# True or false. a) The vector field mathbf{F} {x,y} = frac{ {y,x} }{sqrt{x^2 + y^2}} is a...

## Question:

True or false.

a) The vector field {eq}\mathbf{F} \left \langle {x,y} \right \rangle = \frac{\left \langle {y,x} \right \rangle}{\sqrt{x^2 + y^2}} {/eq} is a radial vector field.

b) The vector field {eq}\mathbf{F} = \left \langle {2,5} \right \rangle {/eq} is conservative (i.e. has a potential function).

## Vector Field:

{eq}\\ {/eq}

Radial Vector Field - A rationally symmetric vector field is a Radial Vector Field i.e. {eq}F(x) {/eq} is radial if {eq}F(x)=f(|x|)x {/eq} with some function {eq}f(r) {/eq} or we can say that if {eq}F(x)=f(|r|)r {/eq}.

## Answer and Explanation:

{eq}\\ {/eq}

(a) TRUE

Explanation - A vector field {eq}F(x) {/eq} is radial if {eq}F(x)=f(|x|)x {/eq} with some function {eq}f(r). {/eq}

(b) Neither TRUE nor FALSE.

Explanation - The given statement is incomplete to decide whether it is true/false.

A vector field {eq}\vec F=P\widehat{i}+Q\widehat{j} {/eq} is conservative if {eq}\dfrac{\delta P}{\delta y}=\dfrac{\delta Q}{\delta x} {/eq}.