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8/8/2019 Surface Integration
http://slidepdf.com/reader/full/surface-integration 1/23
Surface Integration
Eddie Wilson
Department of Engineering Mathematics
University of Bristol
Surface Integration – p.1/ ?
8/8/2019 Surface Integration
http://slidepdf.com/reader/full/surface-integration 2/23
Integration in the plane
x
y
S
f (x, y)δS
S
f (x, y) dS = lim
f (x, y)δS.
Surface Integration – p.2/ ?
8/8/2019 Surface Integration
http://slidepdf.com/reader/full/surface-integration 3/23
Double integration and its order
xy
S
δx
f (x, y) dy
δx
S
f (x, y) dS = lim
f (x, y) dy
δx
,
= f
(x, y
) dy
dx.
Surface Integration – p.3/ ?
8/8/2019 Surface Integration
http://slidepdf.com/reader/full/surface-integration 4/23
Double integration and its order
xy
S
δy
f (x, y) dx
δy
S f (x, y) dS = lim
f (x, y) dx
δy
,
=
f (x, y) dx
dy.
Surface Integration – p.3/ ?
8/8/2019 Surface Integration
http://slidepdf.com/reader/full/surface-integration 5/23
Formulae for surfaces
Examples in the plane
C :=
(x,y,z) such that x2 + y2 ≤ a2, z = 0
x, y ≥ 0, z = 1.
Surface Integration – p.4/ ?
8/8/2019 Surface Integration
http://slidepdf.com/reader/full/surface-integration 6/23
Formulae for surfaces
Examples in the plane
C :=
(x,y,z) such that x2 + y2 ≤ a2, z = 0
x, y ≥ 0, z = 1.
Fully 3D examples:
z = x2
+ y2
, x, y ≥ 0,z = xy
x2 + y2 + z2 = a2
Surface Integration – p.4/ ?
8/8/2019 Surface Integration
http://slidepdf.com/reader/full/surface-integration 7/23
Formulae for surfaces
Examples in the plane
C :=
(x,y,z) such that x2 + y2 ≤ a2, z = 0
x, y ≥ 0, z = 1.
Fully 3D examples:
z = x2
+ y2
, x, y ≥ 0,z = xy
x2 + y2 + z2 = a2
How to write so that 2D nature is clear? c.f. curves
Surface Integration – p.4/ ?
8/8/2019 Surface Integration
http://slidepdf.com/reader/full/surface-integration 8/23
Parametric representation
IDEA: create two extra parameters which trace outsurface. Write all coordinates in new parameters.
Examples:
Surface Integration – p.5/ ?
8/8/2019 Surface Integration
http://slidepdf.com/reader/full/surface-integration 9/23
Parametric representation
IDEA: create two extra parameters which trace outsurface. Write all coordinates in new parameters.
Examples:
BEFORE: x2 + y2 = a2.AFTER: x = a cos θ, y = a sin θ, z = t,θ ∈ [0, 2π), t ∈ (−∞, +∞).
Surface Integration – p.5/ ?
8/8/2019 Surface Integration
http://slidepdf.com/reader/full/surface-integration 10/23
Parametric representation
IDEA: create two extra parameters which trace outsurface. Write all coordinates in new parameters.
Examples:
BEFORE: x2 + y2 = a2.AFTER: x = a cos θ, y = a sin θ, z = t,θ ∈ [0, 2π), t ∈ (−∞, +∞).
BEFORE: x2 + y2 + z2 = a2.AFTER: x = a sin θ cos φ, y = a sin θ sin φ, z = cos θ,
θ ∈ [0, π], φ ∈ [0, 2π).
Surface Integration – p.5/ ?
8/8/2019 Surface Integration
http://slidepdf.com/reader/full/surface-integration 11/23
Parametric representation
IDEA: create two extra parameters which trace outsurface. Write all coordinates in new parameters.
Examples:
BEFORE: x2 + y2 = a2.AFTER: x = a cos θ, y = a sin θ, z = t,θ ∈ [0, 2π), t ∈ (−∞, +∞).
BEFORE: x2 + y2 + z2 = a2.AFTER: x = a sin θ cos φ, y = a sin θ sin φ, z = cos θ,
θ ∈ [0, π], φ ∈ [0, 2π).
