Question

. a) Determine whether the following force field F  is conservative: F ˆ i ˆ j k ˆ  x  y 

Answer 1

A field is a area where the force are applied...

I think you understand

Answer 2

Conservative vector field

DEFINITION. If the line integral of a vector field $\mathrm{\vec{F}}$ depends on $\mathrm{\vec{F}}$, and on the end points $\mathrm{A,\:B}$ but not on the nature of the path joining $\mathrm{A}$ and $\mathrm{B}$, then we call the vector field $\mathrm{\vec{F}}$ a conservative vector field.

Formula. In rectangular cartesian system, the condition of conservative field for the vector field $\mathrm{\vec{F}}$ is

$\quad\quad \mathrm{curl\:\vec{F}=0}$

$\quad i.e.,\:\color{blue}{\mathrm{\nabla\times\vec{F}=\vec{0}}}$.

Solution.

The given force field is

$\quad\quad\mathrm{\vec{F}=x\hat{i}+y\hat{j}}$

To find if $\mathrm{\vec{F}}$ is conservative.

$\therefore\mathrm{\nabla\times\vec{F}}$

$=\left|\begin{array}{ccc}\mathrm{\hat{i}}&\mathrm{\hat{j}}&\mathrm{\hat{k}}\\ \mathrm{\frac{\partial}{\partial x}}&\mathrm{\frac{\partial}{\partial y}}&\mathrm{\frac{\partial}{\partial z}}\\\mathrm{x}&\mathrm{y}&\mathrm{0}\end{array}\right|$

$=\mathrm{\left[\frac{\partial}{\partial y}(0)-\frac{\partial y}{\partial z}\right]\hat{i}-\left[\frac{\partial}{\partial x}(0)-\frac{\partial x}{\partial z}\right]\hat{j}}\\\mathrm{+\left[\frac{\partial y}{\partial x}-\frac{\partial x}{\partial y}\right]\hat{k}}$

$\quad$(Expanding along the first row.)

$\mathrm{=0\:\hat{i}+0\:\hat{j}+0\:\hat{k}}$

$\mathrm{=\vec{0}}$

$\Rightarrow \mathrm{\nabla\times\vec{F}=\vec{0}}$

$\color{red}{\therefore \mathrm{\vec{F}\:is\:conservative.}}$

Vector related problem.

A vector of magnitude $\color{blue}{\mathrm{10}}$ unit is perpendicular to $\color{blue}{\mathrm{\vec{a}=\hat{i}+\hat{j}-\hat{k}}}$ and coplanar with the vector $\color{blue}{\mathrm{\vec{b}=2\hat{i}-\hat{j}-\hat{k}}}$ and $\color{blue}{\mathrm{\vec{c}=\hat{i}+2\hat{j}-\hat{k}}}$ is

- https://brainly.in/question/12666578