-dimensional space, while maintaining key ideas like direction and magnitude.