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Math 53: Fall 2020, UC Berkeley Lecture 21 Copyright: J.A.
Sethian
All rights reserved. You may not
distribute/reproduce/display/post/upload any course materials in
any way, regardless of whether or not a fee is charged, without my
express written consent. You also may not allow anyone else to do
so. If you do so, you will be prosecuted under UC Berkeley student
proceedings Secs. 102.23 and 102.25 [email protected]
Lecture 16.7: Last time—we parameterize as 2D surface living in
3D
u
v
x
z
y
input output
[x(u,v), y(u,v), z(u,v)]
Surface S
(u,v) x
y u
x(u,v)
y(u,v)
z(u,v) z v
So we can describe a point in (x,y,z) output space in terms of
where it came from in (u,v) input space
u
v
input
y output Surface S
dS = surface patch du
dv
And we figured out the Jacobian for the mapping of a surface
patch
So = the tangent vector in output space as u changes with v
fixed
= the tangent vector in output space as v changes with u
fixed
And before, we figured out that
(area of parallelogram swept out by tangent vectors)
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Math 53: Fall 2020, UC Berkeley Lecture 21 Copyright: J.A.
Sethian
All rights reserved. You may not
distribute/reproduce/display/post/upload any course materials in
any way, regardless of whether or not a fee is charged, without my
express written consent. You also may not allow anyone else to do
so. If you do so, you will be prosecuted under UC Berkeley student
proceedings Secs. 102.23 and 102.25 [email protected]
x
z
y output Surface S
dS = surface patch
So, in output space, we can write down an integral of a function
f(x,y,z) over the surface:
f(x(u,v), y(u,v), z(u,v))
So now we have our transformation:
u
v dv
D
Example: Find area of a sphere of radius R where S is sphere
radius R So find:
Spherical coordinates:
Magic Factor (Jacobian)
u=f, v=q (R is constant, since this is the “outside” of the
sphere”)
So, Check this!
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Math 53: Fall 2020, UC Berkeley Lecture 21 Copyright: J.A.
Sethian
All rights reserved. You may not
distribute/reproduce/display/post/upload any course materials in
any way, regardless of whether or not a fee is charged, without my
express written consent. You also may not allow anyone else to do
so. If you do so, you will be prosecuted under UC Berkeley student
proceedings Secs. 102.23 and 102.25 [email protected]
So now we have our transformation:
Example: Find area of a sphere of radius R where S is sphere of
radius R
So find:
Surface Area of Sphere of radius R =
!!!
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Math 53: Fall 2020, UC Berkeley Lecture 21 Copyright: J.A.
Sethian
All rights reserved. You may not
distribute/reproduce/display/post/upload any course materials in
any way, regardless of whether or not a fee is charged, without my
express written consent. You also may not allow anyone else to do
so. If you do so, you will be prosecuted under UC Berkeley student
proceedings Secs. 102.23 and 102.25 [email protected]
So, so far, we know how to
Integrate a function over a 1D object in input space living in
1D output space
x a b
Integrate a function over a 1D object in input space object
living in 2D output space
Integrate a function over a 2D object in input space object
living in 2D output space
x input space
output space
x(t)=t
x input space x=x(t),y=y(t)
u
v
x
y
x=x(u,v),y=y(u,v)
input space output
D
Integrate a function over a 2D object in input space object
living in 3D output space
u
v input
y output
S
S
S
S
D
D
D
LINE INTEGRAL
LINE INTEGRAL
SURFACE INTEGRAL
SURFACE INTEGRAL
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Math 53: Fall 2020, UC Berkeley Lecture 21 Copyright: J.A.
Sethian
All rights reserved. You may not
distribute/reproduce/display/post/upload any course materials in
any way, regardless of whether or not a fee is charged, without my
express written consent. You also may not allow anyone else to do
so. If you do so, you will be prosecuted under UC Berkeley student
proceedings Secs. 102.23 and 102.25 [email protected]
So, can we integrate vector fields over 2D input objects in 3D
output space?
You bet—here goes
y
S
x
z
Def: a surface is oriented if there is a smoothly varying normal
as we move along the surface Let’s see how to find a unit normal
for a surface
We start with the easier case: suppose the surface is the graph
of a function z=g(x,y) y
S
x
z
z=g(x,y)
Remember: the two tangents on the surface at (x,y) are (1,0,gx),
and (0,1,gy)
So, normal is (1,0,gx) x (0,1,gy) = ( -gx, -gy, 1) Unit
normal
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Math 53: Fall 2020, UC Berkeley Lecture 21 Copyright: J.A.
