2250 S2008 7:30 and 10:45 Lecture Record
Updated: Wednesday April 23: 09:33AM, 2008
Today: Thursday April 18: 20:14PM, 2024
| Week 15: 21 Apr, |
23 Apr, |
25 Apr. |
28 Apr. |
| Week 14: 14 Apr, |
16 Apr, |
18 Apr. |
| Week 13: 07 Apr, |
09 Apr, |
11 Apr. |
| Week 12: 31 Mar, |
02 Apr, |
04 Apr. |
| Week 11: 24 Mar, |
26 Mar, |
28 Mar. |
| Week 10: 10 Mar, |
12 Mar, |
14 Mar. |
| Week 9: 03 Mar, |
05 Mar, |
07 Mar. |
| Week 8: 25 Feb, |
27 Feb, |
29 Feb. |
| Week 7: 18 Feb, |
20 Feb, |
22 Feb. |
| Week 6: 11 Feb, |
13 Feb, |
15 Feb. |
| Week 5: 04 Feb, |
06 Feb, |
08 Feb. |
| Week 4: 28 Jan, |
30 Jan, |
01 Feb. |
| Week 3: 22 Jan, |
23 Jan, |
25 Jan. |
| Week 2: 14 Jan, |
16 Jan, |
18 Jan. |
| Week 1: 07 Jan, |
09 Jan, |
11 Jan. |
Week 15, Apr 21,22,23: Sections 7.3, 7.4, exam review
21 Apr:
Collected maple Mechanical
Oscillations L6.1, L6.2, L6.3. Catch-up day for ch10 problems.
Ch6,ch7,ch10 extra credit due by 4pm on 29 April.
Lecture: Laplace methods and examples for systems. Final exam review ch7 and ch10.
Solving x'=Ax when A is a diagonal matrix [ch1 method]
or when A is non-diagonal [ch5 method]. Laplace theory for x'=Ax+F. Resolvent.
Lecture: General systems u'=Au+F and Second order systems
Coupled spring-mass systems of dimension 2 and higher. Railway cars.
Earthquake models.
Complex eigenvalues and how to deal with the expressions for real
solutions in the eigenanalysis method for u''+Au=F.
Projected slides:
Newton integral calculus and Laplace calculus. Laplace method.
(pdf slides)
Second order systems, 3x3 spring-mass model, railway cars, earthquakes. Derivation.
(pdf slides)
Laplace method for systems, resolvent.
(pdf slides)
Laplace theory examples. Forward and backward table. Shifting, parts, s-differentiation theorems.
(pdf slides)
Laplace rules. Includes: Lerch, Linearity,
t-Differentiation, s-Differentiation, Integral, Shifting I and II, Periodic, Convolution.
(pdf slides)
22 Apr: Final Exam review. Chapters 5,4,3 in that order.
23 Apr: Last day of class. Collect 7.3, 7.4.
Ch6,Ch7,Ch10 extra credit due by 4pm on 29 April.
Final exam review chapters 10,7,6 in that order.
Correction: "ramp" should be "step' to agree with literature in engineering.
Lecture: Transfer function and convolution.
Lecture: Brine tank models. Recirculating brine tanks.
Projected Slides:
Brine tank cascade. Brine tank recyling, home heating.
(pdf slides)
The eigenanalysis method for 2x2 and 3x3 systems x'=Ax.
(pdf slides)
Newton integral calculus and Laplace calculus. Laplace method.
(pdf slides)
Laplace method for systems, resolvent.
(pdf slides)
Laplace theory examples. Forward and backward table. Shifting, parts, s-differentiation theorems.
(pdf slides)
Laplace rules. Includes: Lerch, Linearity,
t-Differentiation, s-Differentiation, Integral, Shifting I and II, Periodic, Convolution.
(pdf slides)
Second order systems, 3x3 spring-mass model, railway cars, earthquakes. Derivation.
(pdf slides)
25 Apr: Final Exam day for the 10:45 class, starts at 10:10am in LCB219.
28 Apr: Final Exam day for the 7:30 class, starts at 7:30am in JTB140.
Week 14: Sections 10.1 to 10.4, Midterm 3
14 Apr:
Lecture: Basic Laplace theory. Shift theorem. Parts theorem.
Forward table. Backward table. Extensions of the Table.
Lecture: Solving differential equations by Laplace's method.
Rules and the brief table [Laplace calculus]. Partial fractions.
Using trig identities [sin 2u = 2 sin u cos u, etc].
Partial fraction expansions suited for LaPlace theory.
Solving initial value problems by LaPlace's method. Details of
the backward table and the forward table. Information about the
equivalence of the inverse of L and Lerch's theorem.
Laplace theory references
Intro to Laplace theory. L-notation. Forward and backward table. Examples.
(pdf slides)
Laplace theory examples. Forward and backward table. Shifting, parts, s-differentiation theorems.
(pdf slides)
Laplace theory typeset manuscript 2008
(49 pages pdf)
Heaviside's method
2008, typeset
(pdf)
Ch10 Laplace solutions [scanned], 10.1 to 10.4
(9 pages, 2mb, pdf)
15 Apr: Midterm 3. There are 5 problems.
16 Apr: Collected Page 576, 10.1: 18, 28 and
Page 588, 10.2: 10, 16, 24
Lecture: Examples of forward and backward table calculations. Harmonic oscillator.
Systems and Cramer's rule. Laplace theory tricks with the Shifting
theorem and the s-differentiation theorem.
Partial fractions, Heaviside
method, shortcuts, failsafe [sampling] method for partial fractions,
method of atoms, systems of two
differential equations, Cramer's Rule, matrix inversion methods.
How to
write up a solution which postpones partial fraction evaluation of
constants to the end. Use of inverse Laplace and Lerch's theorem.
Partial fraction methods for complex roots and roots of multiplicity
higher than one.
