Lecture 8 Analyzing the diffusion weighted signal. Room CSB 272 this week! Please install AFNI

Size: px
Start display at page:

Download "Lecture 8 Analyzing the diffusion weighted signal. Room CSB 272 this week! Please install AFNI"

Transcription

1 Lecture 8 Analyzing the diffusion weighted signal Room CSB 272 this week! Please install AFNI

2 Next lecture, DTI For this lecture, think in terms of a single voxel

3 We re still looking only at a single voxel experiment This Last Lecture: Multiple Single diffusion encoding directions to to estimate a a diffusion coefficient tensor DD

4 G x B(x) x B(x) x time time x

5 Phases of diffusing spins 90 TEê2 180 TEê2 echo d d»g» e D = + ( )= 0 G(t)x(t) dt

6 Diffusion phase in a Bipolar Pulse 100 x(t 1 ) x(t 1 ) r x(t 2 ) G(t) G x '(x, t) = G {z} q G x [ x(t 2 ) x(t 1 )] {z } r = q r t

7 The Estimation Problem for Gaussian Diffusion measured signal Z S(') S(b) = =S(0) dx P e(x, bd t) + e i'(x,t) non-diffusion weighted signal (b=0) Gaussian b = q 2 noise object of our desire! pulse sequence parameters G(t) G x q = G = 3 G x t

8 The signal from 1D Gaussian Diffusion s(b i )=s 0 e b id + i s 1.0 where s 0 s(b = 0) b

9 Consider only two measurements and write data in vector form s(b1 ) s(b 2 ) exp( b1 D) = s 0 exp( b 2 D) = s 0 exp apple b1 D b 2 D This clearly generalizes to n measurements

10

11 Recall: gradients add like vectors k x = k x x + k y y = G x tx + G y ty G y (t) t y G x (t) x t spatial modulation of the phase

12 Directional Diffusion Encoding 90 TEê2 180 TEê2 echo d d»gx» e D z y»gy» z y x x G (x,y)

13 Ideal b-matrix G i G j 2 q b ij = Gq ii q i = G i G j j ( where /3) = /3 q i q j

14 Ideal b-matrix G i G j b ij = q i q j where q i = G i = /3

15 The b-matrix b ij ( )= 0 q i (t)q j (t)dt i=(x,y,z) where q = g(t) dt For constant diffusion gradients b ij ( ) =q i q j

16 The NMR signal for 1D Gaussian diffusion s(q, )=s(0) P ( r, )e iq r d r P ( r, ) = 1 4 D e r 2 /(4D ) s(q, )=s(0)e bd

17 The NMR signal for 3D Gaussian diffusion s(q, ) = Z P ( r, )e iq r d r P ( r, ) = 1 p (4 )3 D e rt D 1 r/4 s(q, ) =s(0)e bd

18 b and D qx 2 q x q y q x q z b = q y q x qy 2 q y q z q z q x q z q y qz 2 known D xx D xy D xz D = D yx D yy D yz D zx D zy D zz desired

19 The NMR signal 3D Gaussian diffusion s(q, )=s(0)e bd s(b) = s(0) exp 3 3 b ij D ij i j bd = q 2 D bd = q t D q

20 A single diffusion-weighting direction G x y z G y x G z

21 1 i j bd b ij D ij = q 2 xd xx + q x q y D xy + q x q z D xz + q y q x D yx + q 2 yd yy + q y q z D yz + q z q x D yx + q z q y D zy + q 2 zd zz

22 Rearranging the directions bd = q t D q q = qû q q q q û qû bd = q 2 u t D u

23 The NMR signal bd = q 2 u t D u D s(q, )=s(0)e bd = e q2 D s(q, )=e b D where b = q 2

24 Measuring the Diffusion Tensor S(b, r ) = S(0)e bd + y r = 3 D= Dx 0 0 Dy cos sin x t 2 2 D = r Dr = Dx cos + Dy sin projection of an ellipsoid! not like projection of a vector

25 Measuring the Diffusion Tensor b=1000 b=0 1.0 y x -1 fiber axis S(b, ) = S(0)e bd( ) + D( ) = Dx cos2 + Dy sin2

26 The Shape of Diffusion fiber signal S b ( ) D app ( ) = 1 b log Sb S 0

27 What is the meaning of D? D u t D u It is the projection of D along û D û D

28 Diffusion Tensor is Symmetric D xx D xy D xz D yx D yy D yz D zx D zy D zz = D xx D xy D xz D xy D yy D yz D xz D yz D zz D = D t matrix form D ij = D ji component form

29 1 i j bd b ij D ij = q 2 xd xx + q x q y D xy + q x q z D xz + q y q x D yx + q 2 yd yy + q y q z D yz + q z q x D yx + q z q y D zy + q 2 zd zz 1 i j b ij D ij = q 2 xd xx +2q x q y D xy +2q x q z D xz + q 2 yd yy +2q y q z D yz + q 2 zd zz

30 A computational simplification s(b) =s(0)e bd a trick: write log s(0) s(0) = e s(b) =s(0)e bd = e log s(0) e bd = e bd+log s(0)

31 Estimating the Diffusion Tensor d = D xx D yy D zz D xy D xz D yz log s(0) There are 7 unknowns

32 Estimating the Diffusion Tensor B =(q 2 x,q 2 y,q 2 z, 2q x q y, 2q x q z, 2q y q z, 1)

33 Estimating the Diffusion Tensor log s(b) y = (q 2 x,q 2 y,q 2 z, 2q x q y, 2q x q z, 2q y q z, 1) B t D xx D yy D zz D xy D xz D yz log s(0) D But there are 7 unknowns, so we need 7 equations to solve for them

34 Estimating the Diffusion Tensor y = log s(b 1 ) log s(b 2 ). log s(b n ) We make 7 measurements, each with a different direction

