Question
Verify that each of the following force fields is conservative. Then find, for each, a scalar potential $\phi$ such that $\mathbf{F}=-\nabla \phi$.$$\mathbf{F}=\mathbf{i}-z \mathbf{j}-y \mathbf{k}$$
Step 1
Recall that the curl of a vector field $\mathbf{F}$ is given by $\nabla \times \mathbf{F}$. If the curl is zero, then the field is conservative. Let's calculate the curl of $\mathbf{F}=\mathbf{i}-z \mathbf{j}-y \mathbf{k}$: $$\nabla \times \mathbf{F} = Show more…
Show all steps
Your feedback will help us improve your experience
James Kiss and 82 other educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Verify that each of the following force fields is conservative. Then find, for each, a scalar potential $\phi$ such that $\mathbf{F}=-\mathbf{\nabla} \phi$.$\mathbf{F}=\mathrm{i}-\mathbf{j}-y \mathbf{k}$
VECTOR ANALYSIS
Line integrals
Verify that each of the following force fields is conservative. Then find, for each, a scalar potential $\phi$ such that $\mathbf{F}=-\nabla \phi$. $$\mathbf{F}=\left(3 x^{2} y z-3 y\right) \mathbf{i}+\left(x^{3} z-3 x\right) \mathbf{j}+\left(x^{3} y+2 z\right) \mathbf{k}$$
Vector Analysis
Line Integrals
Verify that each of the following force fields is conservative. Then find, for each, a scalar potential $\phi$ such that $\mathbf{F}=-\nabla \phi$. $$\mathbf{F}=-k \mathbf{r}, \mathbf{r}=\mathbf{i} x+\mathbf{j} y+\mathbf{k} z, \quad k=\mathrm{const.}$$
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD