Multivariable Review 14.1-14.3

A vector field F and a curve C are shown. The work required to move an object along C is

• positive

• negative

• zero

• cannot be determined

Evaluate  $\int_C^{ }\ yds$  , where C is the parabola  $y=x^2$  from (0,0) to (3,9).  (Calc OK)

• 41.590

• 40.5

• 0.606

• -40.5

Evaluate  $\int_C^{ }\ x^3dx$ where C is  $x^2+y^2=1$  with counterclockwise direction.  (No Calc)

• 0

• 1/2

• -1/2

• 1/4

Find a potential function f(x,y,z) for  $\overrightarrow{F}\left(x,y,z\right)=<\sin y,x\cos y,-\sin z>$(x,y,z)=<siny,xcosy,−sinz> 

•  $f\left(x,y,z\right)=x\sin y+\cos z+C$  

•  $f\left(x,y,z\right)=x\cos y+\sin z+C$  

•  $f\left(x,y,z\right)=x\sin y+C$  

• F is not conservative

 Find the work required to move an object through force field $\overrightarrow{F}\left(x,y\right)=<x^2y,e^y>\$ along a path C given by  $\overrightarrow{r}\left(t\right)=<t,-t^2>$  between t=0 and t = 1.  (No Calc)

•  $e^{-1}-\frac{6}{5}$  

•  $e^{-1}-\frac{1}{5}$  

•  $e^1+\frac{4}{5}$  

•  $e^{-1}-\frac{4}{5}$  

 Evaluate $\int_C^{ }\ \overrightarrow{F}d\overrightarrow{r}$ where  $\overrightarrow{F}=<4x^3y^2-2xy^3,2x^4y-3x^2y^2+4y^3>$  and C goes between (0,1) to (2,5).  (Calc OK)

• 524

• Cannot be determined since F is not conservative.

• 1524

• 523

If F is conservative, then it is independent of path and FTC can be applied.

• True

• False