Partial Fractions: How to deal with
complex factors like s^2+4. Heaviside's coverup method and how it works in
the case of complex roots.
Solving second order DE by Laplace. What to expect and how to do it.
Projected slides:
Laplace theory examples. Forward and backward table. Shifting, parts, s-differentiation theorems.
(pdf slides)
Laplace rules. Includes: Lerch, Linearity,
t-Differentiation, s-Differentiation, Integral, Shifting I and II, Periodic, Convolution.
(pdf slides)
18 Apr: Collected Page 597, 10.3: 6,18 and Page 606, 10.4: 22
Lecture: Final exam problems.
Convolution theorem. Convolution examples. How to do 10.4 problems.
More on ramp, sawtooth, staircase, rectified sine. Applied expansions
of periodic functions as Laplace transforms.
Periodic function theorem. Proof of the periodic function theorem.
Laplace of the square wave, derivation, tanh function.
Laplace of the triangular wave, derivation from the integral theorem.
Week 13, Apr 7,9,11 Sections 6.2,7.1,7.2,7.3,
07 Apr: Collected 5.6
Exam Review: midterm 3 problem 3: undetermined coefficients, solving homogeneous DE.
Lecture: Eigenanalysis solved problems, sample exam problems.
How to use determinants and frame sequences to find eigenpairs and P,D packages.
Lecture: Diagonalization. Equivalence of AP=PD to Fourier's model
A(c1 v1 + c2 v2) = c1 lambda1 v1 + c2 lambda2 v2.
Equivalence of AP=PD to the set of equations A v1 = lambda1 v1, A v2
= lambda2 v2.
Eigenvalues from determinants and eigenvectors from frame sequences.
References on Fourier's model, eigenanalysis.
What's eigenanalysis slides 2008 ( pdf)
algebraic eigenanalysis slides 2008 ( pdf)
Eigenanalysis-I manuscript S2008 (typeset 19 pages, 200k pdf)
08 Apr: E. Meucci and L. Zhang, Problems 6.1, 6.2.
Review midterm problems 4, 5 and Fourier's model.
09 Apr: Collected 6.1
Lecture: Complex eigenvalues and eigenvectors.
Answer check for eigenpairs [compute AP and PD, then compare AP=PD].
Lecture: Systems of differential equations, position-velocity
substitution, conversion of scalar equations to vector-matrix systems,
general solution, review problems for exam 3.
Systems of differential equations, conversion of 2x2 scalar linear
equations to vector-matrix systems x'=Ax.
Chapter 7 references:
Slides 2008, solving triangular and non-triangular systems (pdf)
Ch7 Systems of DE slides 7.1,7.2,7.3 solved problems (4 pages, 0.8mb pdf)
Recipes for 1st, 2nd order and 2x2 Systems of DE PDF Document (2 pages, 50kb)
DE systems manuscript S2007 (typeset 69 pages, 970k pdf)
11 Apr: Collected 6.2 and Ch3, Ch4 extra credit.
Lecture: Exam review. Problem 5.
Solving dynamical systems, 2x2 case.
Superposition. Answer check for a general solution. Wronskian test
for independence.
main theorem of eigenanalysis and u'=Au.
Solving u'=Au by eigenanalysis. Exercise solutions for
sections 7.1, 7.3. Sample solutions for solving u'=Au when
A is 2x2, 3x3 and 4x4.
Solving u'=Au in the 2x2 case. Methods 1,2,3,4.
Method 1 is eigenanalysis.
Method 2 is the Chapter 5 method, where x(t) is the solution of a
second order equation with characteristic equation det(A- r I)=0. Then
y(t) is found by using the first DE, and existing formulas for x(t) and
x'(t). This method works when A is not a diagonal matrix.
Method 3 is for diagonal systems u'=Au, in which both differential
equations are growth-decay equations solved by Chapter 1 methods.
Method 4 is solving u'=Au+F by Laplace. Details next week.
Lecture: Introduction to Laplace's method. The method of quadrature
for higher order equations and systems. Calculus for chapter one
quadrature versus the Laplace calculus. The Laplace integrator
dx=exp(-st)dt. The Laplace integral abbreviation L(f(t)) for
the Laplace integral of f(t). Lerch's cancellation law and the fundamental
theorem of calculus.
Def: Direct Laplace transform == Laplace integral ==
int(f(t)exp(-st),t=0..infinity) == L(f(t)).
Linearity. The s-differentiation theorem (d/ds)L(f(t))=L((-t)f(t)).
Laplace theory references week 13.
Intro to Laplace theory. L-notation. Forward and backward table. Examples.
(pdf slides)
Laplace theory examples. Forward and backward table. Shifting, parts, s-differentiation theorems.
(pdf slides)
Laplace theory typeset manuscript 2008
(49 pages pdf)
Heaviside's method
2008, typeset (pdf)
Ch10 Laplace solutions [scanned], 10.1 to 10.4
(9 pages, 2mb, pdf)
Week 12, Mar 31, Apr 2,4: Sections 5.4, 5.6, 6.1, 6.2
31 Mar: Nothing due. Catch-up day.
Problems solved in class: 5.4-20,34 [review] and 5.5-50,62
Lecture 5.4: overdamped, critically damped and under damped behavior, pseudoperiod.
Lecture 5.6: Applications of undetermined coefficients. More fixup rule
examples.
Lecture: Pure resonance and practical resonance.
Damped forced oscillations. Practical resonance
plots.
Lecture 5.6: Wine glass experiment. Tacoma narrows, resonance and vortex shedding.
Soldiers marching in cadence. Theorems on mx''+kx=F0 cos(omega t). Theorems
on mx''+cx'+kx=F0 cos(omega t). Bounded and unbounded solutions. Unique periodic
steady-state solution. Pure resonance omega = sqrt(k/m). Practical resonance
omega = sqrt(k/m - c^2/(2m^2)). Resonance and the fixup rule: omega=sqrt(k/m)
if and only if the fixup rule applies to mx''+kx = F0 cos(omega t).
