Question

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

Short answer

The force field is conservative and the scalar potential $\mathbf{\text{phi}}=\frac{k}{2}(x^2+y^2+z^2)+C$.

Step-by-step solution

- Identify the Force Field

Given force field is $\mathbf{\text{F}}=-k\mathbf{\text{r}}=-k(x\mathbf{\text{i}}+y\mathbf{\text{j}}+z\mathbf{\text{k}})$.

- Verify if the Force Field is Conservative

A force field is conservative if $abla\times \mathbf{\text{F}}=0$. Compute the curl of $\mathbf{\text{F}}$: $abla\times \mathbf{\text{F}}=abla\times (-k(x\mathbf{\text{i}}+y\mathbf{\text{j}}+z\mathbf{\text{k}}))$. Since $k$ is a constant, the partial derivatives of each component will be zero. Hence, $abla\times \mathbf{\text{F}}=0$, confirming that $\mathbf{\text{F}}$ is a conservative force field.

- Set up the Potential Function

To find the scalar potential $\mathbf{\text{phi}}$ such that $\mathbf{\text{F}}=-abla\mathbf{\text{phi}}$: $-abla\mathbf{\text{phi}}=-k(x\mathbf{\text{i}}+y\mathbf{\text{j}}+z\mathbf{\text{k}})$. Therefore, $\frac{\mathbf{\text{d}}\mathbf{\text{phi}}}{\mathbf{\text{d}}x}=kx$, $\frac{\mathbf{\text{d}}\mathbf{\text{phi}}}{\mathbf{\text{d}}y}=ky$, and $\frac{\mathbf{\text{d}}\mathbf{\text{phi}}}{\mathbf{\text{d}}z}=kz$. Integrate each of these equations with respect to $x$, $y$, and $z$, respectively.

- Solve for the Potential Function $\mathbf{\text{phi}}$

Integrate $\frac{\mathbf{\text{d}}\mathbf{\text{phi}}}{\mathbf{\text{d}}x}=kx$: ${\mathbf{\text{phi}}}_1(x)=\frac{k{x}^2}{2}+C(y,z)$; integrate $\frac{\mathbf{\text{d}}\mathbf{\text{phi}}}{\mathbf{\text{d}}y}=ky$: ${\mathbf{\text{phi}}}_2(y)=\frac{k{y}^2}{2}+C(x,z)$; and integrate $\frac{\mathbf{\text{d}}\mathbf{\text{phi}}}{\mathbf{\text{d}}z}=kz$: ${\mathbf{\text{phi}}}_3(z)=\frac{k{z}^2}{2}+C(x,y)$. By verifying the consistency, the combined potential φ with constants accounted for is $\mathbf{\text{phi}}(x,y,z)=\frac{k}{2}(x^2+y^2+z^2)+C$.

Key concepts

scalar potential

A scalar potential is a function whose gradient yields a given vector field. In this exercise, you are asked to find such a function for a given force field $\mathbf{F}$. The scalar potential is denoted by $\varphi$ and satisfies the equation $\mathbf{F}=-abla\varphi$. This implies that the force field can be represented as the negative gradient of the scalar potential. This relationship is key in identifying conservative force fields and simplifying the analysis of physical systems.

vector calculus

Vector calculus is a branch of mathematics focusing on vector fields and differential operators. Concepts such as divergence, gradient, and curl are fundamental in this field. In the context of the problem, vector calculus helps to determine if a force field is conservative, usually by checking if the curl of the vector field is zero. This branch of calculus also provides tools like line integrals and surface integrals to facilitate the computing of quantities associated with vector fields in physics and engineering.

gradient

The gradient of a scalar field $\varphi$ is a vector field that points in the direction of the greatest rate of increase of $\varphi$. Mathematically, it is denoted as $abla\varphi$. In this exercise, the gradient plays a crucial role because the force field $\mathbf{F}$ can be expressed as the negative gradient of the scalar potential. To find the scalar potential, you start by identifying the given force field and then setting up the equations to match $\mathbf{F}=-kx\hat{i}-ky\hat{j}-kz\hat{k}$ to $-abla\varphi$. By integrating partial derivatives, you can determine $\varphi$, confirming that the force field is derived from a potential function.

curl

The curl of a vector field measures the field's tendency to rotate around a point. Mathematically, the curl of $\mathbf{F}$ is represented by $abla\times \mathbf{F}$. In this exercise, to verify if the force field is conservative, you check if its curl is zero: $abla\times \mathbf{F}=0$. A zero curl indicates that the vector field is irrotational, meaning it is conservative. For the given force field, calculating the curl and seeing that it is zero confirms that we can find a scalar potential $\varphi$.

integration in mathematics

Integration is a crucial mathematical process that allows you to compute the antiderivatives of functions. In this exercise, you need to integrate the components of the force field to find the scalar potential $\varphi$. The problem involves integrating the partial derivatives $\frac{\partial \varphi }{\partial x}=kx$, $\frac{\partial \varphi }{\partial y}=ky$, and $\frac{\partial \varphi }{\partial z}=kz$. These integrations yield the potential function $\varphi (x,y,z)=\frac{k}{2}(x^2+y^2+z^2)+C$. Understanding integration techniques is essential to solving such problems in vector calculus and physics.