of 12 /12
Math 53: Fall 2020, UC Berkeley Lecture 015 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 Section 15.10: Change of Variable This lecture is one of those key moments in the course when all of sudden you really understand what is going on! Let’s go back to 1D calculus g(u) u x Then we can rewrite by finding the ``magic factor’’ (1) Let x = g(u) (2) Then dx = g’(u) du So we can write Finally, we need to get the limits of integration correct, so if g(c)= a and g(d)=b, then Example: Let f(x)=(2x) 3 , and suppose we want We can let x = g(u)= (1/2) u. Then dx = g’(u) du = ½ du, so You have been doing this all along

# Then we can rewrite by finding the ``magic factor’’sethian/math53_2020/... · This lecture is one of those key moments in the course when all of sudden you really understand what

others

• View
0

0

Embed Size (px)

### Text of Then we can rewrite by finding the ``magic factor’’sethian/math53_2020/... · This lecture is... Math 53: Fall 2020, UC Berkeley Lecture 015 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]

Section 15.10: Change of Variable This lecture is one of those key moments in the course when all of sudden you really understand what is going on!

Let’s go back to 1D calculus

g(u) u x

Then we can rewrite by finding the ``magic factor’’

(1) Let x = g(u) (2) Then dx = g’(u) du

So we can write

Finally, we need to get the limits of integration correct, so if g(c)= a and g(d)=b, then

Example: Let f(x)=(2x)3, and suppose we want We can let x = g(u)= (1/2) u. Then dx = g’(u) du = ½ du, so

You have been doing this all along Math 53: Fall 2020, UC Berkeley Lecture 015 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]

Section 15.10: Change of Variable

The key to changing variables was the calculation of the magic factor: dx = g’(u) du

g(u) u x Which allowed us to go from

Okay, what if we had a function of two variables?

u

v

x

y

f(u,v)

g(u,v)

Could we find a magic factor?

that allows us to transform variables?

Example: We have already done this for polar coordinates!!!

r

q

x

y

So, for polar coordinate transformations, the magic factor is r

How do we find it in general? Math 53: Fall 2020, UC Berkeley Lecture 015 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]

Section 15.10: Change of Variable

Finding the magic factor general: (by the way, there’s nothing magic about it-it’s just really cool)

u

v

x

y S

(u1,v1) R

(x1,y1)

u

v

x=g(u,v)

y=h(u,v)

g(u,v)

h(u,v)

Step 2: Suppose we have We can write this in general as

T(u,v)= (x,y) Math 53: Fall 2020, UC Berkeley Lecture 015 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]

Section 15.10: Change of Variable

Example: Consider the transformation: x = u2 – v2, y = 2uv

What happens to the unit square 0≤u≤𝟏, 𝟎≤𝒗≤𝟏, under this transformation?

u

v

x

y R

(x1,y1) T (u1,v1)

What does the output region R look like?

Part 1: I am now going to do a funky example to show you that nice regions can be

transformed intro strange ones. Math 53: Fall 2020, UC Berkeley Lecture 015 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: x = u2 – v2, y = 2uv

u

v

x

y R

(x1,y1) T (u1,v1)

What does the output region R look like?

A

B

C

D

(1,0)

(0,1)

(0,0)

(1,1)

Let’s see what happens to the edges of the unit square

Edge A: For edge A, v=0 and 0≤u≤1, so the transformation is x=u2-v2 which = u2; y=2uv which = 0

So, x=u2 and y= 0 is a line from x=0 to x=1 u x

y

A (1,0) (0,0) (1,0) (0,0) A’

Edge B: For edge B, u=1 and 0≤v≤1, so the transformation is x=u2-v2 which = 1-v2; y=2uv which = 2v

So (y/2)=v, so x = 1-v2=1-(y/2)2=1-y2/4 which is a parabola

x

B

(1,0)

(1,1)

(1,0)

(0,2)

A’