Slides projected:
Forced vibrations, undamped case, slides 2008 (pdf)
Forced vibrations, damped case, slides 2008 (pdf)
Forced vibrations and resonance, slides 2008 (pdf)
01 Apr: Meucci and Zhang lecture on 5.4, 5.5 problems and two problems
from the S2007 midterm 3 key. Field questions on 5.4, 5.5, 5.6 as well as
extra credit problems (ch5).
02 Apr: Collected 5.4-20,34. One stapled package.
Problems solved in class: 5.5: 6, 14, 27, 50, 62 [review]
and 5.6: 2, 10, 16 [see web problem notes].
More on resonance, including practical resonance theory.
Wine glass breakage (QuickTime MOV)
Wine glass experiment (12mb mpg 2min video)
Tacoma Narrows Bridge Nov 7, 1940 (18mb mpg 4min video)
Lecture: One more example of undetermined coefficients. Intro to eigenanalysis.
Fourier's model. History.
References on Fourier's model, eigenanalysis.
What's eigenanalysis slides 2008 (pdf)
Eigenanalysis-I manuscript S2007 (typeset 19 pages, 200k pdf)
04 Apr: Collected in class,
Page 346, 5.5: 6, 14, 27, 50, 62 in one stapled package.
Page 357, 5.6: 2, 10, 16 in one stapled package.
Lecture: How to find the variables lambda and v in Fourier's
model using determinants and frame sequences.
Solved in class: examples similar to the problems in 6.1 and 6.2.
Slides and problem notes exist for 6.1 and 6.2 problems. See the web site.
References on Fourier's model, eigenanalysis.
What's eigenanalysis slides 2008 ( pdf)
Eigenanalysis-I manuscript S2007 (typeset 19 pages, 200k pdf)
Week 11, Mar 24,26,28: Sections 5.5, 5.4, 5.6
24 Mar: Collect 5.1 problems 34 to 48.
Lecture: Solving constant coefficient nth order DE by finding its
list of n atoms. Start section 5.5 undetermined coefficients.
References for weeks 10 and 11.
Picard's Theorem for systems, slides 2008 (pdf)
How to solve linear DE, slides 2008 (pdf)
Solving linear DE, examples of orders 1 to 4, slides 2008 (pdf)
Basic Reference: First order constant coefficient recipe, structure of solutions, superposition (slides, 3 pages pdf)
Basic Reference: First order constant coefficient recipe + theory + variation of parameters and undetermined coefficients (typeset, 11 pages pdf)
Second order constant coefficient recipe + theory (typeset, 7 pages pdf)
Undetermined coefficients example, cafe door, pet door, phase-amplitude, resonance slides F2007
Week 11 references for Edwards-Penney section 5.5
Second order variation of parameters (typeset, 6 pages pdf)
Second order undetermined coefficients (typeset, 7 pages pdf)
Higher order linear differential equations. Higher order recipe. Higher order undetermined coefficients. (typeset, 9 pages pdf)
Undetermined coefficients slides Nov 1, 2007(pdf)
Undetermined coefficients slides Nov 8, 2007 (pdf)
Ch5 undetermined coeff illustration (3 pages, pdf)
25 Mar: Questions on exam 2 problems 2,3. Questions on 5.2
problems, 5.3-16, 5.3-32.
Midterm 2, problems 2 and 3 [40 min].
26 Mar: Collect Maple L3.2,L3.3,L3.4,L4.2,L4.3,L4.4. Collect 5.2.
Lecture: Questions on maple lab 5. Undetermined coefficients,
Related atoms, atomRoot function and the fixup rules 2,3,4. How to find
trial solutions quickly. Section 5.5 of Edwards-Penney. A number of
slides were projected in class, two from Monday (how to solve the
homogeneous equation)
How to solve linear DE, slides 2008 (pdf)
Solving linear DE, examples of orders 1 to 4, slides 2008 (pdf)
and one for today (how to solve for y_p by undetermined coefficients).
Undetermined coefficients slides Nov 1, 2007(pdf)
Undetermined coefficients slides Nov 8, 2007 (pdf)
28 Mar: Collect 5.3 and maple 5. If you could not finish
maple labs, then look at the extra credit problems.
Lectur: 5.4 Damped and undamped motion. Pendulum, harmonic oscillations,
spring-mass equation, phase-amplitude conversions from the trig course,
Lecture: 5.5 variation of parameters formula (33).
Second order variation of parameters slides 2008 (pdf)
Second order variation of parameters (typeset, 6 pages pdf)
Lecture: Sections 5.4, 5.6. Forced oscillations.
Forced vibrations, undamped case, slides 2008 (pdf)
Forced vibrations, damped case, slides 2008 (pdf)
Forced vibrations and resonance, slides 2008 (pdf)
Week 10, Mar 10,12,14: Sections 4.6, 4.7, 5.1, 5.2, 5.3
10 Mar: Nothing collected.
Exam review: questions answered about exam 2.
Sample test for problems 1,4,5.
Lecture: Standard basis in R^n. Theorems on independent sets and bases.
Kernel. Nullspace. Image. Column space. Row space.
Equivalent bases.
Review independence of functions: sampling
test and Wronskian test. Review of RREF [RANK] and DETERMINANT test for
independence of fixed vectors.
Orthogonality. General vector spaces.
Lecture: Picard's Theorem and the dimension of the solution space
of a linear constant system of differential equations.
Problem session 4.5, 4.6, 4.7.
Web References:
Lecture slides on Vector spaces, Independence tests. (pdf)
Lecture slides on Basis, Dimension, Rank, kernel, pivot theorem, rowspace, colspace (pdf)
Lecture slides on orthogonality, independence of orthogonal sets, Cauchy-Schwartz, Pythagorean Identity (pdf)
11 Mar: Exam 2 at 7:15 in JTB 140 and 10:30 in LCB 219, proctors Zhang and Meucci. Covers only problems 1,4,5 of exam 2. Problems 2,3 delayed uintil 25 March.
