Nz > 0 S = {z = z(x, y), (x, y) ∈ D ⊂ R2}

�S

�Fd�S=�D

det

⎡⎣Fx Fy Fz

1 0 ∂xz0 1 ∂yz

⎤⎦dxdy �

S

fdS=�D

f(x, y, z(x, y)) ·√1 + ( ∂z

∂x)2 + (∂z

∂y)2 · dxdy

�S

�Fd�S=�D

(Fx ·x+Fy ·y)dφdz S={r=R, (φ, z)∈D}

Sf→ R S =

{(r · cos(φ), r · sin(φ), z(r, φ))∣∣ (r, φ) ∈ D

}�S

fdS =�D

f(x(r, φ), y(r, φ), z(r, φ))√r2 + ( ∂z

∂φ)2 + r2(∂z

∂r)2dr · dφ

S ⊆ S2 = {x2+ y2+ z2 = R2} ⊂ R3 S

��S

�Fd�S�S

fdS

�S

�Fd�S =�D

f · r3sin(θ)dθdφ �F = f · �r

S ={�r = �r(φ1, φ2)

∣∣ (φ1, φ2) ∈ D}

�S

fdS =�D

f(x(φ1, φ2), y(φ1, φ2), z(φ1, φ2)

)(R + r · sin(φ1)

)r · dφ1dφ2

{x2 + y2 + z2 = a2

x2

a2+ z2

b2≤ 1, 0 < b ≤ a

} {z = 2− x2 − y2

z ≥ √x2 + y2

}

{(x, y, z)| x2 + y2 = R2, x < z ≤ 2x}{√x2 + y2 ≤ z ≤ 1} S

�S

(x2 + y2)dS

{x2 + y2 + z2 = 1, −12≤ z ≤

√32, x ≤ y}xy

{x2

a2+ y2

b2+ z2

c2= 1} �F = ( 1

x, 1y, 1z)

0 ← εx, εy, εz {|x| > εx, |y| > εy, |z| > εz}πxz πyz πxy S ⊂ R

3

sign(Ni)∫�S

�F · d�S =∫

πxy(S)

Fz · sign(Nz)dxdy +∫

πyz(S)

Fx · sign(Nx)dydz +∫

πxz(S)

Fy · sign(Ny)dxdz

S = {0 ≤ z =√x2 + y2 ≤ 1} ⊂ R

3 �F = ln(y2 + z2 + 1)x+ ex

z2+1y + (x− y − 1)z

S = {(x, y, z)| x2 + y2 + z2 = R2} ⊂ R3

��S

ayz·dydz+bxz·dxdz+cxy·dxdy(x2+y2+z2)n

�F = (3x− 2y + z)x+ (2x+ 3y − z)y + (x− 3y + z)zS =

{|3x− 2y + z|+ |2x+ 3y − z|+ |x− 3y + z| ≤ 1} ⊂ R

3

S = {x2 + y2 + z2 = 4, z ≥ −√2} �F = (0, x+ 1, 0) rot(�F ){

|x|3 + |y|4 + |z + 14|2 ≤ 4, x2 + y2 − 1 ≤ z

}�F = �r

r3

C2 Vf,g,h→ R ∂V = S V ⊂ R

3∫Scos(N , �v)dS (N , �v)�

∂Vf∇g · d2�S =

�∂V

f∂N g · d2S =�

V

((∇f) · (∇g) + f∆g

)d3V�

∂Vg ·

(f · ∂Nh− h · ∂Nf

)· d2S =

�V

(f∇(g∇h)− h∇(g∇(f))

)d3V

f |∂V f V ∆(f)=0 f∂V f g = f h = 1

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