B’

v

u

Edge C: For edge C, v=1 and 1≥u≥0, so the transformation is x=u2-v2 which = u2 -1; y=2uv which = 2u

So (y/2)=u, so x = u2 -1=(y/2)2-1=y2/4 -1 which is a parabola u x

C (1,1) (0,1)

(1,0) (0,0) A’

C’

x

Edge D: For edge D, u=0 and 1≥v≥0, so the transformation is x=u2-v2 which = -v2; y=2uv which = 0

So, x=-v2 and y= 0 is a line from x=-1 to x=0 u x

y

(0,0) (-1,0)

D’

v

D Math 53: Fall 2020, UC Berkeley Lecture 015 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: x = u2 – v2, y = 2uv So we just found that the uv unit square maps into:

u

v

x

y

(x1,y1)

T (u1,v1)

A

B

C

D

(1,0)

(0,1)

(0,0)

(1,1)

A’

B’ C’

D’

So, how do we find the magic factor such that ?

Here goes: Let’s agree that (x0,y0) = T(u0,v0) Standing at input (u0,v0), edges of the box DuDv get sent to different vectors in xy space

T(u,v)

u

v

The output (x,y) comes from an input (u,v). Then any point (x,y) in output space can be written as vector whose components depend on u and v

Our strategy will be to take the two vectors in the u,v plane making up a DuDv box, and see what they get transformed into. We will then take their cross-product to find the area of the transformed box—which is the magic stretch factor…

T(u,v)

u

v

T(bottom side)

T(left side) Area

[x coordinate of vector = g(u,v) y coordinate = h(u,v)] Math 53: Fall 2020, UC Berkeley Lecture 015 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, how do we find the magic factor such that ?

Here goes: Let’s agree that (x0,y0) = T(u0,v0) Standing at input (u0,v0), edges of the box DuDv get sent to different vectors in xy space

T(u,v)

u

v

The output (x,y) comes from an input (u,v). Then any point (x,y) in output space can be written as vector whose components depend on u and v

Our strategy will be to take the two vectors in the u,v plane making up a DuDv box, and see what they get transformed into. We will then take their cross-product to find the area of the transformed box—which is the magic stretch factor…

T(u,v)

u

v

T(bottom side)

T(left side) Area

[x coordinate of vector = g(u,v) y coordinate = h(u,v)]

Part 2: I am now going to go back to the main question: and find magic factors in general! Math 53: Fall 2020, UC Berkeley Lecture 015 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]

Step 1: Tangent vector holding v fixed, at input (u,v) is

T(u,v)

u

v

T(bottom side)

T(left side) Area

Step 2: Remembering that

u

v

x=g(u,v)

y=h(u,v)

g(u,v)

h(u,v)

Step 3: So, now can find the vectors that make up the bottom and left sides:

bottom side vector =

left side vector =

Step 4: So we can take the cross-product to find the area

Called the Jacobian

Tangent vector holding u fixed, at input (u,v) is Math 53: Fall 2020, UC Berkeley Lecture 015 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]

u

v

x

y

f(u,v)

g(u,v)

So now have the result:

u

v

x

y S

(u1,v1) R

(x1,y1)

Let’s use it right away: Prove that the Jacobian for the polar transformation x=r cos q, y=r sin q is rdrdq

r

q

x=r cos q

y=r sin q

Jacobian is

[R plays the role of u, and q plays the role of v]

Do you remember that for polar coordinates, we had Math 53: Fall 2020, UC Berkeley Lecture 015 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]

u

v

x

y

f(u,v)

g(u,v)

Let’s do another example:

u

v

x

y S

(u1,v1) R

(x1,y1)

Using the change of variables x = u2 – v2, y = 2uv to evaluate where R is the region bounded by the x-axis

and the parabolas y2=4-4x,x≥0 and y2=4-4x, x≤0 x

y

Solution: The Jacobian for this mapping is

..I’ll let you finish it….. Math 53: Fall 2020, UC Berkeley Lecture 015 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]

And now you see how to find the magic factor!!!

x (u,v,w)

y (u,v,w)

z (u,v,w)

u

w z

x

y v

dx dy dz= [magic factor] du dv dw

Form the matrix of partials

Then the magic factor = | det A |

Let’s check it for things we know!

