SURFACE INTEGRAL SUMMARY

PORAMATE (TOM) PRANAYANUNTANA

Key For Surface Integrals

d ~AS = (~rs × ~rt)︸ ︷︷ ︸|J | = ‖~rs × ~rt‖hidden here

dsdt︸︷︷︸dAT

,

s tx yr θz θφ θu v...

...

There are only 2 ways to find

∫S

(~F � d ~AS

):

(1) By parameterization:∫ (~F � d ~AS

)S:~r(s,t),(s,t)∈T in st-plane

=

∫T

(~F � (~rs × ~rt)

)dsdt︸︷︷︸dAT

.

(2) By Geometry (for simple surfaces):∫S

(~F � d ~AS︸︷︷︸

n̂SdAS

)=

∫S

(~F � n̂S

)dAS︸︷︷︸|J |dAT

.

Observe that

d ~AS =

n̂S |J |︷ ︸︸ ︷(~rs × ~rt) dsdt︸︷︷︸

dAT

= n̂S

dAS︷ ︸︸ ︷|J | dsdt︸︷︷︸

dAT

.

Coordinates used ~rs × ~rt (orientation of S)

S : ~r(x, y) =

xy

f(x, y)

~rx × ~ry =

−fx−fy1

(up)

S : ~r(z, θ) =

R cos θR sin θz

where r = R is a constantradius from z-axis

~rθ × ~rz =

xy0

(away from z-axis)

S : ~r(φ, θ) =

R sinφ cos θR sinφ sin θR cosφ

where ρ = R is a constantradius from origin (0, 0, 0)

~rφ × ~rθ =

xyz

R

R2 sinφ (away from (0, 0, 0))

Date: June 28, 2015.

Surface Integral Summary Poramate (Tom) Pranayanuntana

To find AS ,

AS =

∫S

dAS =

∫ ∥∥∥d ~AS∥∥∥S:~r(s,t),(s,t)∈T

=

∫T

dAS︷ ︸︸ ︷‖~rs × ~rt‖︸ ︷︷ ︸

|J|

dsdt︸︷︷︸dAT

.

The flux through a graph of z = f(x, y) above a region R in the xy-plane, oriented upward, is

∫S

(~F � d ~AS

)=

∫R

F1(x, y, f(x, y))F2(x, y, f(x, y))F3(x, y, f(x, y))

�

−fx−fy1

︸ ︷︷ ︸

~nSR

dxdy︸ ︷︷ ︸dAR

=

∫R

F1(x, y, f(x, y))F2(x, y, f(x, y))F3(x, y, f(x, y))

�1√

1 + f2x + f2y

−fx−fy1

︸ ︷︷ ︸

n̂S

√

1 + f2x + f2y︸ ︷︷ ︸|J|

dxdy︸ ︷︷ ︸dAR︸ ︷︷ ︸

dAS

.

The flux through a cylindrical surface S of radius r = R and oriented away from the z-axis is

∫S

(~F � d ~AS

)=

∫T

F1(R, θ, z)F2(R, θ, z)F3(R, θ, z)

�1

R

xy0

︸ ︷︷ ︸~nST︸ ︷︷ ︸n̂S

R︸︷︷︸|J|

dzdθ︸︷︷︸dAT

=

∫T

F1

F2

F3

�

cos θsin θ

0

︸ ︷︷ ︸

n̂S

R︸︷︷︸|J|

dzdθ︸︷︷︸dAT

,

where T is the θz-region corresponding to S.

The flux through a spherical surface S of radius ρ = R and oriented away from the origin is

∫S

(~F � d ~AS

)=

∫T

F1(R,φ, θ)F2(R,φ, θ)F3(R,φ, θ)

�1

R

xyz

︸ ︷︷ ︸

n̂S

R2 sinφ dφdθ

=

∫T

F1

F2

F3

�

sinφ cos θsinφ sin θ

cosφ

︸ ︷︷ ︸

n̂S

R2 sinφ︸ ︷︷ ︸|J|

dφdθ︸ ︷︷ ︸dAT

,

where T is the θφ-region corresponding to S.

June 28, 2015 Page 2 of 2