lecture 16.7: last time we ... - math.berkeley.edusethian/math53_2020/...102.23 and 102.25...

Math 53: Fall 2020, UC Berkeley Lecture 21 Copyright: J.A. Sethian

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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)



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!



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 =

!!!



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


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


u

v input

y output

S

S

S

S

D

D

D

LINE INTEGRAL

LINE INTEGRAL

SURFACE INTEGRAL

SURFACE INTEGRAL



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




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




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



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:



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?



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





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!).





The flux of the curl of a vector field through the surface enclosed by that

boundary

Which immediately leads to a remarkable statement:

=



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!!!

lecture 16.7: last time we ... - math.berkeley.edusethian/math53_2020/...102.23 and 102.25...

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