Surface Integration

8/8/2019 Surface Integration

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Surface Integration

Eddie Wilson

[email protected]

Department of Engineering Mathematics

University of Bristol

Surface Integration – p.1/ ?



Integration in the plane

x

y

S

f (x, y)δS

S

f (x, y) dS = lim

f (x, y)δS.




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.




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.




Formulae for surfaces

Examples in the plane

C :=

(x,y,z) such that x2 + y2 ≤ a2, z = 0

x, y ≥ 0, z = 1.






C :=


x, y ≥ 0, z = 1.

Fully 3D examples:

z = x2

+ y2

, x, y ≥ 0,z = xy

x2 + y2 + z2 = a2






C :=


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




Parametric representation

IDEA: create two extra parameters which trace outsurface. Write all coordinates in new parameters.

Examples:






Examples:

BEFORE: x2 + y2 = a2.AFTER: x = a cos θ, y = a sin θ, z = t,θ ∈ [0, 2π), t ∈ (−∞, +∞).






Examples:


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).






Examples:


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/ ?



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 =?




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






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




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


H t l l t I

f( ) dS



How to calculate I =S f (x) dS


H t l l t I

f( ) dS




1: Express surface in parametric form:

r = r(t1, t2) for a1 < t1 < b1, a2 < t2 < b2.


H t l l t I

f( ) dS





r = r(t1, t2) for a1 < t1 < b1, a2 < t2 < b2.

2: Find area of infinitesimal element:

dS = ∂ r

∂t1×

∂ r

∂t2 dt1 dt2.


H t l l t I

f( ) dS





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.


Fl integrals



Flux integrals

Let u(x) be a velocity field.

u

n

δA δS = cos θ δA

θ


Flux integrals



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.


Coming soon



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

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