35 The B-matrix B t = tensor dimensions ˆq 1,x 2 ˆq 1,y 2 ˆq 2 1,z ˆq 1,xˆq 1,y ˆq 1,xˆq 1,z ˆq 1,y ˆq 1,z 1 ˆq 2,x 2 ˆq 2,y 2 ˆq 2,z 2 ˆq 2,xˆq 1,y ˆq 2,xˆq 2,z ˆq 2,y ˆq 2,z ˆq n,x 2 ˆq n,y 2 ˆq n,z 2 ˆq n,xˆq 1,y ˆq n,xˆq n,z ˆq n,y ˆq n,z 1 gradient directions q j,k = g k = /3 j th direction

36 Angular measurements

37 Estimating the Diffusion Tensor log s(b 1 ) ˆq 1,x 2 ˆq 1,y 2 ˆq 2 1,z ˆq 1,xˆq 1,y ˆq 1,xˆq 1,z ˆq 1,y ˆq 1,z 1 log s(b 2 ).. = ˆq 2,x 2 ˆq 2,y 2 ˆq 2,z 2 ˆq 2,xˆq 1,y ˆq 2,xˆq 2,z ˆq 2,y ˆq 2,z log s(b n ) ˆq n,x 2 ˆq n,y 2 ˆq n,z 2 ˆq n,xˆq 1,y ˆq n,xˆq n,z ˆq n,y ˆq n,z 1 D xx D yy D zz D xy D xz D yz log s(0) y B t d

38 Least Squares The matrix equation y = Bd has dimensions [n 1] = [n m][m 1]

39 Estimating the Diffusion Tensor Solving for the diffusion tensor is reduced to finding the solution to the matrix equation y = Bd diffusion tensor elements data b-matrix

40 Estimating the Diffusion Tensor Matrix equation y = B t d data b-matrix diffusion tensor elements Matrix solution d = B + y pseudo-inverse

41 Least Squares The least-squares solution to the matrix equation y = Ax is? ˆx = A 1 y? NO!

42 Least Squares The least-squares solution to the matrix equation y = Ax is ˆx = A + y (note that ˆx = A 1 y) where A + (A t A) 1 A t This is called the pseudo-inverse of A

43 Estimating the Diffusion Tensor In practice D calculated with 3dDWItoDT (AFNI) eigensystem calculated by: [evals,evecs] = eig(d)

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»

More information

A. H. Hall, 33, 35 &37, Lendoi

A. H. Hall, 33, 35 &37, Lendoi 7 X x > - z Z - ----»»x - % x x» [> Q - ) < % - - 7»- -Q 9 Q # 5 - z -> Q x > z»- ~» - x " < z Q q»» > X»? Q ~ - - % % < - < - - 7 - x -X - -- 6 97 9

More information

LOWELL WEEKLY JOURNAL

LOWELL WEEKLY JOURNAL Y -» $ 5 Y 7 Y Y -Y- Q x Q» 75»»/ q } # ]»\ - - $ { Q» / X x»»- 3 q $ 9 ) Y q - 5 5 3 3 3 7 Q q - - Q _»»/Q Y - 9 - - - )- [ X 7» -» - )»? / /? Q Y»» # X Q» - -?» Q ) Q \ Q - - - 3? 7» -? #»»» 7 - / Q

More information

Two Posts to Fill On School Board

Two Posts to Fill On School Board Y Y 9 86 4 4 qz 86 x : ( ) z 7 854 Y x 4 z z x x 4 87 88 Y 5 x q x 8 Y 8 x x : 6 ; : 5 x ; 4 ( z ; ( ) ) x ; z 94 ; x 3 3 3 5 94 ; ; ; ; 3 x : 5 89 q ; ; x ; x ; ; x : ; ; ; ; ; ; 87 47% : () : / : 83

More information

OWELL WEEKLY JOURNAL

OWELL WEEKLY JOURNAL Y \»< - } Y Y Y & #»»» q ] q»»»>) & - - - } ) x ( - { Y» & ( x - (» & )< - Y X - & Q Q» 3 - x Q Y 6 \Y > Y Y X 3 3-9 33 x - - / - -»- --

More information

Closed-Form Solution Of Absolute Orientation Using Unit Quaternions

Closed-Form Solution Of Absolute Orientation Using Unit Quaternions Closed-Form Solution Of Absolute Orientation Using Unit Berthold K. P. Horn Department of Computer and Information Sciences November 11, 2004 Outline 1 Introduction 2 3 The Problem Given: two sets of corresponding

More information

' Liberty and Umou Ono and Inseparablo "

' Liberty and Umou Ono and Inseparablo 3 5? #< q 8 2 / / ) 9 ) 2 ) > < _ / ] > ) 2 ) ) 5 > x > [ < > < ) > _ ] ]? <

More information

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Preliminary Math Concept of Stress Stress Components Equilibrium

More information

LOWELL WEEKLY JOURNAL

LOWELL WEEKLY JOURNAL Y G y G Y 87 y Y 8 Y - $ X ; ; y y q 8 y $8 $ $ $ G 8 q < 8 6 4 y 8 7 4 8 8 < < y 6 $ q - - y G y G - Y y y 8 y y y Y Y 7-7- G - y y y ) y - y y y y - - y - y 87 7-7- G G < G y G y y 6 X y G y y y 87 G

More information

M E 320 Professor John M. Cimbala Lecture 10

M E 320 Professor John M. Cimbala Lecture 10 M E 320 Professor John M. Cimbala Lecture 10 Today, we will: Finish our example problem rates of motion and deformation of fluid particles Discuss the Reynolds Transport Theorem (RTT) Show how the RTT