12 Mar: Collected 4.5-8,22,28 and 4.6-2.
Lecture: Proofs involving subspaces for vector spaces V whose
data item packages are functions.
Lecture: Definition of atom. Independence of atoms.
Method of atoms in partial fractions. Sampling in partial fractions. Heaviside's coverup method.
Solution space theorem for linear differential equations.
Picard's Theorem for higher order DE and systems. Dimension of the solution space.
Structure of solutions.
Solutions to 4.7-10,22,24.
Week 10 references.
Picard's Theorem for systems, slides 2008 (pdf)
How to solve linear DE, slides 2008 (pdf)
Solving linear DE, examples of orders 1 to 4, slides 2008 (pdf)
Basic Reference: First order constant coefficient recipe, structure of solutions, superposition (slides, 3 pages pdf)
Basic Reference: First order constant coefficient recipe + theory + variation of parameters and undetermined coefficients (typeset, 11 pages pdf)
Ch5. Constant coefficient recipe (typeset, 2 pages, pdf)
Second order constant coefficient recipe + theory (typeset, 7 pages pdf)
Undetermined coefficients, cafe door, pet door, phase-amplitude, resonance slides F2007
14 mar: Collected 4.7-10,22,24.
Lecture: Second order and higher order differential Equations.
Picard theorem for second order equations, superposition, solution
space structure, dimension of the solution set. Quadratic equations
again. Constant-coefficient second order homogeneous differential
equations. Spring-mass DE and RLC-circuit DE derivations.
Electrical-mechanical analogy.
Euler's theorem. Complex roots and
the formula exp(i theta)=cos
(theta) + i sin(theta). How to solve homogeneous
equations by searching for a list of n distinct atoms that are solutions
to the equation. Specific examples for first, second and higher order
equations. Common errors in solving higher order equations.
Solved examples like the 5.1,5.2,5.3 problems.
Identifying atoms in linear combinations. Solving more complicated
homogeneous equations.
Higher order constant equations, homogeneous and
non-homogeneous structure. Superposition. Solution space structure.
Week 9, Mar 3,5,7: Sections 4.3, 4.4, 4.5, 4.6, 4.7
03 Mar: Collected 4.1-18,22,30. Didn't do 3.6-64? Do the
extra credit from ch3.
Extra credit Ch3 due March 12.
Lecture: Independence and dependence. Algebraic tests. Geometric tests.
Kernel theorem. Not a subspace theorem. Solutions of problems 4.1, 4.2.
More on the toolkit. Vectors as packages of data items. Examples
of vector packaging in applications. The kernel: sols of Ax=0.
Web References:
Lecture slides on Vector spaces, Independence tests. (pdf)
Lecture slides on Basis, Dimension, Rank, kernel, pivot theorem, rowspace, colspace (pdf)
04 Mar: Meucci and Zhang conduct exam 2 review. Questions on 4.3 problems.
Discuss midterm problems 1,4,5.
05 Mar: 4.2 problems due.
Exam review continues from Tuesday: problem 3.
Lecture: More on independence. General solutions with a minimal
number of terms. Basis == independence + span. Independence of atoms.
Function spaces. Wronskian test. The pivot theorem. Rank test.
Determinant test.
07 Mar: Problems 4.3, 4.4 due.
Exam review continues from Tuesday: problems 1,2,3.
Examples in class for Problem 1, midterm 2. To make your own copy of the
projected material, click here.
Lecture: Basis == independence + span. Dimension. Basis for linear
system Ax=0 from the last frame algorithm. Partial derivatives and bases.
The pivot theorem proof. Proof rank(A)=rank(A^T).
Week 8: Feb 25,27,29: Sections 3.6, 4.1, 4.2, 4.3
25 Feb: Collect Page 194, 3.5: 14, 26, 38.
Chapter 3 and maple lab 2 extra credit problems
click here
Lecture: Section 3.6
Determinant slides 2008 (pdf)
Ch3 Page 149+ slides, Exercises 3.1 to 3.6 (6 pages, 890k pdf)
Determinants and Cramer's Rule (typeset 186k pdf)
Methods for computing a determinant. Sarrus' rule, Cofactor
expansion, four rules for determinants. In-class examples.
Hybrid methods to compute a determinant. In-class example.
Special theorems for determinants having a zero row, duplicates rows or
proportional rows. How to use the 4 rules.
Elementary matrices and determinants. Determinant product rule.
26 Feb: Meucci and Zhang: review 3.6 problems,
maple lab 2 questions, lecture on maple lab 3 and maple lab 4.
Details on problems 3.6: 6, 18, 30, 38, 64.
27 Feb: Due are Page 194, 3.5: 14, 26, 38 and the
last problem of Maple lab 2 are due. If you
are not done with lab 2, then do the two lab 2 extra credit problems, 7 day limit.
See the ch3 extra credit problems.
Due today: All chapter 2 extra credit problems, covers 2.1, 2.2, 2.3. Other
sections in chapter 2 appear in later extra credit packages.
Lecture: Cofactor expansion. Hybrid methods. Frame sequences and
determinants. Formula for det(A) in terms of swap and mult operations.
How to compute determinants of sizes 3x3, 4x4, 5x5 and higher.
Cramer's rule. Adjugate matrix. In-class examples on Cramer's rule for
2x2 and 3x3. How to form minors, cofactors and the adjugate matrix.
Cofactor rule and the adjugate matrix. How to find det(A) from A and
adj(A). Cofactor rules imbedded in the formula det(A)I = A adj(A).
Determinant slides 2008 (pdf)
Examples in class: (1) solve a 2x2 system by Cramers rule. (2) Find entry in row 3, col 2
of the inverse of A = adj(A)/det(A) as a quotient of 2 determinants. (3)
Find det(A) from A and adj(A).
