Problem 6 - Calculus Early Transcendentals

Question

Determine whether \(\mathbf{F}\) is a conservative vector field. If so, find a potential function for it. $$ \mathbf{F}(x, y)=x \ln y \mathbf{i}+y \ln x \mathbf{j} $$

Step-by-Step Solution

Answer

The vector field \( \mathbf{F} \) is not conservative.

Step 1: Analyze the Components

The vector field \( \mathbf{F}(x, y)=x \ln y \mathbf{i}+y \ln x \mathbf{j} \) has two components: \( P(x, y) = x \ln y \) and \( Q(x, y) = y \ln x \). To check if \( \mathbf{F} \) is conservative, we will use the partial derivatives of \( P \) and \( Q \).

Step 2: Compute Partial Derivatives

Calculate the partial derivative \( \frac{\partial P}{\partial y} \): \[ \frac{\partial }{\partial y}(x \ln y) = \frac{x}{y} \] Calculate the partial derivative \( \frac{\partial Q}{\partial x} \): \[ \frac{\partial }{\partial x}(y \ln x) = \frac{y}{x} \]

Step 3: Verify Equality of Mixed Derivatives

A vector field is conservative if the mixed partial derivatives \( \frac{\partial P}{\partial y} \) and \( \frac{\partial Q}{\partial x} \) are equal. We compare:\[ \frac{x}{y} \quad \text{and} \quad \frac{y}{x} \]Since these are not equal, \( \mathbf{F} \) is not conservative.

Key Concepts

Potential Function

A potential function is a concept that plays a crucial role when dealing with conservative vector fields. A vector field is considered conservative if it is the gradient of some scalar potential function. This means that the vector field can be expressed as the derivative of a single function with respect to all its variables, capturing the idea that potential functions describe force fields derived from energy principles. When a vector field is conservative, the line integral between two points depends only on the endpoints, not on the path taken.

If a potential function exists for a vector field \( \mathbf{F} \), then there exists a scalar function \( f(x, y) \) such that \( abla f = \mathbf{F} \). This function \( f(x, y) \) would satisfy the conditions:

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

If you can find such \( f(x, y) \), the vector field is conservative, and \( f \) is the potential function.

Partial Derivatives

Partial derivatives are a fundamental tool in multivariable calculus which allow us to investigate how a function changes as we vary one of its variables independently of the others. Consider a function \( f(x, y) \) of two variables, its partial derivatives are:

• \( \frac{\partial f}{\partial x} \): this derivative measures the rate of change of \( f \) with respect to \( x \), treating \( y \) as a constant.
• \( \frac{\partial f}{\partial y} \): this derivative measures the change of \( f \) with respect to \( y \), considering \( x \) a constant.

In the case of the vector field \( \mathbf{F}(x, y) = x \ln y \mathbf{i} + y \ln x \mathbf{j} \), computing partial derivatives helps analyze if this field is conservative. The calculation of \( \frac{\partial P}{\partial y} \) and \( \frac{\partial Q}{\partial x} \) is a pivotal step in determining the characteristics of the vector field.

Mixed Partial Derivatives

The mixed partial derivatives criterion is essential for assessing whether a vector field is conservative. Mixed partial derivatives refer to the derivatives of order two or higher involving different variables. For a function \( f(x, y) \), the mixed partial derivatives are \( \frac{\partial^2 f}{\partial x \partial y} \) and \( \frac{\partial^2 f}{\partial y \partial x} \).

• For a vector field to be conservative, the mixed partial derivatives of its component functions must be equal. This is expressed mathematically using Clairaut's theorem, which states that if the second mixed partial derivatives are continuous, they are equal.

In the given exercise, the mixed partial derivatives \( \frac{\partial P}{\partial y} = \frac{x}{y} \) and \( \frac{\partial Q}{\partial x} = \frac{y}{x} \) were compared to check this condition. Since they are not equal, the vector field \( \mathbf{F} \) is not conservative. This demonstrates the importance of equality in mixed partial derivatives to identify conservative fields.