More information

Ordinary Least Squares and its applications

Ordinary Least Squares and its applications Ordinary Least Squares and its applications Dr. Mauro Zucchelli University Of Verona December 5, 2016 Dr. Mauro Zucchelli Ordinary Least Squares and its applications December 5, 2016 1 / 48 Contents 1

More information

g(t) = f(x 1 (t),..., x n (t)).

g(t) = f(x 1 (t),..., x n (t)). Reading: [Simon] p. 313-333, 833-836. 0.1 The Chain Rule Partial derivatives describe how a function changes in directions parallel to the coordinate axes. Now we shall demonstrate how the partial derivatives

More information

County Council Named for Kent

County Council Named for Kent \ Y Y 8 9 69 6» > 69 ««] 6 : 8 «V z 9 8 x 9 8 8 8?? 9 V q» :: q;; 8 x () «; 8 x ( z x 9 7 ; x >«\ 8 8 ; 7 z x [ q z «z : > ; ; ; ( 76 x ; x z «7 8 z ; 89 9 z > q _ x 9 : ; 6? ; ( 9 [ ) 89 _ ;»» «; x V

More information

and A L T O S O L O LOWELL, MICHIGAN, THURSDAY, OCTCBER Mrs. Thomas' Young Men Good Bye 66 Long Illness Have Sport in

and A L T O S O L O LOWELL, MICHIGAN, THURSDAY, OCTCBER Mrs. Thomas' Young Men Good Bye 66 Long Illness Have Sport in 5 7 8 x z!! Y! [! 2 &>3 x «882 z 89 q!!! 2 Y 66 Y $ Y 99 6 x x 93 x 7 8 9 x 5$ 4 Y q Q 22 5 3 Z 2 5 > 2 52 2 $ 8» Z >!? «z???? q > + 66 + + ) ( x 4 ~ Y Y»» x ( «/ ] x ! «z x( ) x Y 8! < 6 x x 8 \ 4\

More information

PEAT SEISMOLOGY Lecture 2: Continuum mechanics

PEAT SEISMOLOGY Lecture 2: Continuum mechanics PEAT8002 - SEISMOLOGY Lecture 2: Continuum mechanics Nick Rawlinson Research School of Earth Sciences Australian National University Strain Strain is the formal description of the change in shape of a

More information

GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS

GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS I Main Topics A Why deal with tensors? B Order of scalars, vectors, and tensors C Linear transformation of scalars and vectors (and tensors) II Why

More information

Educjatipnal. L a d ie s * COBNWALILI.S H IG H SCHOOL. I F O R G IR L S A n B k i n d e r g a r t e n.

Educjatipnal. L a d ie s * COBNWALILI.S H IG H SCHOOL. I F O R G IR L S A n B k i n d e r g a r t e n. - - - 0 x ] - ) ) -? - Q - - z 0 x 8 - #? ) 80 0 0 Q ) - 8-8 - ) x ) - ) -] ) Q x?- x - - / - - x - - - x / /- Q ] 8 Q x / / - 0-0 0 x 8 ] ) / - - /- - / /? x ) x x Q ) 8 x q q q )- 8-0 0? - Q - - x?-

More information

Q SON,' (ESTABLISHED 1879L

Q SON,' (ESTABLISHED 1879L ( < 5(? Q 5 9 7 00 9 0 < 6 z 97 ( # ) $ x 6 < ( ) ( ( 6( ( ) ( $ z 0 z z 0 ) { ( % 69% ( ) x 7 97 z ) 7 ) ( ) 6 0 0 97 )( 0 x 7 97 5 6 ( ) 0 6 ) 5 ) 0 ) 9%5 z» 0 97 «6 6» 96? 0 96 5 0 ( ) ( ) 0 x 6 0

More information

Diffusion Tensor Imaging (DTI): An overview of key concepts

Diffusion Tensor Imaging (DTI): An overview of key concepts Diffusion Tensor Imaging (DTI): An overview of key concepts (Supplemental material for presentation) Prepared by: Nadia Barakat BMB 601 Chris Conklin Thursday, April 8 th 2010 Diffusion Concept [1,2]:

More information

LOWELL WEEKLY JOURNAL.

LOWELL WEEKLY JOURNAL. Y 5 ; ) : Y 3 7 22 2 F $ 7 2 F Q 3 q q 6 2 3 6 2 5 25 2 2 3 $2 25: 75 5 $6 Y q 7 Y Y # \ x Y : { Y Y Y : ( \ _ Y ( ( Y F [ F F ; x Y : ( : G ( ; ( ~ x F G Y ; \ Q ) ( F \ Q / F F \ Y () ( \ G Y ( ) \F

More information

Tensor Visualization. CSC 7443: Scientific Information Visualization

Tensor Visualization. CSC 7443: Scientific Information Visualization Tensor Visualization Tensor data A tensor is a multivariate quantity Scalar is a tensor of rank zero s = s(x,y,z) Vector is a tensor of rank one v = (v x,v y,v z ) For a symmetric tensor of rank 2, its

More information

Orientation Distribution Function for Diffusion MRI

Orientation Distribution Function for Diffusion MRI Orientation Distribution Function for Diffusion MRI Evgeniya Balmashnova 28 October 2009 Diffusion Tensor Imaging Diffusion MRI Diffusion MRI P(r, t) = 1 (4πDt) 3/2 e 1 4t r 2 D 1 t Diffusion time D Diffusion

More information

Rigid body simulation. Once we consider an object with spatial extent, particle system simulation is no longer sufficient

Rigid body simulation. Once we consider an object with spatial extent, particle system simulation is no longer sufficient Rigid body dynamics Rigid body simulation Once we consider an object with spatial extent, particle system simulation is no longer sufficient Rigid body simulation Unconstrained system no contact Constrained

