Problem 15 - Advanced Engineering Mathematics

Question

In Problems, determine whether the given vector field is a conservative field. If so, find a potential function \(\phi\) for \(\mathbf{F}\). $$ \mathbf{F}(x, y)=\left(x^{3}+y\right) \mathbf{i}+\left(x+y^{3}\right) \mathbf{j} $$

Step-by-Step Solution

Answer

The vector field is conservative; the potential function is \( \phi(x, y) = \frac{x^4}{4} + xy + \frac{y^4}{4} + C \).

Step 1: Check if the Vector Field is Conservative

A vector field \( \mathbf{F}(x, y) = P(x, y)\mathbf{i} + Q(x, y)\mathbf{j} \) is conservative if \[ \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}. \]For \( \mathbf{F}(x, y) = (x^3 + y)\mathbf{i} + (x + y^3)\mathbf{j} \), we have:- \( P(x, y) = x^3 + y \) - \( Q(x, y) = x + y^3 \) Calculate the partial derivatives: - \( \frac{\partial P}{\partial y} = 1 \) - \( \frac{\partial Q}{\partial x} = 1 \).Since \( \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} \), the vector field is conservative.

Step 2: Integrate to Find the Potential Function

To find the potential function \( \phi(x, y) \), we integrate \( P(x, y) \) and \( Q(x, y) \): 1. Integrate \( P(x, y) = x^3 + y \) with respect to \( x \): \[ \int (x^3 + y) \, dx = \frac{x^4}{4} + xy + g(y) \] where \( g(y) \) is an arbitrary function of \( y \).2. Integrate \( Q(x, y) = x + y^3 \) with respect to \( y \) and check consistency with the terms from the previous integration: \[ \int (x + y^3) \, dy = xy + \frac{y^4}{4} + h(x). \] From consistency, we know: \( g'(y) = y^3 \), thus \( g(y) = \frac{y^4}{4} + C \). Therefore, \( \phi(x, y) = \frac{x^4}{4} + xy + \frac{y^4}{4} + C \) where \( C \) is a constant.

Key Concepts

Potential Function

In vector calculus, a potential function is a scalar function whose gradient results in a given vector field. When a vector field is conservative, it means the field can be represented as the gradient of some potential function \( \phi \). To find this potential function, we often use the method of partial integration as distinct from simple integration.
For the given vector field \( \mathbf{F}(x, y) = (x^3 + y)\mathbf{i} + (x + y^3)\mathbf{j} \), the process involves integrating components of the vector field:

• First, integrate \( P(x, y) = x^3 + y \) with respect to \( x \).
• This yields \( \frac{x^4}{4} + xy + g(y) \), where \( g(y) \) is an unknown function of \( y \).
• Next, integrate \( Q(x, y) = x + y^3 \) with respect to \( y \) to check for consistency.
• The result \( xy + \frac{y^4}{4} + h(x) \) gives clues on what \( g(y) \) must be.

By consistency, we derive that \( g(y) = \frac{y^4}{4} + C \), producing the potential function \( \phi(x, y) = \frac{x^4}{4} + xy + \frac{y^4}{4} + C \). This potential function fully describes the vector field in terms of its scalar equivalent.

Vector Calculus

Vector calculus is an extension of traditional calculus that involves vector fields rather than single-variable functions. It incorporates multiple dimensions and includes operations like divergence, curl, and gradient. Understanding these operations helps interpret the behavior and properties of vector fields.
In the context of this exercise, the vector field \( \mathbf{F}(x, y) \) can be represented as \( P(x, y)\mathbf{i} + Q(x, y)\mathbf{j} \) where:

• \( P(x, y) = x^3 + y \), indicating the \( x \)-component of the field.
• \( Q(x, y) = x + y^3 \), indicating the \( y \)-component of the field.

Determining if a vector field is conservative involves checking conditions such as the partial derivative equality: \( \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} \). This condition must hold true for there to be a corresponding potential function. If it does, as in this exercise, the field can be described via the gradient of a potential function \( \phi \).
Being familiar with vector calculus equips you with the tools to analyze fields in physics and engineering that involve diverse quantities like force, velocity, and electromagnetic fields.

Partial Derivatives

Partial derivatives are a fundamental concept in multivariable calculus, which allow us to analyze functions of two or more variables. They show how a function changes in relation to changes in one variable while keeping other variables constant.
For instance, in the vector field \( \mathbf{F}(x, y) = (x^3 + y)\mathbf{i} + (x + y^3)\mathbf{j} \):

• To check if the field is conservative, compute \( \frac{\partial P}{\partial y} \) for \( P(x, y) = x^3 + y \), resulting in 1.
• Also compute \( \frac{\partial Q}{\partial x} \) for \( Q(x, y) = x + y^3 \), which also results in 1.

Since \( \frac{\partial P}{\partial y} \) equals \( \frac{\partial Q}{\partial x} \), the field is conservative, implicating a potential function. This derivative equality is central for checking conservativeness in vector fields.
Understanding partial derivatives helps in visualizing how a function behaves over a spatial domain, offering insights into dynamic systems in mathematics, science, and engineering.