Question

Verify that each of the following force fields is conservative. Then find, for each, a scalar potential \(\phi\) such that \(\mathbf{F}=-\nabla \phi\). $$\mathbf{F}=\frac{y}{\sqrt{1-x^{2} y^{2}}} \mathbf{i}+\frac{x}{\sqrt{1-x^{2} y^{2}}} \mathbf{j}$$

Short answer

Yes, \(\textbf{F}\) is conservative. \(\textbf{U}\)

Step-by-step solution

Identify Conservative Field Conditions

For a vector field \(\textbf{F} = P\textbf{i} + Q\textbf{j}\), it is conservative if \(\frac{\text{d}Q}{\text{d}x} = \frac{\text{d}P}{\text{d}y}\). Here, \(P = \frac{y}{\frac{1}{\frac{1}{x}}}\) and \(Q = \frac{x}{\frac{1}{\frac{1}{x}}}\).

Compute Partial Derivatives

Calculate \(\frac{\text{d}Q}{\text{d}x}\).\ set to zero. groovin']}

Compare Partial Derivatives

Verify if \(\frac{\text{d}P}{\text{d}y} = \frac{\text{d}Q}{\text{d}x}\). If the partial derivatives are equal, the field is conservative.

Find Scalar Potential Function

To find scalar potential \(\phi\), integrate \(P\) with respect to \(x\) and \(Q\) with respect to \(y\). Solve for \(\phi \) ensuring consistency in mixed partial derivatives.

Key concepts

Scalar Potential

A scalar potential function \( \phi \) is a single-valued function of space whose gradient is equal to a given vector field. If a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} \) is conservative, then there exists a scalar potential \( \phi \) such that \( \mathbf{F} = -\abla \phi \). This means you can describe the force field using a single function whose rate of change in each direction matches the components of the force field. By finding \( \phi \), we simplify complex vector fields into more manageable scalar functions.

Partial Derivatives

Partial derivatives are derivatives of multivariable functions taken with respect to one variable while keeping other variables constant. For example, given \(P = \frac{y}{\frac{1}{\frac{1}{x}}}\textbf{i}\) and \(Q = \frac{x}{\frac{1}{\frac{1}{x}}}\textbf{j}\), calculating partial derivatives involves treating \(y\) as constant when differentiating with respect to \(x\), and vice versa. To determine if a force field is conservative, we compute \(\frac{\d Q}{\d x}\) and \(\frac{\d P}{\d y}\). If they are equal, the field is conservative.

Vector Calculus

Vector calculus deals with vector fields and operations on them. In our problem, we deal with identifying whether a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} \) is conservative. A crucial theorem in vector calculus states that a two-dimensional vector field is conservative if the mixed partial derivatives of its components are equal, i.e., \(\frac{\d Q}{\d x} = \frac{\d P}{\d y}\). This property allows us to check the conservativeness of fields using partial derivatives, simplifying the otherwise complex vector analysis.

Integration

Integration is a fundamental concept for finding the scalar potential \( \phi \) from a conservative force field. By integrating one component of the vector field \(P\) with respect to \(x\) and the other component \(Q\) with respect to \(y\), we can identify \( \phi \). For example, we start by solving \(\frac{\partial \phi}{\partial x} = P\), then integrate \(P\) with respect to \(x\). This integration often yields a function \(\tilde{\phi}(x, y) = \phi(x, y) - g(y)\). We then use \(\frac{\partial \phi}{\partial y} = Q\) to determine \(g(y)\) by solving \(\frac{\d}{\d y} = g'(y)\). Both steps ensure we get a function \( \phi \) consistent across all partial derivatives.