More information

" W I T H M: A. L I G E T O ' W ^ P L D IST O ISTE -A-IsTD G H! A-I^IT Y IPO PL A.LI-i. :

 W I T H M: A. L I G E T O ' W ^ P L D IST O ISTE -A-IsTD G H! A-I^IT Y IPO PL A.LI-i. : : D D! Y : V Y JY 4 96 J z z Y &! 0 6 4 J 6 4 0 D q & J D J» Y j D J & D & Y = x D D DZ Z # D D D D D D V X D DD X D \ J D V & Q D D Y D V D D? q ; J j j \V ; q» 0 0 j \\ j! ; \?) j: ; : x DD D J J j ;

More information

Rigid body dynamics. Basilio Bona. DAUIN - Politecnico di Torino. October 2013

Rigid body dynamics. Basilio Bona. DAUIN - Politecnico di Torino. October 2013 Rigid body dynamics Basilio Bona DAUIN - Politecnico di Torino October 2013 Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October 2013 1 / 16 Multiple point-mass bodies Each mass is

More information

Chapter 2 Governing Equations

Chapter 2 Governing Equations Chapter Governing Equations Abstract In this chapter fundamental governing equations for propagation of a harmonic disturbance on the surface of an elastic half-space is presented. The elastic media is

More information

Polynomials. In many problems, it is useful to write polynomials as products. For example, when solving equations: Example:

Polynomials. In many problems, it is useful to write polynomials as products. For example, when solving equations: Example: Polynomials Monomials: 10, 5x, 3x 2, x 3, 4x 2 y 6, or 5xyz 2. A monomial is a product of quantities some of which are unknown. Polynomials: 10 + 5x 3x 2 + x 3, or 4x 2 y 6 + 5xyz 2. A polynomial is a

More information

LOWELL. MICHIGAN, OCTOBER morning for Owen J. Howard, M last Friday in Blodpett hospital.

LOWELL. MICHIGAN, OCTOBER morning for Owen J. Howard, M last Friday in Blodpett hospital. G GG Y G 9 Y- Y 77 8 Q / x -! -} 77 - - # - - - - 0 G? x? x - - V - x - -? : : - q -8 : : - 8 - q x V - - - )?- X - - 87 X - ::! x - - -- - - x -- - - - )0 0 0 7 - - 0 q - V -

More information

MATH 19520/51 Class 5

MATH 19520/51 Class 5 MATH 19520/51 Class 5 Minh-Tam Trinh University of Chicago 2017-10-04 1 Definition of partial derivatives. 2 Geometry of partial derivatives. 3 Higher derivatives. 4 Definition of a partial differential

More information

oenofc : COXT&IBCTOEU. AU skaacst sftwer thsa4 aafcekr will be ehat«s«ai Bi. C. W. JUBSSOS. PERFECT THBOUGH SDFFEBISG. our

oenofc : COXT&IBCTOEU. AU skaacst sftwer thsa4 aafcekr will be ehat«s«ai Bi. C. W. JUBSSOS. PERFECT THBOUGH SDFFEBISG. our x V - --- < x x 35 V? 3?/ -V 3 - ) - - [ Z8 - & Z - - - - - x 0-35 - 3 75 3 33 09 33 5 \ - - 300 0 ( -? 9 { - - - -- - < - V 3 < < - - Z 7 - z 3 - [ } & _ 3 < 3 ( 5 7< ( % --- /? - / 4-4 - & - % 4 V 2

More information

Neatest and Promptest Manner. E d i t u r ami rul)lihher. FOIt THE CIIILDIIES'. Trifles.

Neatest and Promptest Manner. E d i t u r ami rul)lihher. FOIt THE CIIILDIIES'. Trifles. » ~ $ ) 7 x X ) / ( 8 2 X 39 ««x» ««! «! / x? \» «({? «» q «(? (?? x! «? 8? ( z x x q? ) «q q q ) x z x 69 7( X X ( 3»«! ( ~«x ««x ) (» «8 4 X «4 «4 «8 X «x «(» X) ()»» «X «97 X X X 4 ( 86) x) ( ) z z

More information

Integration - Past Edexcel Exam Questions

Integration - Past Edexcel Exam Questions Integration - Past Edexcel Exam Questions 1. (a) Given that y = 5x 2 + 7x + 3, find i. - ii. - (b) ( 1 + 3 ) x 1 x dx. [4] 2. Question 2b - January 2005 2. The gradient of the curve C is given by The point

More information

L bor y nnd Union One nnd Inseparable. LOW I'LL, MICHIGAN. WLDNHSDA Y. JULY ), I8T. liuwkll NATIdiNAI, liank

L bor y nnd Union One nnd Inseparable. LOW I'LL, MICHIGAN. WLDNHSDA Y. JULY ), I8T. liuwkll NATIdiNAI, liank G k y $5 y / >/ k «««# ) /% < # «/» Y»««««?# «< >«>» y k»» «k F 5 8 Y Y F G k F >«y y

More information

Homework 1/Solutions. Graded Exercises

Homework 1/Solutions. Graded Exercises MTH 310-3 Abstract Algebra I and Number Theory S18 Homework 1/Solutions Graded Exercises Exercise 1. Below are parts of the addition table and parts of the multiplication table of a ring. Complete both

More information

FMIA. Fluid Mechanics and Its Applications 113 Series Editor: A. Thess. Moukalled Mangani Darwish. F. Moukalled L. Mangani M.

FMIA. Fluid Mechanics and Its Applications 113 Series Editor: A. Thess. Moukalled Mangani Darwish. F. Moukalled L. Mangani M. FMIA F. Moukalled L. Mangani M. Darwish An Advanced Introduction with OpenFOAM and Matlab This textbook explores both the theoretical foundation of the Finite Volume Method (FVM) and its applications in

More information

P A L A C E P IE R, S T. L E O N A R D S. R a n n o w, q u a r r y. W WALTER CR O TC H, Esq., Local Chairman. E. CO O PER EVANS, Esq.,.

