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aruv | Student

a. Let S be the given surface with its boundary C, which path. Then Stoke's Theorem relates a line integral and surface integral of some vector field F i.e.,

`int_C vecF.d vecr=int int_S curlvecF.dvec S`

We have given vector field F=<yz,-xz,z^3> and surface

S={(x,y,z): z=-sqrt(x^2+y^2) and `-2<=z<=0}`

We have



Let transform this equation in parametric form





`=> r=2`


`vecr(t)=2cos(t)vec i+2sin(t)vecj-2veck`

`dvecr=(-2sin(t) vec i +2cos(t) vecj) dt`

`vecF=(2sin(t)(-2)) veci -(2cos(t))(-2)vec j+ (-2)^3 vec k`

`=-4sin(t)veci+4cos(t) vecj -8vec k`




```dvecS=(gradg)/||gradg|| ||gradg|| dA`


`gradg=x/sqrt(x^2+y^2) veci+y/sqrt(x^2+y^2) vecj+vec k`




Thus substitute these values in surface integral and evalute. We will

get  8pi.

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