29 FebDue are Page 212, 3.6: 6, 18, 30, 38, 64.
Lecture: Chapter 4, sections 4.1 and 4.2.
Web references for chapter 4.
Lecture slides on Vector spaces, Independence tests. (pdf)
Lecture slides on Basis, Dimension, Rank, kernel, pivot theorem, rowspace, colspace (pdf)
Vector space, Independence, Basis, Dimension, Rank (typeset pdf)
Ch4 Page 237+ slides, Exercises 4.1 to 4.4, some 4.9 (4 pages, 480k pdf)
Vectors==packages of data items.
Examples of vectors: digital photos, Fourier coefficients, Taylor
coefficients, sols to DE like y=2exp(-x^2).
The toolkit of 8 properties (Thm 1, 4.2). Vector
spaces. Subspaces. Parking lot analogy. Data recorder example and data conversion to fit
physical models. Subspace criterion. Kernel theorem for Ax=0.
Example in class: Apply the subspace theorem to
x+y+z=0, writing it as Ax=0, then apply the kernel theorem (thm 2 page 239 of Edwards-Penney).
Lecture: Chapter 4, section 4.3
More on subspaces: detection of subspaces and data sets that are not
subspaces. Use of theorems 1,2 in section 4.2. Problems 4.1, 4.2 solved
in class.
Week 7, Feb 18,20,22
Sections 3.3, 3.4, 3,5, 3.6
18 Feb: President's Day holiday. No classes.
19 Feb: L. Zhang, review 3.4 problems 20, 30, 34, 40, maple lab 2 questions, maple lab 3 numerical work.
Under construction 20Feb
Maple Lab 3, Numerical Solutions
Maple Lab 3 Spring 2007 (pdf)
Maple L3 snips Spring 2007 (maple text)
Maple Worksheet files: In Mozilla firefox, save to disk using
right-mouseclick and then "Save link as...".
Some browsers require SHIFT and then mouse-click. Open the saved
file in xmaple.
Maple L3 snips worksheet Spring 2007 (maple .mws)
Numerical DE coding hints (txt)
The actual symbolic solution derivation and answer check are
submitted as L3.1. Confused? Follow the details in the next link.
Sample symbolic solution report for 2.4-3 (pdf 1 page, 120k)
Do not submit any work for L3.1 with decimals! Only 1.4 methods are
to appear.
The numerical work, Euler, Heun, RK4 parts are L3.2, L3.3, L3.4.
Confused about what to put in your L3.2 report? Do the same
as what appears in the sample report for 2.4-3 (below).
Sample Report for 2.4-3. Includes symbolic solution report. (pdf 3 pages, 350k)
Sample maple code for Euler, Heun, RK4 (maple .mws)
Sample maple code for exact/error reporting (maple .mws)
Additional reference, probably not needed:
Report details on 2.4,2.5,2.6 prob 6 (pdf)
20 Feb: Problems 3.3 due.
[Wednesday, all of 3.4 problems will be due, 3.4-20,30,34,40]
Lecture:
Discussion of Cayley-Hamilton theorem [3.4-29] and how to solve problem 3.4-30.
Ch1 extra credit due Friday. Ch2 extra credit due next Wed after that.
Lecture: How to compute the inverse matrix from inverse = adjugate/determinant
and also by frame sequences.
Web Reference: Construction of inverses. Theorems on inverses.
slides on rref inverse method S2008
22 Feb: Ch1 extra credit due. Please work on 3.6 and 4.1 problems.
.
About problem 3.5-44: This problem is the basis for the
fundamental theorem on elementary matrices (see below). While 3.5-44 is
a difficult technical proof, the extra credit problems on this subject
replace the proofs by a calculation. See Xc3.5-44a and Xc3.5-44b.
Digitial photos and matrices.
Web Reference: Image sensors, digitalphotos, checkerboard analogy,
visualization of matrix addition and scalar multiplication.
Digital photos and matrix operations slides S2008
Lecture: Elementary matrices.
Theorem: rref(A)=(product of elementary matrices)A.
Web Reference: Elementary matrices
Elementary matrix slides S2008
Lecture: Introduction to 3.6 determinant theory and Cramer's rule.
Lecture: Adjugate formula for the inverse. Review of Sarrus' Rules.
slides for 3.6 determinant theory
Ch3 Page 149+ slides, Exercises 3.1 to 3.6 (6 pages, 890k pdf)
Determinants and Cramer's Rule (typeset, 11 pages, 140k pdf)
Determinant slides 2008 (pdf)
Week 6, Feb 11,13,15: Sections 3.3, 3.4, 3.5
11 Feb: Due today, symbolic sol L3.1, L4.1.
Review of linear equations: Rank, Nullity, dimension, 3 possibilities, elimination algorithm.
Slides on rank, nullity, elimination algorithm 11FEb2008 (pdf)
Slides on the 3 possibilities, rank, sytems with symbol k 11Feb2008 (pdf)
Slides on three possibilities with symbol k,14Feb2007 (3 pages, pdf)
Start of class: Exam 1 review, problems 2,3.
Lecture: 3.2, 3.3. Intro to matrices and matrix models for linear equations. Matrix multiply Ax for x a vector.
Fixed vectors, physics vectors i,j,k, engineering vectors (arrows), Gibbs vectors.
Parallelogram law. Head-tail rule.
12 Feb, Midterm 1 at 7:10am in JTB 140 or 10:30 in LCB 219.
13 Feb: Page 162, 3.2: 10, 18, 24
In 3,2 solutions, back-substitution should be presented as combo
operations in a frame sequence, not as isolated algebraic jibberish.
Lecture: The 8-property toolkit for vectors. Vector spaces. Read 4.1
in Edwards-Penney, especially the 8 properties pages 223-226 [227-233 can be read later].
