of 12 /12
Math 53: Fall 2020, UC Berkeley Lecture 06 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 14.4: Tangent Planes and linear approximations (Review) Let’s recall 1D Calculus: y=f(x) x y x=x 0 y=f(x 0 ) Tangent line with slope f’(x 0 ) going through the point (x 0 ,f(x 0 )) The tangent line is given by We want to construct a similar idea for functions of two (or more variables): The Tangent Plane y y z 2D Input space 1D Output space (x 0 ,y 0 ) Tangent plane Tangent line at (x 0 ,f(x 0 )) touches the graph y=f(x0) at only one point in a near (x 0 ,f(x 0 )) Tangent plane at (x 0 ,y 0 ,f(x 0 ,y 0 )) touches the graph z=f(x 0 ,y 0 ) at only one point in a near (x 0 ,y 0 ,f(x 0 ,y 0 )) dx dy Tangent vector = (dx,dy) = (1, dy/dx)= (1, f’(a))

# 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

others

• View
1

0

Embed Size (px)

### Text of Section 14.4: Tangent Planes and linear approximations (Review)... Math 53: Fall 2020, UC Berkeley Lecture 06 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 14.4: Tangent Planes and linear approximations (Review)

Let’s recall 1D Calculus: y=f(x)

x

y

x=x0

y=f(x0)

Tangent line with slope f’(x0) going through the point (x0,f(x0))

The tangent line is given by

We want to construct a similar idea for functions of two (or more variables): The Tangent Plane

y

y

z

2D Input space

1D Output space

(x0,y0)

Tangent plane

Tangent line at (x0,f(x0)) touches the graph y=f(x0) at only one point in a near (x0,f(x0))

Tangent plane at (x0,y0,f(x0,y0)) touches the graph z=f(x0,y0) at only one point in a near (x0,y0,f(x0,y0))

dx

dy

Tangent vector = (dx,dy) = (1, dy/dx)= (1, f’(a)) Math 53: Fall 2020, UC Berkeley Lecture 06 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 14.4: Tangent Planes and linear approximations

y

y

z 1D Output space

(x0,y0)

Tangent plane

Tangent plane at (x0,y0,f(x0,y0)) touches the graph z=f(x0,y0) at only one point in a near (x0,y0,f(x0,y0))

Formula for the tangent plane

Question: Suppose f(x,y) = x3y4 Approximate f(1.1, 1.2) using the tangent plane at (x0=1, y0=1)

Solution: Step 1: to use the tangent plane, we need to find several things: f(x,y) = x3y4 so f(x0,y0) = f(1,1) = 1314 = 1 fx(x,y)= 3x2 y4 so fx(x0,y0) = 3 (x0)2 (y0)4 = 3(1)2 (1)4 = 3 fy(x,y)= x3 4y3 so fy(x0,y0) = (x0)3 4 (y0)3 = (1)3 4(1)3 = 4 Step 2: So the tangent plane is: z – 1 = 3 (x-1) + 4 (y-1) Step 3: Evaluate the tangent plane at input (1.1, 1.2): z = 1+ 3 (1.1-1) + 4 (1.2-1) = 2.1 Math 53: Fall 2020, UC Berkeley Lecture 06 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 14.4: Tangent Planes and linear approximations

2d: Formula for the tangent plane

1d: Slope of a tangent line

Why stop there?

3d: Formula for the tangent “hyperplane”

“n”d: Formula for the tangent “hyperplane”

(No—I can’t draw it!) Math 53: Fall 2020, UC Berkeley Lecture 06 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 14.5: The Chain Rule

This is beautiful—but it requires some conceptual thinking. Let’s go!

Begin by recalling the 1D chain rule:

g(t) f(x) t x y

Then if we ask “how does the output y change as we change the input t”,

What does mean when more than one variable is involved as input? Many different diagrams to draw!

Example 1:

f(x,y) g(z) z w x

y

What is dw/dx? What is dw/dy?

Example 2:

x(t) f(x,y)

x w t What is dw/dt?

y(t) y

The book has many examples of diagrams. They are *all* the same thing! Math 53: Fall 2020, UC Berkeley Lecture 06 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 14.5: The Chain Rule x(t)

f(x,y) x

w t What is dw/dt? y(t) y

Claim:

First question: Why do I write “d” on the left but partial signs on the right?

