Question

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

Short answer

The force field is conservative and the scalar potential is \( \phi = -z^2 \cosh y + C \).

Step-by-step solution

- Identify if the Force Field is Conservative

To verify if a force field is conservative, check if the curl of the force field is zero. For a vector field \( \mathbf{F} = P(x,y,z)\mathbf{i} + Q(x,y,z)\mathbf{j} + R(x,y,z)\mathbf{k} \), calculate the curl \( \abla \times \mathbf{F} \): \[abla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right)\mathbf{i} - \left(\frac{\partial R}{\partial x} - \frac{\partial P}{\partial z}\right)\mathbf{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\mathbf{k}\].Substitute \( P = 0 \), \( Q = z^2 \sinh y \), and \( R = 2z \cosh y \).

- Calculate Partial Derivatives

Calculate the partial derivatives involved in the curl expression: \( \frac{\partial R}{\partial y} = \frac{\partial (2z \cosh y)}{\partial y} = 2z \sinh y \) \( \frac{\partial Q}{\partial z} = \frac{\partial (z^2 \sinh y)}{\partial z} = 2z \sinh y \) \( \frac{\partial R}{\partial x} = \frac{\partial (2z \cosh y)}{\partial x} = 0 \) \( \frac{\partial P}{\partial z} = \frac{\partial 0}{\partial z} = 0 \) \( \frac{\partial Q}{\partial x} = \frac{\partial (z^2 \sinh y)}{\partial x} = 0 \) \( \frac{\partial P}{\partial y} = \frac{\partial 0}{\partial y} = 0 \)

- Verify the Curl is Zero

Substitute the partial derivatives into the curl expression: \( \abla \times \mathbf{F} = (2z \sinh y - 2z \sinh y)\mathbf{i} - (0 - 0)\mathbf{j} + (0 - 0)\mathbf{k} = 0 \).Since the curl is zero, the force field \( \mathbf{F} \) is conservative.

- Determine the Scalar Potential \( \phi \)

Since \( \mathbf{F} = - \abla \phi \), integrate the components of \( \mathbf{F} \) to find \( \phi \). Given \( \mathbf{F} = z^2 \sinh y \mathbf{j} + 2z \cosh y \mathbf{k} \), set up partial differential equations: \[\frac{\partial \phi}{\partial y} = - z^2 \sinh y \] Integrate with respect to \( \phi \): \[\frac{\partial \phi}{\partial y} = - z^2 \sinh y \Rightarrow \phi = - z^2 \cosh y + f(z)\]Then \[\frac{\partial \phi}{\partial z} = - 2z \cosh y + f'(z)\] Matching components, \( f'(z) = 0 \), so \( f(z) \) is a constant.

- Write the Final Scalar Potential \( \phi \)

Combine results to find \( \phi \): \[ \phi = -z^2 \cosh y + C \] Therefore, the scalar potential is \( \phi = -z^2 \cosh y + C \).

Key concepts

scalar potential

A scalar potential, often represented by \( \phi\ \), is a scalar function where the gradient \( - \abla \phi\ \) equals the given vector field \( \mathbf{F} \). Essentially, it provides a way to express a vector field as the gradient of a scalar field. This concept is very useful in physics and engineering.
In the given exercise, we need to determine the scalar potential for the force field \( \mathbf{F} = z^{2} \sinh y \mathbf{j} + 2z \cosh y \mathbf{k} \).
The steps involve:

• Checking if the curl of \( \mathbf{F} \) is zero (confirming it's conservative)
• Using integration to find the scalar potential \( \phi\ \).

Our final \( \phi\ \) is:
\[ \phi = -z^{2} \cosh y + C \] where \( C \) is a constant of integration.

curl of a vector field

The curl of a vector field helps determine if it is conservative. In simple terms, the curl measures the rotation or 'twisting' of a vector field. For a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \), we calculate the curl as:
\[ \abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right)\mathbf{ i} - \left( \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z}\right)\mathbf{ j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\mathbf{ k} \]
To verify if \( \mathbf{F} \) is conservative, the curl must be zero. In the exercise, after substituting \( P = 0, Q = z^2 \sinh y, and R = 2z \cosh y \), we found that \( \abla \times \mathbf{F} = 0 \), proving \( \mathbf{F} \) is conservative.

partial derivatives

Partial derivatives represent how a function changes as one of its variables changes while keeping other variables constant. They are fundamental in vector calculus. For example, in the exercise, to compute the curl, we calculate several partial derivatives:

• \( \frac{\partial(2z \cosh y)}{\partial y} = 2z \sinh y \)
• \( \frac{\partial(z^2 \sinh y)}{\partial z} = 2z \sinh y \)
• \( \frac{\partial(2z \cosh y)}{\partial x} = 0 \)

These partial derivatives help us confirm that the field is conservative and determine the scalar potential \( \phi \).

integration in vector calculus

Integration in vector calculus involves finding an antiderivative or potential function for a given vector field. In the exercise, we integrate the components of \( \mathbf{F} \) to discover \( \phi \). The steps include:

• Setting up integrals for each component, such as \( \frac{\partial \phi}{\partial y} = -z^{2} \sinh y \).
• Solving these integrals gives functions that, when summed correctly, provide the scalar potential.

For our force field, after integrating, we obtain \[ \phi = -z^2 \cosh y + C \] where \( C \) is a constant.