BEFORE: z = xy, x ≥ 0
AFTER: x = t1, y = t2, z = t1t2,t1 ≥ 0, t2 ∈ (−∞, +∞).Surface Integration – p.5/ ?
8/8/2019 Surface Integration
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Principle of surface integration
−1
−0.5
0
0.5
1 −1
−0.5
0
0.5
10
1
2
yx
z
S
S
f (x) dS =?
Surface Integration – p.6/ ?
8/8/2019 Surface Integration
http://slidepdf.com/reader/full/surface-integration 13/23
Principle of surface integration
−1
−0.5
0
0.5
1 −1
−0.5
0
0.5
10
1
2
yx
z
S
x
area δS
S
f (x) dS = lim f (x) δS
Surface Integration – p.6/ ?
8/8/2019 Surface Integration
http://slidepdf.com/reader/full/surface-integration 14/23
8/8/2019 Surface Integration
http://slidepdf.com/reader/full/surface-integration 15/23
Infinitesimal surface element
n δS =
∂ r
∂t1×
∂ r
∂t2
δt1 δt2
δS
r(t1, t2)
r(t1, t2 + δt2)
r(t1 + δt1, t2 + δt2)
r(t1 + δt1, t2)
∂ r
∂t1δt1
∂ r
∂t2
δt2
Surface Integration – p.7/ ?
8/8/2019 Surface Integration
http://slidepdf.com/reader/full/surface-integration 16/23
Infinitesimal surface element
δS =
∂ r
∂t1×
∂ r
∂t2
δt1 δt2
δS
r(t1, t2)
r(t1, t2 + δt2)
r(t1 + δt1, t2 + δt2)
r(t1 + δt1, t2)
∂ r
∂t1δt1
∂ r
∂t2
δt2
Surface Integration – p.7/ ?
H t l l t I
f( ) dS
8/8/2019 Surface Integration
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How to calculate I =S f (x) dS
Surface Integration – p.8/ ?
H t l l t I
f( ) dS
8/8/2019 Surface Integration
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How to calculate I =S f (x) dS
1: Express surface in parametric form:
r = r(t1, t2) for a1 < t1 < b1, a2 < t2 < b2.
Surface Integration – p.8/ ?
H t l l t I
f( ) dS
8/8/2019 Surface Integration
http://slidepdf.com/reader/full/surface-integration 19/23
How to calculate I =S f (x) dS
1: Express surface in parametric form:
r = r(t1, t2) for a1 < t1 < b1, a2 < t2 < b2.
2: Find area of infinitesimal element:
dS = ∂ r
∂t1×
∂ r
∂t2 dt1 dt2.
Surface Integration – p.8/ ?
H t l l t I
f( ) dS
8/8/2019 Surface Integration
http://slidepdf.com/reader/full/surface-integration 20/23
How to calculate I =S f (x) dS
1: Express surface in parametric form:
r = r(t1, t2) for a1 < t1 < b1, a2 < t2 < b2.
2: Find area of infinitesimal element:
dS = ∂ r
∂t1×
∂ r
∂t2 dt1 dt2.
3: Work out standard double integral:
I = b
2
a2
b1
a1
f (r(t1, t2)) ∂ r
∂t1× ∂ r
∂t2
dt1 dt2.
Surface Integration – p.8/ ?
Fl integrals
8/8/2019 Surface Integration
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Flux integrals
Let u(x) be a velocity field.
u
n
δA δS = cos θ δA
θ
Surface Integration – p.9/ ?
Flux integrals
8/8/2019 Surface Integration
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Flux integrals
Let u(x) be a velocity field.
u
n
δA δS = cos θ δA
θ
Flow rate through S =
S
u · ndS,
= b2a2
b1a1
u(x(t1, t2)) ∂ x
∂t1 ×∂ x
∂t2
dt1 dt2.
Surface Integration – p.9/ ?
Coming soon
8/8/2019 Surface Integration
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Coming soon . . .
Stokes theorem
How to relate line and surface integrals
Green’s theorem in the plane
Conservative forces
Path independence of work integral
Curl-free force fields
Gauss divergence theoremHow to relate surface and volume integrals
Flux integrals and conservation laws
Pressure integrals and resultant forces
Surface Integration – p.10/ ?