Sethian
All rights reserved. You may not
distribute/reproduce/display/post/upload any course materials in
any way, regardless of whether or not a fee is charged, without my
express written consent. You also may not allow anyone else to do
so. If you do so, you will be prosecuted under UC Berkeley student
proceedings Secs. 102.23 and 102.25 [email protected]
So, can we integrate vector fields over 2D input objects in 3D
output space?
Let’s see how to find a unit normal for a general case
= the tangent vector in output space as u changes with v
fixed
= the tangent vector in output space as v changes with u
fixed
So then, the unit normal to surface is just
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Math 53: Fall 2020, UC Berkeley Lecture 21 Copyright: J.A.
Sethian
All rights reserved. You may not
distribute/reproduce/display/post/upload any course materials in
any way, regardless of whether or not a fee is charged, without my
express written consent. You also may not allow anyone else to do
so. If you do so, you will be prosecuted under UC Berkeley student
proceedings Secs. 102.23 and 102.25 [email protected]
So, can we integrate vector fields over 2D input objects in 3D
output space?
We are now ready to define the surface integral over a vector
field
Definition: the flux of a vector field through a surface S is
defined as Flux
Q. What does this mean geometrically?
x
z
y Surface S
dS = surface patch A. It’s the amount of the vector field
flowing through the surface
LOTS OF FLOW THROUGH THE SURFACE VERY LITTLE FLOW THROUGH THE
SURFACE
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Math 53: Fall 2020, UC Berkeley Lecture 21 Copyright: J.A.
Sethian
All rights reserved. You may not
distribute/reproduce/display/post/upload any course materials in
any way, regardless of whether or not a fee is charged, without my
express written consent. You also may not allow anyone else to do
so. If you do so, you will be prosecuted under UC Berkeley student
proceedings Secs. 102.23 and 102.25 [email protected]
Flux
We can work harder and write the flux in terms of the
mapping
x
z
y output Surface S
dS = surface patch
So, in output space, we can write down an integral of a function
f(x,y,z) over the surface:
f(x(u,v), y(u,v), z(u,v))
u
v dv
D
So:
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Math 53: Fall 2020, UC Berkeley Lecture 21 Copyright: J.A.
Sethian
All rights reserved. You may not
distribute/reproduce/display/post/upload any course materials in
any way, regardless of whether or not a fee is charged, without my
express written consent. You also may not allow anyone else to do
so. If you do so, you will be prosecuted under UC Berkeley student
proceedings Secs. 102.23 and 102.25 [email protected]
Example: Find the flux of vector field through unit sphere
So, we have the mapping: u=f, v=q
Go check that
Something—who knows?
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Math 53: Fall 2020, UC Berkeley Lecture 21 Copyright: J.A.
Sethian
All rights reserved. You may not
distribute/reproduce/display/post/upload any course materials in
any way, regardless of whether or not a fee is charged, without my
express written consent. You also may not allow anyone else to do
so. If you do so, you will be prosecuted under UC Berkeley student
proceedings Secs. 102.23 and 102.25 [email protected]
Okay, what do you do with all this?
Remember Green’s theorem
y
x
And we had a fancy way to write Green’s theorem: Suppose (no z
component) Then we have that
So now we can write Green’s theorem as
Let’s interpret the above in a new way:
Collecting the tangential component of a vector field
around a closed curve =
The flux of the vector field through the surface enclosed by
that
boundary
-
Math 53: Fall 2020, UC Berkeley Lecture 21 Copyright: J.A.
Sethian
All rights reserved. You may not
distribute/reproduce/display/post/upload any course materials in
any way, regardless of whether or not a fee is charged, without my
express written consent. You also may not allow anyone else to do
so. If you do so, you will be prosecuted under UC Berkeley student
proceedings Secs. 102.23 and 102.25 [email protected]
Collecting the tangential component of a vector field
around a closed curve =
The flux of the curl of a vector field through the surface
enclosed by that
boundary
Now comes something amazing: Stokes’ Theorem The same thing is
true in 3D!!!
(this is supposed to be a surface in 3D!).
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Math 53: Fall 2020, UC Berkeley Lecture 21 Copyright: J.A.
Sethian
All rights reserved. You may not
distribute/reproduce/display/post/upload any course materials in
any way, regardless of whether or not a fee is charged, without my
express written consent. You also may not allow anyone else to do
so. If you do so, you will be prosecuted under UC Berkeley student
proceedings Secs. 102.23 and 102.25 [email protected]
Collecting the tangential component of a vector field
around a closed curve =
The flux of the curl of a vector field through the surface
enclosed by that
boundary
Which immediately leads to a remarkable statement:
=
Collecting the tangential component of a vector field
around a closed curve =
The flux of the curl of the vector field through the surface
enclosed by that
boundary
The flux of the curl of a vector field through the surface
enclosed by
that boundary
These have to be the same!!!