Question

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

Short answer

\[ \mathbf{F} = y \mathbf{i} + x \mathbf{j} + \mathbf{k} \] is conservative. The potential function is \[ \phi = -yx - z + C \].

Step-by-step solution

Determine Conservativeness

To verify if the force field \(\mathbf{F}=y\mathbf{i}+x\mathbf{j}+\mathbf{k}\) is conservative, check if the curl of \(\mathbf{F}\) is zero. In three dimensions, the components of the curl are given by \(abla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right)\mathbf{i} + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right)\mathbf{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right)\mathbf{k} \).

Compute the Partial Derivatives

Compute the partial derivatives needed for the curl: \- \(\frac{\partial F_z}{\partial y} = \frac{\partial 1}{\partial y} = 0\)\- \(\frac{\partial F_y}{\partial z} = \frac{\partial x}{\partial z} = 0\)\- \(\frac{\partial F_x}{\partial z} = \frac{\partial y}{\partial z} = 0\)\- \(\frac{\partial F_z}{\partial x} = \frac{\partial 1}{\partial x} = 0\)\- \(\frac{\partial F_y}{\partial x} = \frac{\partial x}{\partial x} = 1\)\- \(\frac{\partial F_x}{\partial y} = \frac{\partial y}{\partial y} = 1\)

Check the Curl

Using the partial derivatives obtained, the curl is: \(abla \times \mathbf{F} = \left( 0 - 0 \right)\mathbf{i} + \left( 0 - 0 \right)\mathbf{j} + \left( 1 - 1 \right)\mathbf{k} = 0\). Since the curl is zero, the force field \(\mathbf{F}\) is conservative.

Find the Potential Function

Since the force field is conservative, there exists a scalar potential function \(\phi\) such that \(\mathbf{F} = -abla \phi\). This means \(\frac{\partial \phi}{\partial x} = -y\), \(\frac{\partial \phi}{\partial y} = -x\), \(and \frac{\partial \phi}{\partial z} = -1\).

Integrate to Find \(\phi\)

Integrate \(\frac{\partial \phi}{\partial x} = -y\) with respect to \(x\): \(\phi = -yx + g(y,z)\), where \(g(y,z)\) is a function of \(y\) and \(z\) only.\ Next, integrate the result with respect to \(y\): \(-x = \frac{\partial (-yx + g(y,z))}{\partial y} = -x + \frac{\partial g(y,z)}{\partial y}\). Hence, \(\frac{\partial g(y,z)}{\partial y} = 0 \) implying \(g(y,z) = h(z)\), a function of \(z\) only.\ Finally, integrate with respect to \(z\): \(\frac{\partial \phi}{\partial z} = \frac{\partial (-yx + h(z))}{\partial z} = h'(z) = -1\) yielding \(h(z) = -z + C\).

Combine to Find \(\phi\)

Combining all parts, the scalar potential is: \(\phi = -yx - z + C\).

Key concepts

Scalar Potential

In physics and mathematics, a scalar potential is a scalar field whose gradient gives a particular vector field. For a given conservative force field \(\textbf{F}\), the scalar potential \(\textbackslashphi\) satisfies the equation \(\textbf{F} = -\textbackslashnabla \textbackslashphi\). This means that the force field can be derived from the potential function. For example, if we have a force field \(\textbf{F} = y\textbf{i} + x\textbf{j} + \textbf{k}\), we find a scalar potential \(\textbackslashphi\) such that taking the negative gradient of \(\textbackslashphi\) results in \(\textbf{F}\). In practice, you find \(\textbackslashphi\) by integrating the partial derivatives of \(\textbackslashphi\) based on \(\textbf{F}\)'s components.

Vector Calculus

Vector calculus is a branch of mathematics that focuses on vector fields and their manipulation through operations such as differentiation and integration. It's essential to understand concepts like the gradient, divergence, and curl when working with vector fields. A vector field assigns a vector to each point in space, and in problems regarding conservative vector fields, we often deal with the curl of a vector field to check conservativeness. Vector calculus allows you to perform complex operations that help in understanding the physical significance and behavior of vector fields.

Partial Derivatives

Partial derivatives are used in multivariable calculus to express how a function changes as one of its variables is varied, holding the other variables constant. For a function \(\textbackslashphi(x, y, z)\), the partial derivative with respect to \(x\) is denoted as \(\textbackslashfrac{\textbackslashpartial\textbackslashphi}{\textbackslashpartial x}\). In the context of verifying conservative force fields, we compute the partial derivatives to find the curl. For example, \(\textbackslashfrac{\textbackslashpartial F_y}{\textbackslashpartial z}\) and \(\textbackslashfrac{\textbackslashpartial F_z}{\textbackslashpartial y}\) are needed to check if \(\textbackslashtextbf{F}\) is conservative.

Curl of a Vector Field

The curl of a vector field \(\textbackslashtextbf{F}\) measures the field's tendency to rotate around a point. It's given by the cross product of the gradient operator with the vector field, denoted as \(\textbackslashnabla \times \textbackslashtextbf{F}\). If the curl is zero everywhere in the domain, the vector field is conservative. For \(\textbackslashtextbf{F} = y\textbackslashtextbf{i} + x\textbackslashtextbf{j} + \textbackslashtextbf{k}\), calculating the curl involves taking partial derivatives such as \(\textbackslashfrac{\textbackslashpartial F_z}{\textbackslashpartial y}\) and \(\textbackslashfrac{\textbackslashpartial F_y}{\textbackslashpartial z}\). If all these derivatives result in zeros, then the curl is zero indicating the field is conservative.