P A L A C E P IE R, S T. L E O N A R D S. R a n n o w, q u a r r y. W WALTER CR O TC H, Esq., Local Chairman. E. CO O PER EVANS, Esq.,. ? ( # [ ( 8? [ > 3 Q [ ««> » 9 Q { «33 Q> 8 \ \ 3 3 3> Q»«9 Q ««« 3 8 3 8 X \ [ 3 ( ( Z ( Z 3( 9 9 > < < > >? 8 98 ««3 ( 98 < # # Q 3 98? 98 > > 3 8 9 9 ««««> 3 «>

More information

LOWELL WEEKLY JOURNAL

LOWELL WEEKLY JOURNAL G $ G 2 G ««2 ««q ) q «\ { q «««/ 6 «««««q «] «q 6 ««Z q «««Q \ Q «q «X ««G X G ««? G Q / Q Q X ««/«X X «««Q X\ «q «X \ / X G XX «««X «x «X «x X G X 29 2 ««Q G G «) 22 G XXX GG G G G G G X «x G Q «) «G

More information

A Memorial. Death Crash Branch Out. Symbol The. at Crossing Flaming Poppy. in Belding

A Memorial. Death Crash Branch Out. Symbol The. at Crossing Flaming Poppy. in Belding - G Y Y 8 9 XXX G - Y - Q 5 8 G Y G Y - - * Y G G G G 9 - G - - : - G - - ) G G- Y G G q G G : Q G Y G 5) Y : z 6 86 ) ; - ) z; G ) 875 ; ) ; G -- ) ; Y; ) G 8 879 99 G 9 65 q 99 7 G : - G G Y ; - G 8

More information

Compatible Systems and Charpit s Method

Compatible Systems and Charpit s Method MODULE 2: FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 28 Lecture 5 Compatible Systems Charpit s Method In this lecture, we shall study compatible systems of first-order PDEs the Charpit s method for solving

More information

CH.3. COMPATIBILITY EQUATIONS. Multimedia Course on Continuum Mechanics

CH.3. COMPATIBILITY EQUATIONS. Multimedia Course on Continuum Mechanics CH.3. COMPATIBILITY EQUATIONS Multimedia Course on Continuum Mechanics Overview Introduction Lecture 1 Compatibility Conditions Lecture Compatibility Equations of a Potential Vector Field Lecture 3 Compatibility

More information

12. Stresses and Strains

12. Stresses and Strains 12. Stresses and Strains Finite Element Method Differential Equation Weak Formulation Approximating Functions Weighted Residuals FEM - Formulation Classification of Problems Scalar Vector 1-D T(x) u(x)

More information

Lecture 4: Least Squares (LS) Estimation

Lecture 4: Least Squares (LS) Estimation ME 233, UC Berkeley, Spring 2014 Xu Chen Lecture 4: Least Squares (LS) Estimation Background and general solution Solution in the Gaussian case Properties Example Big picture general least squares estimation:

More information

Linear Algebra. Chapter 8: Eigenvalues: Further Applications and Computations Section 8.2. Applications to Geometry Proofs of Theorems.

Linear Algebra. Chapter 8: Eigenvalues: Further Applications and Computations Section 8.2. Applications to Geometry Proofs of Theorems. Linear Algebra Chapter 8: Eigenvalues: Further Applications and Computations Section 8.2. Applications to Geometry Proofs of Theorems May 1, 2018 () Linear Algebra May 1, 2018 1 / 8 Table of contents 1

More information

LOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort

LOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort - 7 7 Z 8 q ) V x - X > q - < Y Y X V - z - - - - V - V - q \ - q q < -- V - - - x - - V q > x - x q - x q - x - - - 7 -» - - - - 6 q x - > - - x - - - x- - - q q - V - x - - ( Y q Y7 - >»> - x Y - ] [

More information

Review Questions for Test 3 Hints and Answers

Review Questions for Test 3 Hints and Answers eview Questions for Test 3 Hints and Answers A. Some eview Questions on Vector Fields and Operations. A. (a) The sketch is left to the reader, but the vector field appears to swirl in a clockwise direction,

More information

Local Chapter. Mr raised the que stion of what is ad't. deliver the s, nnun. You are cor- c Mr 1 n d. "P**"' iropiie.

Local Chapter. Mr raised the que stion of what is ad't. deliver the s, nnun. You are cor- c Mr 1 n d. P**' iropiie. D D D? M G D Y M 2 99 M «4 \ & M? x q M M GM M \ M! 94 - G? \ M M q > G -? Y - M - - - z - > M Z >? - M» > M M - > G! /? - «\- - < x - M-! z - M M M \- - x 7 x GG q M _ ~ > M > # > > M - -

More information

Numerical Modelling in Geosciences. Lecture 6 Deformation

Numerical Modelling in Geosciences. Lecture 6 Deformation Numerical Modelling in Geosciences Lecture 6 Deformation Tensor Second-rank tensor stress ), strain ), strain rate ) Invariants quantities independent of the coordinate system): - First invariant trace:!!

More information

DIFFUSION MAGNETIC RESONANCE IMAGING

DIFFUSION MAGNETIC RESONANCE IMAGING DIFFUSION MAGNETIC RESONANCE IMAGING from spectroscopy to imaging apparent diffusion coefficient ADC-Map anisotropy diffusion tensor (imaging) DIFFUSION NMR - FROM SPECTROSCOPY TO IMAGING Combining Diffusion

More information

LOWELL, MICHIGAN, NOVEMBER 27, Enroute to Dominican Republic

LOWELL, MICHIGAN, NOVEMBER 27, Enroute to Dominican Republic LDG L G L Y Y LLL G 7 94 D z G L D! G G! L $ q D L! x 9 94 G L L L L L q G! 94 D 94 L L z # D = 4 L ( 4 Q ( > G D > L 94 9 D G z ] z ) q 49 4 L [ ( D x ] LY z! q x x < G 7 ( L! x! / / > ( [ x L G q x!