Cylindrical coordinates: x (r, q,z)

y (r, q,z)

z (r, q,z)

r

q

z z=z

x=r cos q

y=r sin q

det A= r cos q cos q – ( - r sin q sin q ) = r !!!

dx dy dz = r dr dq dz Cartesian element

Cylindrical element

Geometrically, we got Math 53: Fall 2020, UC Berkeley Lecture 015 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]

Spherical coordinates: x (r, q,z)

y (r, q,z)

z (r, q,z)

r

q

z z= r cos f

x=r sin f cos q

y=r sin f sin q

det A= r2 sin f!!!

dx dy dz = r2 sin f dr dq df And geometrically the book got

Here goes!

I leave it to you to check that:

And remembering that if then det A= a (ei-fh) – b (di-fg) + c (dh-eg) ##### sofe 05 sethian - Princeton Plasma Physics LaboratoryJohn Sethian and Steve Obenschain Plasma Physics Division Naval Research Laboratory Washington, DC 21 st Symposium on Fusion Engineering
Documents ##### John Sethian Naval Research Laboratory April 4, 2002 Electra title pageElectra NRL J. Sethian M. Friedman M. Myers S. Obenschain R. Lehmberg J. Giuliani
Documents ##### Valentinian and Sethian Apocalyptic Traditions*rfrey/PDF/166/Judaism Christianity... · Valentinian and Sethian Apocalyptic Traditions* HAROLD W. ATTRIDGE ... frequently discussed,
Documents ##### Lecture 16.7: Last time we ... - math.berkeley.edusethian/math53_2020/...102.23 and 102.25 [email protected] Lecture 16.7: Last time—we parameterize as 2D surface living
Documents ##### Sethian Gnosticism By Joan Ann Lansberry · Sethian Gnosticism By Joan Ann Lansberry . ... Gnostics seek transformative ... fourth day of the fourth month of the summer season of
Documents ##### Section 14.1: Review of the Idea of Level Setssethian/math53_2020/... · so, you will be prosecuted under UC Berkeley student proceedings Secs. 102.23 and 102.25 [email protected]
Documents ##### Image Processing - Segmentation Variational Models Osher-Sethian Level-Set Framework + topologically flexible Osher-Sethian Level-Set Framework + topologically
Documents ##### COSMO - Dynamical Core Rewrite Approach, Rewrite and Status Tobias Gysi
Documents ##### Section 14.4: Tangent Planes and linear approximations (Review) …sethian/math53_2020/lectures/... · 2020. 9. 15. · Third question: Why do we add the contributions from the two
Documents ##### Question 1A: Find all critical points of the function ...sethian/math53_2020/lectures/lecture_53… · yy = 2, f xy = 1, so D = f xx f yy – (f xy)2 = (2)(2) -1 = 3 So (1/3, -2/3)
Documents ##### Crystal Growth and Dendritic Solidiï¬cation James A. Sethian John
Documents ##### Osher and Sethian 1988 Fronts Propagating With Curvature Dependent Speed
Documents ##### Rewrite, rewrite, rewrite, rewrite, rewrite, · Rewrite systems are sets of directed equations used to compute by repeatedly replacing ... In this paper, we consider systems that
Documents ##### From Linear Temporal Logic Properties to Rewrite Propositionsmembers.femto-st.fr/sites/femto-st.fr.pierre-cyril... · 2.1 Rewrite Words & Maximal Rewrite Words A comprehensive survey
Documents ##### Lecture 6 : Level Set Method. Introduction Developed by –Stanley Osher (UCLA) –J. A. Sethian (UC Berkeley) Books –J.A. Sethian: Level Set Methods and
Documents ##### Rewrite With Fractional Exponents. Rewrite with fractional exponent:
Documents Documents