Problem 16 - Advanced Engineering Mathematics

Question

Determine whether the given vector field is a conservative field. If so, find a potential function \(\phi\) for \(\mathbf{F}\). $$ \mathbf{F}(x, y)=2 e^{2 y} \mathbf{i}+x e^{2 y} \mathbf{j} $$

Step-by-Step Solution

Answer

The vector field is not conservative; no potential function exists.

Step 1: Identify Components of the Vector Field

The given vector field is \( \mathbf{F}(x, y) = 2e^{2y}\mathbf{i} + xe^{2y}\mathbf{j} \). The components are \( P(x, y) = 2e^{2y} \) and \( Q(x, y) = xe^{2y} \).

Step 2: Check for Conservation Using Curl

A vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} \) is conservative if \( \frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y} \). Compute \( \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(xe^{2y}) = e^{2y} \) and \( \frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(2e^{2y}) = 4e^{2y} \). Since \( e^{2y} eq 4e^{2y} \), the field is not conservative.

Step 3: Conclude with Final Answer

Since the curl condition \( \frac{\partial Q}{\partial x} eq \frac{\partial P}{\partial y} \) is not satisfied, the vector field is not conservative. Therefore, there is no potential function \( \phi \).

Key Concepts

Potential Function

In vector calculus, a potential function, usually denoted by \( \phi \), is a scalar function whose gradient (denoted by \( abla \phi \)) is equal to a vector field \( \mathbf{F} \). If such a function \( \phi \) exists, then \( \mathbf{F} \) is called a conservative vector field.

One of the key characteristics of a conservative field is that the field's work around any closed path is zero. That means, mathematically, there are no 'loops' where energy is lost.

For example, if \( \mathbf{F} = \mathbf{i} + \mathbf{j} \), then \( abla \phi = \mathbf{i} + \mathbf{j} \), and a potential function could be \( \phi(x, y) = x + y \). In this case, the partial derivatives \( \frac{\partial \phi}{\partial x} = 1 \) and \( \frac{\partial \phi}{\partial y} = 1 \) match the vector field components.
If no such potential function exists, as seen in the given vector field \( \mathbf{F}(x, y) = 2e^{2y}\mathbf{i} + xe^{2y}\mathbf{j} \), then the field is not conservative.

Vector Calculus

Vector calculus is a branch of mathematics focusing on vector fields and differentiable functions. It's crucial for studying physical phenomena like electromagnetism and fluid flow. Vector calculus includes operations such as gradient, divergence, and curl.

In the context of this problem, we used the concept of the curl. This operation helps determine if a vector field is conservative. The curl of a 2D vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} \) involves checking if \( \frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y} \). This condition means that the vector field has no rotational components, thus could have a potential function.
Understanding these operations is key to solving vector calculus problems efficiently. Learning to calculate derivatives and integrals of scalar and vector fields will give you insight into fields like physics and engineering.

Vector Field Components

A vector field in two dimensions is often represented by \( \mathbf{F} = P(x, y)\mathbf{i} + Q(x, y)\mathbf{j} \), where \( P \) and \( Q \) are the field's components.

In our example, the vector field given is \( \mathbf{F}(x, y) = 2e^{2y}\mathbf{i} + xe^{2y}\mathbf{j} \). Here:

• \( P(x, y) = 2e^{2y} \)


• \( Q(x, y) = xe^{2y} \)

These components serve different roles and influence how the vector field behaves. When checking for conservativeness, we compute partial derivatives like \( \frac{\partial Q}{\partial x} \) and \( \frac{\partial P}{\partial y} \) to see if they are equal.
If they are not equal, as with \( e^{2y} eq 4e^{2y} \), the vector field has some rotation and cannot be conservative. Understanding vector field components enables dissecting complex fields into more manageable pieces that help in solving mathematical and physical problems.