More information

First Order ODEs, Part I

First Order ODEs, Part I Craig J. Sutton craig.j.sutton@dartmouth.edu Department of Mathematics Dartmouth College Math 23 Differential Equations Winter 2013 Outline 1 2 in General 3 The Definition & Technique Example Test for

More information

Chem8028(1314) - Spin Dynamics: Spin Interactions

Chem8028(1314) - Spin Dynamics: Spin Interactions Chem8028(1314) - Spin Dynamics: Spin Interactions Malcolm Levitt see also IK m106 1 Nuclear spin interactions (diamagnetic materials) 2 Chemical Shift 3 Direct dipole-dipole coupling 4 J-coupling 5 Nuclear

More information

NMR Advanced methodologies to investigate water diffusion in materials and biological systems

NMR Advanced methodologies to investigate water diffusion in materials and biological systems NMR Advanced methodologies to investigate water diffusion in materials and biological systems PhD Candidate _Silvia De Santis PhD Supervisors _dott. Silvia Capuani _prof. Bruno Maraviglia Outlook Introduction:

More information

Part 8: Rigid Body Dynamics

Part 8: Rigid Body Dynamics Document that contains homework problems. Comment out the solutions when printing off for students. Part 8: Rigid Body Dynamics Problem 1. Inertia review Find the moment of inertia for a thin uniform rod

More information

REVIEW OF DIFFERENTIAL CALCULUS

REVIEW OF DIFFERENTIAL CALCULUS REVIEW OF DIFFERENTIAL CALCULUS DONU ARAPURA 1. Limits and continuity To simplify the statements, we will often stick to two variables, but everything holds with any number of variables. Let f(x, y) be

More information

TAM3B DIFFERENTIAL EQUATIONS Unit : I to V

TAM3B DIFFERENTIAL EQUATIONS Unit : I to V TAM3B DIFFERENTIAL EQUATIONS Unit : I to V Unit I -Syllabus Homogeneous Functions and examples Homogeneous Differential Equations Exact Equations First Order Linear Differential Equations Reduction of

More information

AB-267 DYNAMICS & CONTROL OF FLEXIBLE AIRCRAFT

AB-267 DYNAMICS & CONTROL OF FLEXIBLE AIRCRAFT FLÁIO SILESTRE DYNAMICS & CONTROL OF FLEXIBLE AIRCRAFT LECTURE NOTES LAGRANGIAN MECHANICS APPLIED TO RIGID-BODY DYNAMICS IMAGE CREDITS: BOEING FLÁIO SILESTRE Introduction Lagrangian Mechanics shall be

More information

Rotational Motion. Chapter 4. P. J. Grandinetti. Sep. 1, Chem P. J. Grandinetti (Chem. 4300) Rotational Motion Sep.

Rotational Motion. Chapter 4. P. J. Grandinetti. Sep. 1, Chem P. J. Grandinetti (Chem. 4300) Rotational Motion Sep. Rotational Motion Chapter 4 P. J. Grandinetti Chem. 4300 Sep. 1, 2017 P. J. Grandinetti (Chem. 4300) Rotational Motion Sep. 1, 2017 1 / 76 Angular Momentum The angular momentum of a particle with respect

More information

Without fully opening the exam, check that you have pages 1 through 12.

Without fully opening the exam, check that you have pages 1 through 12. Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 12. Show all your work on the standard

More information

Chain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics

Chain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics 3.33pt Chain Rule MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Spring 2019 Single Variable Chain Rule Suppose y = g(x) and z = f (y) then dz dx = d (f (g(x))) dx = f (g(x))g (x)

More information

COMP 175 COMPUTER GRAPHICS. Lecture 04: Transform 1. COMP 175: Computer Graphics February 9, Erik Anderson 04 Transform 1

COMP 175 COMPUTER GRAPHICS. Lecture 04: Transform 1. COMP 175: Computer Graphics February 9, Erik Anderson 04 Transform 1 Lecture 04: Transform COMP 75: Computer Graphics February 9, 206 /59 Admin Sign up via email/piazza for your in-person grading Anderson@cs.tufts.edu 2/59 Geometric Transform Apply transforms to a hierarchy

More information

Chapter 9: Differential Analysis

Chapter 9: Differential Analysis 9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control

More information

No calculators, cell phones or any other electronic devices can be used on this exam. Clear your desk of everything excepts pens, pencils and erasers.

No calculators, cell phones or any other electronic devices can be used on this exam. Clear your desk of everything excepts pens, pencils and erasers. Name: Section: Recitation Instructor: READ THE FOLLOWING INSTRUCTIONS. Do not open your exam until told to do so. No calculators, cell phones or any other electronic devices can be used on this exam. Clear

More information

MA102: Multivariable Calculus

MA102: Multivariable Calculus MA102: Multivariable Calculus Rupam Barman and Shreemayee Bora Department of Mathematics IIT Guwahati Differentiability of f : U R n R m Definition: Let U R n be open. Then f : U R n R m is differentiable

More information

General Relativity ASTR 2110 Sarazin. Einstein s Equation

General Relativity ASTR 2110 Sarazin. Einstein s Equation General Relativity ASTR 2110 Sarazin Einstein s Equation Curvature of Spacetime 1. Principle of Equvalence: gravity acceleration locally 2. Acceleration curved path in spacetime In gravitational field,

