Multiple Choices and True$/$False Questions

Choose the correct answer for each question

$(1)$ The vector field vec$(F)(x,y)=(x^2-yx,y^2-xy)$ is conservative

$($a$)$ True

$($b$)$ False

$(2)$ The derivative of a vector function is obtained by differentiating each component functions

$($a$)$ True

$($b$)$ False

$(3)$ grad:$(grad(gradf(x,y)))$ is a vector field

$($a$)$ True

$($b$)$ False

$(4)$ The domain of the function $\sqrt[\phantom{2}]{1-x^2-y^2}*ln(x^2+y^2-1)$ is empty

$($a$)$ True

$($b$)$ False

$(5)$ If vec$(F)$ is a vector field on $R^3,$ then the curl of vec$(F)$ is a scalar field

$($a$)$ True

$($b$)$ False

$(6)$ The vector field vec$(F)=-\frac{y}{x^2+y^2}$hat$(i)+\frac{x}{x^2+y^2}$hat$(j)$ satisfies $\frac{\text{d}\text{e}\text{l}\text{Q}}{\text{d}\text{e}\text{l}\text{x}}=\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{y}}$ throughout its domain. Is ${\displaystyle \underset{C_1}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r)={\displaystyle \underset{C_2}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r)?$

$($a$)$ True

$($b$)$ False

$(7)$ If $f$ has continuous partial derivatives on $R^3,$ then grad$*(gradf)=0$

$($a$)$ True

$($b$)$ False

$(8)$ For every vector field vec$(F)in{R}^3,$grad$(grad*vec(F))=0$

$($a$)$ True

$($b$)$ False

$(9)$ If the vector field vec$(F)$ is conservative on the open region $D$ then line integrals of vec$(F)$ are path$-$independent on $D,$ regardless of the shape of $D.$

$($a$)$ True

$($b$)$ False