Problem 16 - Advanced Engineering Mathematics

Question

In Problems, 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 \( \mathbf{F}(x, y) \) is not conservative.

Step 1: Check for Curl of the Vector Field

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

Step 2: Conclusion

Since the curl of the vector field is not zero, \( \mathbf{F}(x, y)=2 e^{2 y} \mathbf{i}+x e^{2 y} \mathbf{j} \) is not a conservative vector field.

Key Concepts

Vector Calculus

Vector Calculus is a field of mathematics concerned with multivariable functions and the manipulation of vectors and scalars. It provides essential tools for understanding physical quantities in space, like velocity and force.

• Vector Field: A vector field assigns a vector to every point in space. In two dimensions, this might be a function \( \mathbf{F}(x, y) = P(x, y)\mathbf{i} + Q(x, y)\mathbf{j} \). Here, each point \((x, y)\) is associated with a vector consisting of components \(P(x, y)\) and \(Q(x, y)\).
• Gradient: The gradient of a scalar function represents the rate and direction of change in a scalar field, forming a vector field.

Vector calculus is vital in electromagnetism, fluid dynamics, and engineering. Understanding how to manipulate these functions helps to solve real-world problems involving forces and movements.

Curl of a Vector Field

The Curl of a Vector Field represents its rotation at any point. It is a vector operation used in three-dimensional space.

• Mathematical Expression: For a vector field \(\mathbf{F} = P\mathbf{i} + Q\mathbf{j}\), the curl is calculated as \(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\).
• Conservative Field: A vector field is conservative if its curl is zero everywhere. In such cases, the field has no rotational effect.

In the context of the given problem, if the curl \(abla \times \mathbf{F}\) is zero, a potential function exists, indicating the field is conservative. However, if the curl is not zero, as shown by differing derivatives of \(Q\) and \(P\), the field exhibits some kind of rotation or spiraling, indicating non-conservativeness.

Potential Function

A Potential Function is a scalar function whose gradient gives the vector field; it effectively "generates" the vector field. For a field to have a potential function, it must be conservative.

• Finding Potential Function: If a vector field \(\mathbf{F}\) is conservative, its potential function \(\phi\) satisfies \(\mathbf{F} = abla \phi\).
• Path Independence: In conservative fields, the integral of the field over a path depends only on the endpoints, allowing for easier calculations of work and energy.

In summary, if a vector field's curl is zero, it has a potential function. However, in this exercise, our vector field didn't fulfill the criteria as the curl was non-zero, thus no potential function can be calculated. The hunt for potential functions helps find scalar representations of vector behaviors, simplifying complex spatial problems.