More information

LOWELL WEEKI.Y JOURINAL

LOWELL WEEKI.Y JOURINAL / $ 8) 2 {!»!» X ( (!!!?! () ~ x 8» x /»!! $?» 8! ) ( ) 8 X x /! / x 9 ( 2 2! z»!!»! ) / x»! ( (»»!» [ ~!! 8 X / Q X x» ( (!»! Q ) X x X!! (? ( ()» 9 X»/ Q ( (X )!» / )! X» x / 6!»! }? ( q ( ) / X! 8 x»

More information

CIS 4930/6930: Principles of Cyber-Physical Systems

CIS 4930/6930: Principles of Cyber-Physical Systems CIS 4930/6930: Principles of Cyber-Physical Systems Chapter 2: Continuous Dynamics Hao Zheng Department of Computer Science and Engineering University of South Florida H. Zheng (CSE USF) CIS 4930/6930:

More information

Diff. Eq. App.( ) Midterm 1 Solutions

Diff. Eq. App.( ) Midterm 1 Solutions Diff. Eq. App.(110.302) Midterm 1 Solutions Johns Hopkins University February 28, 2011 Problem 1.[3 15 = 45 points] Solve the following differential equations. (Hint: Identify the types of the equations

More information

Chapter 9: Differential Analysis of Fluid Flow

Chapter 9: Differential Analysis of Fluid Flow of Fluid Flow Objectives 1. Understand how the differential equations of mass and momentum conservation are derived. 2. Calculate the stream function and pressure field, and plot streamlines for a known

More information

16.2. Line Integrals

16.2. Line Integrals 16. Line Integrals Review of line integrals: Work integral Rules: Fdr F d r = Mdx Ndy Pdz FT r'( t) ds r t since d '(s) and hence d ds '( ) r T r r ds T = Fr '( t) dt since r r'( ) dr d dt t dt dt does

More information

Review for Exam 1. (a) Find an equation of the line through the point ( 2, 4, 10) and parallel to the vector

Review for Exam 1. (a) Find an equation of the line through the point ( 2, 4, 10) and parallel to the vector Calculus 3 Lia Vas Review for Exam 1 1. Surfaces. Describe the following surfaces. (a) x + y = 9 (b) x + y + z = 4 (c) z = 1 (d) x + 3y + z = 6 (e) z = x + y (f) z = x + y. Review of Vectors. (a) Let a

More information

Basic Equations of Elasticity

Basic Equations of Elasticity A Basic Equations of Elasticity A.1 STRESS The state of stress at any point in a loaded bo is defined completely in terms of the nine components of stress: σ xx,σ yy,σ zz,σ xy,σ yx,σ yz,σ zy,σ zx,andσ

More information

CHAPTER 7 DIV, GRAD, AND CURL

CHAPTER 7 DIV, GRAD, AND CURL CHAPTER 7 DIV, GRAD, AND CURL 1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: (1 ϕ = ( ϕ, ϕ,, ϕ x 1 x 2 x n

More information

MANY BILLS OF CONCERN TO PUBLIC

MANY BILLS OF CONCERN TO PUBLIC - 6 8 9-6 8 9 6 9 XXX 4 > -? - 8 9 x 4 z ) - -! x - x - - X - - - - - x 00 - - - - - x z - - - x x - x - - - - - ) x - - - - - - 0 > - 000-90 - - 4 0 x 00 - -? z 8 & x - - 8? > 9 - - - - 64 49 9 x - -

More information

MATH 31BH Homework 5 Solutions

MATH 31BH Homework 5 Solutions MATH 3BH Homework 5 Solutions February 4, 204 Problem.8.2 (a) Let x t f y = x 2 + y 2 + 2z 2 and g(t) = t 2. z t 3 Then by the chain rule a a a D(g f) b = Dg f b Df b c c c = [Dg(a 2 + b 2 + 2c 2 )] [

More information

Practice problems for Exam 1. a b = (2) 2 + (4) 2 + ( 3) 2 = 29

Practice problems for Exam 1. a b = (2) 2 + (4) 2 + ( 3) 2 = 29 Practice problems for Exam.. Given a = and b =. Find the area of the parallelogram with adjacent sides a and b. A = a b a ı j k b = = ı j + k = ı + 4 j 3 k Thus, A = 9. a b = () + (4) + ( 3)

More information

MATH The Chain Rule Fall 2016 A vector function of a vector variable is a function F: R n R m. In practice, if x 1, x n is the input,

MATH The Chain Rule Fall 2016 A vector function of a vector variable is a function F: R n R m. In practice, if x 1, x n is the input, MATH 20550 The Chain Rule Fall 2016 A vector function of a vector variable is a function F: R n R m. In practice, if x 1, x n is the input, F(x 1,, x n ) F 1 (x 1,, x n ),, F m (x 1,, x n ) where each

More information

The Product Operator Formalism

The Product Operator Formalism 2 The Product Operator Formalism 1. INTRODUCTION In this section we will see that the density matrix at equilibrium can be expressed in terms of the spin angular momentum component I z of each nucleus.

More information

Course 2BA1: Hilary Term 2007 Section 8: Quaternions and Rotations

Course 2BA1: Hilary Term 2007 Section 8: Quaternions and Rotations Course BA1: Hilary Term 007 Section 8: Quaternions and Rotations David R. Wilkins Copyright c David R. Wilkins 005 Contents 8 Quaternions and Rotations 1 8.1 Quaternions............................ 1 8.

