Question

Verify that each of the following force fields is conservative. Then find, for each, a scalar potential \(\phi\) such that \(\mathbf{F}=-\mathbf{\nabla} \phi\).\(\mathbf{F}=\mathrm{i}-\mathbf{j}-y \mathbf{k}\)

Short answer

The force field is conservative and the scalar potential is \(\phi = yz - x + C\).

Step-by-step solution

Understand the Force Field

The given force field is \(\mathbf{F} = \mathbf{i} - \mathbf{j} - y \mathbf{k}\). A force field \(\mathbf{F}\) is conservative if there exists a scalar potential function \(\phi\) such that \(\mathbf{F} = -\mathbf{abla} \phi\).

Verify the Force Field is Conservative

To verify that the force field is conservative, check if the curl of \(\mathbf{F}\) is zero, i.e., \(abla \times \mathbf{F} = \mathbf{0}\). Compute the curl as follows:\[abla \times \mathbf{F} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right)\mathbf{i} + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right)\mathbf{j} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)\mathbf{k}\] Use the components: \ F_x = 1, \ F_y = -1, \ F_z = -y. \ abla \times \mathbf{F} = \left(\frac{\partial (-y)}{\partial y} - \frac{\partial (-1)}{\partial z}\right)\mathbf{i} + \left(\frac{\partial (1)}{\partial z} - \frac{\partial (-y)}{\partial x}\right)\mathbf{j} + \left(\frac{\partial (-1)}{\partial x} - \frac{\partial (1)}{\partial y}\right)\mathbf{k} = 0+0+0 \ Therefore, \(\mathbf{F}\) is conservative.

Find the Scalar Potential

Given \(\mathbf{F} = \mathbf{i} - \mathbf{j} - y \mathbf{k}\), to find the scalar potential \(\phi\) such that \(\mathbf{F} = -\mathbf{abla} \phi\), integrate the components: \(F_x = -\frac{\partial \phi}{\partial x} = 1 \Rightarrow \phi = -x + f(y,z)\), \(F_y = -\frac{\partial \phi}{\partial y} = -1 \Rightarrow \frac{\partial \phi}{\partial y} = 1 \Rightarrow \phi = y - x + g(z),\),\ F_z = -\frac{\partial \phi}{\partial z} = -y \Rightarrow \frac{\partial \phi}{\partial z} = y \Rightarrow \phi = yz - x + C. Hence, the scalar potential function \(\phi\) is \(\phi = yz - x + C\)

Key concepts

scalar potential

A scalar potential \(\phi\) is a function from which we can derive a vector field by taking its negative gradient. In simpler terms, it helps to describe a force field in terms of a single scalar function. For a force field \(\mathbf{F}\), if we can find \(\phi\) such that \(\mathbf{F} = -\mathbf{\abla} \phi\), then \(\mathbf{F}\) is said to be conservative. \

In our example, given \(\mathbf{F} = \mathbf{i} - \mathbf{j} - y \mathbf{k}\), we need to find \(\phi\) such that \(\mathbf{F} = -\mathbf{\abla} \phi\). This involves integrating the components of \(\mathbf{F}\) step by step to reconstruct the scalar potential function \(\phi\). \

Understanding this concept is crucial, as many physical phenomena, such as gravitational and electric fields, can be described using a scalar potential, making analysis simpler and more intuitive.

vector calculus

Vector calculus deals with vector fields and scalar fields, involving operations like differentiation and integration of vectors. It provides the tools to analyze and solve problems involving force fields, fluid flow, electromagnetic fields, and more.

In our exercise, we utilize the concept of the gradient, \(\mathbf{\abla} \phi\), which transforms a scalar function into a vector field. Specifically, if we have a scalar potential \(\phi\), the gradient \(\mathbf{\abla} \phi\) gives a vector pointing in the direction of the steepest increase in \(\phi\).

To verify that \(\mathbf{F}\) is conservative, we also use the operation called 'curl', another fundamental tool in vector calculus, to check if the field has any rotational component. These operations are crucial for understanding and working with vector fields in physics, engineering, and other applied sciences.

curl of a vector field

The curl of a vector field measures the rotation or the 'twisting' of the field at a point. Mathematically, it is expressed as \(\mathbf{\abla} \times \mathbf{F}\). If the curl of a vector field \(\mathbf{F}\) is zero everywhere, the field is said to be conservative.

In our exercise, we compute the curl of \(\mathbf{F} = \mathbf{i} - \mathbf{j} - y \mathbf{k}\). Using the formula for curl, \[\mathbf{\abla} \times \mathbf{F} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right)\mathbf{i} + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right)\mathbf{j} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)\mathbf{k}\]
The computation shows that the curl is zero, confirming that \(\mathbf{F}\) is conservative.

integration of vector fields

To find the scalar potential \(\phi\) from a given conservative vector field \(\mathbf{F}\), we integrate the components of \(\mathbf{F}\). For \(\mathbf{F} = \mathbf{i} - \mathbf{j} - y \mathbf{k}\), we need to solve for \(\phi\)'s partial derivatives to reconstruct the function.

This process includes:

• Integrating \(F_x\) with respect to \x\ to find part of \(\phi\).
• Using \(F_y\) to find another part of \(\phi\).
• Finally, using \(F_z\) to complete \(\phi\).

For our field,

\(F_x = -\frac{\partial \phi}{\partial x} = 1 \Rightarrow \phi = -x + f(y,z)\)

\(F_y = -\frac{\partial \phi}{\partial y} = -1 \Rightarrow \frac{\partial \phi}{\partial y} = 1 \Rightarrow \phi = y - x + g(z)\)

\(F_z = -\frac{\partial \phi}{\partial z} = -y \Rightarrow \frac{\partial \phi}{\partial z} = y \Rightarrow \phi = yz - x + C\)

Hence, the scalar potential is \(\phi = yz - x + C\), where \(C\) is a constant. This step-wise integration is key to solving for scalar potentials in vector fields.