Problem 117 - Calculus Volume 3

Question

For the following exercises, determine whether the vector field is conservative and, if so, find a potential function. $$ \mathbf{F}(x, y)=\left(e^{x} \cos y\right) \mathbf{i}+6\left(e^{x} \sin y\right) \mathbf{j} $$

Step-by-Step Solution

Answer

The vector field is not conservative.

Step 1: Check for Conservative Conditions

A vector field \( \mathbf{F}(x, y) = P(x, y) \mathbf{i} + Q(x, y) \mathbf{j} \) is conservative if there exists a function \( f \) such that \( abla f = \mathbf{F} \). This implies that \( \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} \). Here, \( P(x, y) = e^{x} \cos y \) and \( Q(x, y) = 6e^{x} \sin y \).

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

The vector field is not conservative.

Key Concepts

Potential Function

In vector calculus, a **potential function** plays a major role when dealing with **conservative vector fields**. If you have a vector field and it's deemed conservative, it means there exists a potential function, often denoted as \( f(x, y) \) or \( f(x, y, z) \). This function represents a scalar field whose gradient vector is equivalent to the given vector field.
For a **vector field** \( \mathbf{F}(x, y) = P(x, y) \mathbf{i} + Q(x, y) \mathbf{j} \) to be conservative, there must be such a potential function \( f \) that satisfies the condition \( abla f = \mathbf{F} \). This implies:
\[ \frac{\partial f}{\partial x} = P(x, y) \quad \text{and} \quad \frac{\partial f}{\partial y} = Q(x, y) \]
To find \( f \), you integrate \( P(x, y) \) with respect to \( x \) and \( Q(x, y) \) with respect to \( y \), checking consistency between the two integrations. This provides a single scalar function that can describe the behavior of the vector field across the plane or space.

Gradient Fields

If a vector field can be expressed as the gradient of some scalar potential function, such a field is called a **gradient field**. It means the vector field arises from a scalar field through differentiation process and can be expressed as \( \mathbf{F} = abla f \).
A **gradient field** is always conservative because the components of the vector field correspond to the partial derivatives of the potential function. This ensures the cross-partial derivatives of \( P \) and \( Q \) satisfy the equality \( \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} \), which is a condition for being conservative. - **Gradient**: The gradient \( abla f \) of a function \( f \) in two dimensions is given by \( \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right) \).- **Conservative nature**: A field being conservative means it has no curl, which makes it path-independent; the integral around any closed loop will be zero.
When checking if a vector field constitutes a gradient field, one essentially is verifying for this conservativeness.

Vector Calculus

Let's dive into the fascinating world of **vector calculus**, a branch of mathematics that explores differentiation and integration of vector fields. Vector calculus is fundamental for understanding fields in physics, engineering, and any discipline involving spatial analyses.
When discussing **vector calculus**, concepts such as vector fields, divergence, and curl come into play. A **vector field** assigns a vector to each point in space, often visualizing physical quantity like force or velocity fields. **Conservative vector fields** are special cases of vector fields where the field can also be described by a scalar potential function. - **Divergence**: measures the magnitude of a source or sink at a given point in a vector field. - **Curl**: measures the tendency to rotate about a point. Conservative fields, besides being curl-free (i.e., having zero curl), are deeply associated with path integrals which are integral parts of vector calculus. This means integrating over these fields depends only on the start and end points, not on the path taken.
To determine if a vector field is conservative, students must apply vector calculus principles: ensuring the curl is zero or that it satisfies certain partial derivative equalities of its components. Understanding these concepts is crucial for applying vector calculus in real-world problem-solving and physics applications.