Translation of equation models to (augmented) matrix models and back.
Combo, swap and multiply for matrix models. Frame sequences for matrix
models. Computer algebra systems and error-free frame sequences.
How to program maple to make a frame sequence without errors.
Review of rref, rank, nullity, dimension with examples.
Review of vector models is in the slide set
Slides on vector models and vector spaces 2008 (pdf)
Problem 3.2-24: See
Slides on three possibilities with symbol k,14Feb2007 (3 pages, pdf)
Beamer Slides on three possibilities with symbol k, Sept2007 (9 pages,pdf)
See also Example 10 in
Linear equations, no matrices, DRAFT Feb2008 (typeset, 44 pages, pdf)
Prepare 3.3 problems 8, 18 for next time. Please use frame sequences
to display the solution, as in today's lecture examples. It will be a
sequence of augmented matrices. Yes, you may use maple to make the frame
sequence and to do the asnwer check [rref(A);].
Slides and examples for chapter 3
Linear equations, reduced echelon, three rules (typeset, 7 pages, pdf)
Three rules, frame sequence, maple syntax (typeset, 7 pages, pdf, 12 Oct 2006)
Snapshot sequence and general solution, 3x3 system (1 handwritten page, pdf, 28-Sep-2006)
Slides on three possibilities with symbol k,14Feb2007 (3 pages, pdf)
Ch3 Page 149+ slides, Exercises 3.1 to 3.6 (6 pages, 890k pdf)
Elementary matrices, vector spaces, slides (8 pages, pdf, 12 Oct 2006)
Slides on matrix Operations (pdf)
Typeset references for ch3 and ch4
Linear equations, no matrices, DRAFT 2008 (typeset, 44 pages, pdf)
vectors and matrices (typeset, 14 pages)
Matrix equations (typeset, 12 pages)
Determinants and Cramer's Rule (typeset, 11 pages, 140k pdf)
15 Feb: Due today, Page 170, 3.3: 8, 18
and L1.1, L1.2 the first maple lab. Answer checks should also use the
online FAQ.
Problems submitted without frame sequence details are incomplete. Yes,
you can use maple to create the sequence and do the answer check.
Lecture: 3.3 and 3.4, Vector form of the solution to a linear
system. Matrix multiply and the equation Ax=b.
Slides on matrix Operations (pdf)
Answer to the question: What did I just do, when I found rref(A)?
Equation ideas can be used on a matrix A. View matrix A
as the set of coefficients of a homogeneous linear system Ax=0. The augmented
matrix B for this homogeneous system would be the given matrix with a
column of zeros appended: B=aug(A,0).
L2.1 discussed today. Transparencies projected for L2.1 and L2.4 solutions.
Happens only in class, no web solutions available.
Due next Week: 3.4-18,30,36,40. See FAQ 3.4 for details.
Problems 3.4-17 to 3.4-22 are homogeneous systems Ax=0 with A in
reduced echelon form. Apply the last frame agorithm then write the
general solution in vector form.
Problem 3.4-29 is used in Problem 3.4-30. The result is the Cayley-Hamilton Theorem,
a famous theorem of linear algebra which is the basis for solving systems of differential equations.
Problem 3.4-40 is the superposition principle for the matrix equation Ax=b. It is the analog of the differential equation
relation y=y_h + y_p.
Week 5, Feb 4,6,8: Sections 2.4,2.5, 2.6, 3.1, 3.2
04 Feb:
Review: study the slides on partial fractions.
Due today, Page 106, 2.3: 10, 20
For more details on the 2.3 problems,
Click Here.
Continue the Friday lecture on numerical methods.
Discussed y'=3x^2-1, y(0)=2 with solution y=x^3-x+2. Dot tables, connect the dots graphic.
How to draw a graphic without knowing the solution equation for y.
Main example y'=srqt(x)exp(x^2), y(0)=2. Making the dot table by approximation
of the integral of F(x). Rect, Trap, Simp rules and their accuracy of 1,2,4 digits resp.
Example for your study: The problem y'=x+1, y(0)=1 has a dot table with
x=0, 0.25, 0.5, 0.75, 1 and y= 1, 1.25, 1.5625, 1.9375, 2.375. The exact
solution y=1/2+(x+1)^2/2 has values y=1, 1.28125, 1.625, 2.03125,
2.5000. Try to determine how the dot table was constructed and identify which rule [Rect, Trap, Simp] was applied.
Symbolic solution, no numerics, maple L3.1, L4.1 due next Monday.
Discussion of Euler, Heun, RK4 algorithms. Computer implementations.
Numerical work maple L3.2-L3.4, L4.2-L4.4 will be submitted after the spring break.
All discussion of maple programs will be based in the Tuesday session.
There will be one additional presentation of maple lab details in the main lecture.
References for numerical methods:
Numerical DE slides 2008 (14 slides pdf)
Numerical Solution of First Order DE (typeset, 19 pages, 220k pdf)
Sample Report for 2.4-3 (pdf 3 pages, 350k)
ch2 Numerical Methods Slides, Exercises 2.4-5,2.5-5,2.6-5 plus Rect, Trap, Simp rules (5 pages, pdf)
05 Feb: Exam 1 review, questions and examples on problems 1,2,3,4,5.
How to present the solutions to L3.1, L4.1.
06Feb: Nothing due today, catch-up day, switching chapters.
Maple lab symbolic sol L3.1, L4.1 will be due Monday. Exam 1 next Tuesday.
Lecture: 3.1, frame sequences, combo, swap, multiply, geometry
Prepare 3.1 problems for Friday.