More information

In this section, mathematical description of the motion of fluid elements moving in a flow field is

In this section, mathematical description of the motion of fluid elements moving in a flow field is Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small

More information

Jim Lambers MAT 280 Summer Semester Practice Final Exam Solution. dy + xz dz = x(t)y(t) dt. t 3 (4t 3 ) + e t2 (2t) + t 7 (3t 2 ) dt

Jim Lambers MAT 280 Summer Semester Practice Final Exam Solution. dy + xz dz = x(t)y(t) dt. t 3 (4t 3 ) + e t2 (2t) + t 7 (3t 2 ) dt Jim Lambers MAT 28 ummer emester 212-1 Practice Final Exam olution 1. Evaluate the line integral xy dx + e y dy + xz dz, where is given by r(t) t 4, t 2, t, t 1. olution From r (t) 4t, 2t, t 2, we obtain

More information

Unit IV State of stress in Three Dimensions

Unit IV State of stress in Three Dimensions Unit IV State of stress in Three Dimensions State of stress in Three Dimensions References Punmia B.C.,"Theory of Structures" (SMTS) Vol II, Laxmi Publishing Pvt Ltd, New Delhi 2004. Rattan.S.S., "Strength

More information

a s*:?:; -A: le London Dyers ^CleanefSt * S^d. per Y ard. -P W ..n 1 0, , c t o b e e d n e sd *B A J IllW6fAi>,EB. E D U ^ T IG r?

a s*:?:; -A: le London Dyers ^CleanefSt * S^d. per Y ard. -P W ..n 1 0, , c t o b e e d n e sd *B A J IllW6fAi>,EB. E D U ^ T IG r? ? 9 > 25? < ( x x 52 ) < x ( ) ( { 2 2 8 { 28 ] ( 297 «2 ) «2 2 97 () > Q ««5 > «? 2797 x 7 82 2797 Q z Q (

More information

Multiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015

Multiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015 Multiple Integrals and Vector Calculus (Oxford Physics) Ramin Golestanian Synopsis and Problem Sets; Hilary 215 The outline of the material, which will be covered in 14 lectures, is as follows: 1. Introduction

More information

NDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16.

NDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16. CAVITY INSPECTION NDT&E Methods: UT VJ Technologies NDT&E Methods: UT 6. NDT&E: Introduction to Methods 6.1. Ultrasonic Testing: Basics of Elasto-Dynamics 6.2. Principles of Measurement 6.3. The Pulse-Echo

More information

Electromagnetism II Lecture 7

Electromagnetism II Lecture 7 Electromagnetism II Lecture 7 Instructor: Andrei Sirenko sirenko@njit.edu Spring 13 Thursdays 1 pm 4 pm Spring 13, NJIT 1 Previous Lecture: Conservation Laws Previous Lecture: EM waves Normal incidence

More information

Lecture 13 - Wednesday April 29th

Lecture 13 - Wednesday April 29th Lecture 13 - Wednesday April 29th jacques@ucsdedu Key words: Systems of equations, Implicit differentiation Know how to do implicit differentiation, how to use implicit and inverse function theorems 131

More information

MA 201: Partial Differential Equations Lecture - 2

MA 201: Partial Differential Equations Lecture - 2 MA 201: Partial Differential Equations Lecture - 2 Linear First-Order PDEs For a PDE f(x,y,z,p,q) = 0, a solution of the type F(x,y,z,a,b) = 0 (1) which contains two arbitrary constants a and b is said

More information

Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell Heat Transfer Heat transfer rate by conduction is related to the temperature gradient by Fourier s law. For the one-dimensional heat transfer problem in Fig. 1.8, in which temperature varies in the y-

More information

Lagrange Multipliers

Lagrange Multipliers Optimization with Constraints As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each

More information

Name: SOLUTIONS Date: 11/9/2017. M20550 Calculus III Tutorial Worksheet 8

Name: SOLUTIONS Date: 11/9/2017. M20550 Calculus III Tutorial Worksheet 8 Name: SOLUTIONS Date: /9/7 M55 alculus III Tutorial Worksheet 8. ompute R da where R is the region bounded by x + xy + y 8 using the change of variables given by x u + v and y v. Solution: We know R is

More information

Maxima and Minima. (a, b) of R if

Maxima and Minima. (a, b) of R if Maxima and Minima Definition Let R be any region on the xy-plane, a function f (x, y) attains its absolute or global, maximum value M on R at the point (a, b) of R if (i) f (x, y) M for all points (x,

More information

Lecture 8. Stress Strain in Multi-dimension

Lecture 8. Stress Strain in Multi-dimension Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]

More information

Rotational & Rigid-Body Mechanics. Lectures 3+4

Rotational & Rigid-Body Mechanics. Lectures 3+4 Rotational & Rigid-Body Mechanics Lectures 3+4 Rotational Motion So far: point objects moving through a trajectory. Next: moving actual dimensional objects and rotating them. 2 Circular Motion - Definitions

More information

Useful Formulae ( )

Useful Formulae ( ) Appendix A Useful Formulae (985-989-993-) 34 Jeremić et al. A.. CHAPTER SUMMARY AND HIGHLIGHTS page: 35 of 536 A. Chapter Summary and Highlights A. Stress and Strain This section reviews small deformation

More information

Numerical Implementation of Transformation Optics

Numerical Implementation of Transformation Optics ECE 5322 21 st Century Electromagnetics Instructor: Office: Phone: E Mail: Dr. Raymond C. Rumpf A 337 (915) 747 6958 rcrumpf@utep.edu Lecture #16b Numerical Implementation of Transformation Optics Lecture

More information

2.1 NUMERICAL SOLUTION OF SIMULTANEOUS FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS. differential equations with the initial values y(x 0. ; l.

2.1 NUMERICAL SOLUTION OF SIMULTANEOUS FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS. differential equations with the initial values y(x 0. ; l. Numerical Methods II UNIT.1 NUMERICAL SOLUTION OF SIMULTANEOUS FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS.1.1 Runge-Kutta Method of Fourth Order 1. Let = f x,y,z, = gx,y,z be the simultaneous first order

More information