Question

5. State Stokes' theorem and verify it for F (32, 2r, y) with S being the open paraboloid z = 2+y with height 4. With which simpler surface could you replace the paraboloid for the same contour? verify Stokes the orem to

Answer 1

ANSWER F 32,2M the efen paraboleid with S height 4 2 xty with Stake's theorem open hwrfnee houndel by a clesed ewrve ad vecton F be any seeto faint dunetiom fartial derivatves. Then Contimuons instorder heving enri F.ds F.dT S s a unitnenmal 1veelan at an peint ef S dram hene n ミんt handed sereur wewld in the Sense in chih the bense of the deseribton ot e amre wohen netated the boundarg C of S is a Here in this feroblem 2 = 4 plane of nadins 2 and centre at in the こnele the aigieafoint (0,0,4) y=2n,2=4)0く丈く2F. Ret M=42 Cosオ lhe the parametrie eguations tc Then, f F.dF=すく32,2x,ソ>·くdx,よ7, d) , らく2,4eのちスん)く nt, 2cast, dt 2 TT dt + 8 -24 sin ,2T 2TT Aintdt t 4 ) &t ニ-24 24x 0 t 4[t. +3 [ 2] = 8 1T 2 1T li)

1-0)-30-3) + 32 2y 3, - the wrloceS: 2= n ty <-2a,-2y, 1) = <1.3,2)-,7, dA SS Curl F.nds R. tstere Rs the prejeton n 2 4,ie,on the plome ot c mRwe cam we M= Pcos, y = Yhin 2 =4 And dA r dr de 2T2 1,3,2)-2cs, -2ri, > 21 &-6rsing + 2r = Jne - 2TT (-2es -6inp 2IT 2c086 + 4 . 2T - 1G 3 27T j Cers dg48 3 4do + 2 TT 216 x0 -4xO + 3 3 8 1T li) 11

Foren i)(i) verified wwe can see that Stoke's thearem phere hinpler surface like We cmld have Used ty'+2= 4 nstead tthe parabalaid to verify Stoke 's thesrem with the same cotnr C