References for chapter 3
Slides on Linear equations, reduced echelon, three rules (pdf)
Slides on Linear equations, unique solution case (pdf)
Slides on Linear equations, no solution case, signal equations (pdf)
Slides on Linear equations, infinitely many solution case, last frame algorithm (pdf)
Three rules, frame sequence, maple syntax (typeset, 7 pages, pdf, 12 Oct 2006)
Frame sequence and general solution, 3x3 system (1 page, pdf, 28-Sep-2006)
Linear algebra, no matrices, DRAFT 8Feb2008 (typeset, 44 pages, pdf)
Slides on three possibilities with symbol k,14Feb2007 (3 pages, pdf)
vectors and matrices (typeset, 14 pages)
Matrix equations (typeset, 12 pages)
Ch3 Page 149+ slides, Exercises 3.1 to 3.6 (6 pages, 890k pdf)
Elementary matrices, vector spaces, slides (8 pages, pdf, 12 Oct 2006)
Determinants and Cramer's Rule (typeset, 11 pages, 140k pdf)
08 Feb:
Due today, Page 152, 3.1: 4, 18, 26
Lecture: 3.1, 3.2, 3.3, frame sequences, general solution, three possibilities.
A detailed account of the three possibilities. How to solve a linear system
using the tookit [swap, combo, mult] and frame sequences, for the unique solution case,
no solution case and infinitely many solution case. Examples.
Slides for this lecture.
Slides on Linear equations, unique solution case (pdf)
Slides on Linear equations, no solution case, signal equations (pdf)
Slides on Linear equations, infinitely many solution case, last frame algorithm (pdf)
Sample solution L3.1 (jpg)
Week 4, : Sections 2.2, 2.3, 2.4, 2.5
28 Jan: Due today, 1.5-6,18,22,34
Lecture on stability theory. Discussion of 2.1-6,16.
Reading on partial fractions [we study (1) sampling, (2) method of atoms, (3) Heaviside cover-up]:
Partial Fraction Theory 2008(125k pdf)
Lecture on 2.1, 2.2 problems. How to construct phase line
diagrams. How to make phase plots. Discussion of 2.2-10,18.
Covered in class for 2.1, 2.2: theory of autonomous DE y'=f(y),
stability, funnel, spout, phase diagram, asymptotic stability, unstable,
equil solution, verhulst models with harvesting term.
Work on problems 2.1-6,16 and ask questions on Tuesday.
stability theory, phase diagrams,
calculus tools, DE tools, partial fraction methods.
Next: 2.3 and numerical DE topics.
Due 05 Feb, Page 106, 2.3: 10, 20
Lecture on midterm 1 problem 5, in conjunction with 2.2-18.
Introduction to Newton models for falling bodies and projectiles.
References for 2.1, 2.2, 2.3:
Autonomous DE slides 2008 (pdf)
Newton models, projectile slides 2008 (pdf)
Earth to the moon slides 2008 (pdf)
Verhulst logistic equation (typeset, 5 pages, pdf)
Phase Line and Bifurcation Diagrams (includes "Stability, Funnel, Spout, and bifurcation") (typeset, 6 pages, 161k pdf)
ch2 sections 1,2,3 Slides: 2.1-6, 2.1-16 (rabbit), 2.1-38, 2.2-4, 2.2-10, 2.3-9, 2.3-27+Escape velocity (8 pages, pdf)
ch2 DEplot maple example 1 for exercises 2.2, 2.3 (1 page, 1k)
ch2 DEplot maple example 2 for exercises 2.2, 2.3 (1 page, 1k)
Heaviside partial fraction method (4 pages, 86k)
Heaviside's method and Laplace theory (153k pdf)
Partial Fraction Theory 2008(125k pdf)
29 Jan: Discuss maple 2 in lab.
Content [Meucci, Zhang]:
Maple labs discussed: 1,2. Sources on www. Problems discussed from 2.1 to 2.3.
Midterm problems 3,4 discussed. Midterm sample was distributed in class earlier.
Links for maple lab 2:
maple Lab 2 S2008 (pdf)
maple worksheet text Lab 2 S2008
For more on superposition y=y_p_ + y_h, see Theorem 2 in the link
Linear DE part I (8 pages pdf)
For more about home heating models, read the following link.
Linear equation applications, brine tanks, home heating (typeset, 12 pages, pdf)
30 Jan:
Due today, Page 86, 2.1-6,16. Next time: 2.2-10,18
Free fall with no air resistance F=0.
Linear air resistance models F=kx'.
Nonlinear air resistance models F=kx'|x'|.
Lecture slides on the reading assignment for 2.3, and the work of Isaac
Newton on ascent and descent models for kinematics with air resistance.
Newton models, projectile slides 2008 (pdf)
Problem notes for 2.3-10. 2.3-20 are available,
Click Here.
Intro to the Jules Verne problem and its solution.
Earth to the moon slides 2008 (pdf)
01 Feb: Due today 2.2-10,18. All of 2.3 and Maple Lab 1 is due on Monday 11 Feb: Intro maple L1.1, L1.2.
If you are unable to turn in this lab, then see the Ch2 Extra Credit problems,
which contains 2 problems like L1.1 and L1.2.
Slides on the Jules Verne problem [reading: 2.3].
Earth to the moon slides 2008 (pdf)
Problems discussed: 2.3-10 and 2.3-20.
Problem notes for 2.3-10. 2.3-20 including sample maple code:
Chapter 2, 2.3-10,20,22 notes S2007
Reading assignment: proofs of 2.3 theorems in the textbook and derivation of
details for the rise and fall equations with air resistance.
Lectures begin for 2.4, 2.5, 2.6 topics on numerical solutions.
Numerical DE slides 2008 (14 slides pdf)
Introduction to
numerical solutions of quadrature problems y'=F(x), y(x0)=y0.
The examples used in maple labs 3, 4 are
y'=-2xy, y(0)=2, y=2exp(-x^2) and y'=(1/2)(y-1)^2, y(0)=2,
y=(x-4)/(x-2). Web notes (item 2 in the references below) contain the
examples y'=3x^2-1, y(0)=2, y=x^3-x+2 and y'=exp(x^2), y(0)=2 with
solution y=int(F,0..x)+y0, F(x)=exp(x^2).
