University of Illinois at Urbana Champaign Fall
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4
pages
English
Documents
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- pointing away
- divergence theorem seems
- urbana-champaign fall
- y6 ?
- normal vector
- ∂f ∂x
Publié par
Langue
English
1.
sin(2πt) 2γ x(t) = t y(t) = e z(t) = ln(1 + t ) 0 ≤ t ≤ 1Z
2 z 2 zt (3x +3yz +e )dx+(3xz +3y )dy +(3xy +xe )dz
γ
Z
∂f ∂f ∂f3 3 zf(x,y,z) = x +y + 3xyz +xe dx+ dy + dz
∂x ∂y ∂zγ
f(B)−f(A) A,B B
t = 1 A t = 0
Z
2 z 2 z(3x +3yz +e )dx+(3xz +3y )dy +(3xy +xe )dz = (1+1+3ln(2)+2)−(0+1+0+0) = 3+3ln(2) .
γ
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2 2S x≥ 0 y ≥ 0 x +y = 1 0≤ z ≤ 2ZZ
3 z 2 2~ ~F(x,y,z) = (x +sin(yz)+e ,x+z ,3zy −x) F ·~ndσ
S
∂F ∂F ∂Fx y z 2 2~(F) = + + = 3x +0+3y .
∂x ∂y ∂z
2 2 2x +y = r
ZZ Z Z Z Z Z2 1 π/2 2 1
π π 3 3π2 3~F ·~ndσ = 3r ·rdrdθdz = 3r drdz = ·2· = .
2 2 4 4S z=0 r=0 θ=0 z=0 r=0
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2 2 2S 1 (x,y,z) x +y +z = 1 z≥ 0ZZ
(2xy +z)dσ
S
x = sin(ϕ)cos(θ) y = sin(ϕ)sin(θ)
πz = cos(ϕ) 0≤ θ ≤ 2π 0≤ ϕ≤ dσ2
2−sin(θ)sin(ϕ) cos(θ)cos(ϕ) cos(θ)sin (ϕ)
∂P ∂P 2 × = cos(θ)sin(ϕ) × sin(θ)cos(ϕ) = sin(θ)sin (ϕ)
∂θ ∂ϕ
0 −sin(ϕ) cos(ϕ)sin(ϕ)
sin(ϕ)
Z Z Z Zπ/2 2π π/2 2π sin(2ϕ)3 3I = 2sin (ϕ)cos(θ)sin(θ)+sin(ϕ)cos(ϕ) dθdϕ = sin(2θ)sin (ϕ)+ dθdϕ
2ϕ=0 θ=0 ϕ=0 θ=0
Z π/2 sin(2ϕ)
I = 2π dϕ = π .
2ϕ=0
Correction.,theisofresoothet:ecductv;thisComputeofusinsisco(15,pisointhets)this,Letdeterminebwe.theLetuppuseerusualhemisphereystemofsphericalradiusordinates,,i.e;thedomainset,ofcross-propcomputeoinwtsforsuc.htothethatneedtegralnoeandW.Themagnitude4.
3S x+y +z = 1 x≥ 0 y≥ 0 z≥ 0 ZZ
2~ ~ ~F F(x,y,z) = (x+y +z,y−1,z) F ·~ndσ
S
3z = 1−x−y
3x,y 0≤ x 0≤ y x+y ≤ 1
2 2 2 3 2 2 3~F·~ndσ =±(x+y +z,y−1,z)·(1,3y ,1)dxdy = (x+y +z+3y −3y +z)dxdy = (−x−2y +y +2)dxdy
3f(x,y,z) = x+y +z
S z x,y f
f S f
f
3Z Z Z1 1−y 1 3 2 (1−y )2 3 2 3 3 3 3
I = −x−2y +y +2 dxdy = − −2y (1−y )+y (1−y )+2(1−y ) dy
2y=0 x=0 y=0
Z 1 3 3 3 3 2 1 206 2 5
I = − y −2y +2y dy = − − + = .
2 2 2 14 3 3 21y=0
areallyfromneedorigintothemak(beEvaOneskbetc,hsurface(whicincrease).goingowaouldisb,ebhardnormalin.thattcase)a:assimplyit,wourissurfaceoishoiceanyequip;otennaturaltial.ofectorthefromfunctionpisethatsurfaceonets).theuseisioriginathefromomthervingfe'llynatuallywobtainain,factseoionitorieniThesandnormalentodomaintherepresengradientotit;Oneat.adntheyorigin,paoindotoontedting,,equationwhentheoinLetincreasest(withecapfornormalh)sthegoingincreaseswtoyo,thewhicishsamemehaansgivthatWthediscussgradienitclass.ofensurface,,isepthatoinurtingtegralup.that,Also,theoncomesavstraighabttlinetfromofthecoriginobtainsto.a,pboingivtforontheourtation,cartesiantheusefunctionhereenisissoconhastinCorrection.uallyComputeincreasing.yHenedeeldncevtheenormalandptheoinytingwatingesn'toinwseeaectoryvfromwiththenoriginoriis,the,oneofgivTenethis,bpuponeoingradienb(15ytheh
