Conservative

Not conservative

The process of finding the potential function involves the integration and differentiation.

If the potential function does not exist, then the vector field is not conservative

The potential function can be a single function f(x).

The potential function for the gravitational vector field is$f\left(x,y,z\right)=\frac{mMG}{\sqrt{x^2+y^2+z^2}}$

Line integral  $\int_C^{ }f\left(x,y\right)\ ds$  gives the length of the curve C

Line integral is similar to a single integral except that instead of integrating over x axis, we integrate over a curve C

If the curve C is not smooth, then we have to split the curve into several segments before we determine the line integral

If C is a circle on xy-plane, then the line integral  $\int_C^{ }f\left(x,y\right)ds$  where  $f\left(x,y\right)\ge0$  gives the surface area of a circular cylinder.

 $\frac{3}{2}$  

 $\frac{7}{4}$  

 $2$  

 $\frac{9}{4}$