Answer: because w depends on only one variable “t”, hence it’s a simple derivative

Third question: Why do we add the contributions from the two paths? First, an example

Second question: What is the meaning of the two terms that are added?:

Answer: they represent contributions from two different paths through the diagram

x(t) f(x,y)

x w t

y(t) y Math 53: Fall 2020, UC Berkeley Lecture 06 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 14.5: The Chain Rule x(t)

f(x,y) x

w t What is dw/dt? y(t) y

Claim:

Before proving this, let’s do an example

If w(x,y)=x3y + 3xy4 and x=sin(2t) and y=cos(t), find

Solution: Math 53: Fall 2020, UC Berkeley Lecture 06 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 14.5: The Chain Rule

If w(x,y)=x3y + 3xy4 and x=sin(2t) and y=cos(t), find Solution (old way)--substitute:

Note—you could do this the “old way”

Again:

They are the same! And the new way is much easier in complicated cases Math 53: Fall 2020, UC Berkeley Lecture 06 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 14.5: The Chain Rule

Your turn: w(x,y)=xy3 – x2y x(t) = t2 + 1 and y(t) =t2-1 Find

Solution: Math 53: Fall 2020, UC Berkeley Lecture 06 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 14.5: The Chain Rule (a flaky proof)

x(t) f(x,y)

x w t What is dw/dt?

y(t) y

Claim:

Proof: Step 1: Recall our tangent approximation plane to f(x,y) at an input point (a,b):

Step 2: So we can call w= f(x,y), ∆x = x-a, ∆y = y-b Step 3: Rewriting, we have

Step 4: Divide both sides by ∆t

Step 5: Take the limit as ∆t goes to zero:

This is not much of a proof—but gives the idea of why the terms are added Math 53: Fall 2020, UC Berkeley Lecture 06 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 14.5: The Chain Rule (onwards!)

x(s,t) f(x,y)

x w

s What is

y(s,t) y

A more complicated case

t

Remember:

is “how output w changes as input s changes, holding all other inputs constant”

x(s,t) f(x,y)

x w

s

y(s,t) y t

Step 1: Find all paths through the diagram: for

Path contributions path path

Similarly Math 53: Fall 2020, UC Berkeley Lecture 06 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 14.5: The Chain Rule (your turn)

If w(x,y) = ex sin y and x = st2 and y = s2t find Math 53: Fall 2020, UC Berkeley Lecture 06 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 14.5: The Chain Rule (one more for you)

x1

x2

x3

x4

u1(x1,x2,x3,x4)

u2(x1,x2,x3,x4)

u3(x1,x2,x3,x4)

u4(x1,x2,x3,x4)

z v1(z)

v2(z)

w

What is

Again—evaluate all possible paths:

u4

u3

u2

u1

v1

v2

w(v1,v2) ##### 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 ##### 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 ##### A Coupled Level Set Projection Method Applied to …sethian/2006/Papers/sethian...A Coupled Level Set Projection Method Applied to Ink Jet Simulation Jiun-Der Yu∗ Epson Research
Documents ##### Section 16:3 The Fundamental Theorem of Calculus for Line …sethian/math53_2020/lectures/... · 2020. 10. 29. · Step 2: Pick (a,y) at same y level as the endpoint, giving a straight
Documents ##### Crystal Growth and Dendritic Solidiï¬cation James A. Sethian John
Documents ##### 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
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 ##### Image Processing - Segmentation Variational Models Osher-Sethian Level-Set Framework + topologically flexible Osher-Sethian Level-Set Framework + topologically
Documents ##### Crystal Growth and Dendritic Solidiﬁcation James A. Sethian …sethian/2006/Papers/sethian.crystal.pdf · Crystal Growth and Dendritic Solidiﬁcation James A. Sethian Department
Documents ##### Valentinian and Sethian Apocalyptic Traditions*rfrey/PDF/166/Judaism... · 2005-01-31 · Valentinian and Sethian Apocalyptic Traditions* HAROLD W. ATTRIDGE The paper reexamines the
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 Documents