Intro to Rect, Trap, Simp rules
from calculus and Euler, Heun, RK4 rules
from this course.
References for numerical methods:
Numerical DE slides 2008 (14 slides pdf)
Numerical Solution of First Order DE (typeset, 19 pages, 220k pdf)
Sample Report for 2.4-3 (pdf 3 pages, 350k)
ch2 Numerical Methods Slides, Exercises 2.4-5,2.5-5,2.6-5 plus Rect, Trap, Simp rules (5 pages, pdf)
The work for 2.4, 2.5, 2.6 is in maple lab 3 and maple lab 4. Details for lab 3:
Maple Lab 3, Numerical Solutions
Maple Lab 3 S2008 (pdf)
Maple L3 snips S2008 (maple text)
Maple Worksheet [.mws] files: In Mozilla firefox, save to disk using
right-mouseclick and then "Save link as...".
Some browsers require SHIFT and then mouse-click. Open the saved
file in xmaple or maple.
Maple L3 snips worksheet Spring 2008 (maple .mws)
Numerical DE coding hints (txt)
The actual symbolic solution derivation and answer check are
submitted as L3.1. Confused? Follow the details in the next link.
Sample symbolic solution report for 2.4-3 (pdf 1 page, 120k)
Do not submit any work for L3.1 with decimals! Only 1.4 methods are
to appear.
The numerical work using Euler, Heun, RK4 appears in L3.2, L3.3, L3.4.
Confused about what to put in your L3.2 report? Do the same
as what appears in the sample report for 2.4-3 (below).
Sample Report for 2.4-3. Includes symbolic solution report. (pdf 3 pages, 350k)
Download all .mws maple worksheets to disk, then run in maple.
Sample maple code for Euler, Heun, RK4 (maple .mws)
Sample maple code for exact/error reporting (maple .mws)
Additional reference, probably not needed:
Report details on 2.4,2.5,2.6 prob 6 (pdf)
Week 3, Jan 23,25: Sections 1.5, 2.1, 2.2.
22 Jan: Meucci and Zhang: Discussion 1.4 exercises. Questions
about maple lab 1. Problem 2 midterm review.
23 Jan: Collect in class Page 41, 1.4: 6, 10. Next time Page 41, 1.4: 18, 22, 26
Some solutions for 1.4-6,12,18,22,26.
Linear integrating factor method 1.5. Application to y'+2y=1. Testing
linear DE y'=f(x,y) by f_y independent of y. Examples of linear equations
and non-linear equations. Integrating factor Lemma. Main theorem on linear DE
and explicit general solution.
Three linear examples: y'+(1/x)y=1, y'+y=x, y'+2y=1.
On 1.5-34: The units should be taken as millions of cubic feet.
The textbook gives the initial value problem x'=r_i c_i - (r_0/v)x,
x(0)=x_0. The initial value is x_0 =
(0.25/100)8000, the output rate is r_0=500, and the tank volume is V=8000.
Please determine the value for the
input concentration, constant c_i.
You should obtain r_ic_i=1/4.
Then solve the initial value problem.
The book's answer t = 16 ln 4 = 22.2 days is correct.
Due next, Page 54, 1.5: 8, 18
References for linear DE:
Linear integrating factor method, Section 1.5, slides (pdf)
Linear DE method, Section 1.5 slides: 1.5-3,5,11,33+Brine mixing (9 pages, pdf)
Linear DE part I (Integrating Factor Method), (typeset, 8 pages, pdf)
Linear DE part II (Variation of Parameters, Undetermined Coefficients), (typeset, 7 pages, pdf)
How to do a maple answer check for y'=y+2x (TEXT 1k)
Linear first order slides, integrating factor method (2 pages, pdf)
Reference slides for separable DE.
Separable Equations 2007 slides, separability test, tests I and II (6 pages, pdf)
Separable Equations 2008 slides, separability test, tests I and II (9 pages, pdf)
Separable Equations manuscript, classification (pdf)
25 Jan:
Collected in class, Page 41, 1.4: 18, 22, 26
Section 1.5. General solution of the homogeneous equation. Superposition
principle. See FAQ and slides for exercises 1.5-3, 1.5-5, 1.5-11, 1.5-33.
Some more class discussion of 1.5-34.
Introduction to 2.1, 2.2 topics: autonomous DE, partial fraction methods,
Newton's laws.
References for 2.1, 2.2, 2.3:
Autonomous DE slides 2008 (pdf)
Newton models, projectile slides 2008 (pdf)
Earth to the moon slides 2008 (pdf)
Verhulst logistic equation (typeset, 5 pages, pdf)
Phase Line and Bifurcation Diagrams (includes "Stability, Funnel, Spout, and bifurcation") (typeset, 6 pages, 161k pdf)
ch2 sections 1,2,3 Slides: 2.1-6,16,38, 2.2-4,10, 2.3-9,27+Escape velocity (8 pages, pdf)
ch2 DEplot maple example 1 for exercises 2.2, 2.3 (1 page, 1k)
ch2 DEplot maple example 2 for exercises 2.2, 2.3 (1 page, 1k)
Week 2, Jan 14,16,18: Sections 1.4, 1.5, 2.1.
14 Jan: Collected in class, Page 16, 1.2-10.
Theory of equations review, including quadratic equations, Factor and
root theorem, division algorithm, recovery of the quadratic from its
roots.
Classification of y'=f(x,y): quadrature, separable, linear. Venn
diagram of classes. Examples of various types. Implicit and explicit
definitions. Equilibrium solution. Algorithm for solving a separable
equation.
For 1.3-14, a discussion of background material on functions and
continuity
Click here.
For the write-up of 1.3-14 see part (a) of the link
Picard-Lindelof and Peano Existence Example (1 page, pdf).
See also examples in the summary of Peano, Picard and direction Fields [Jan 2008]
Peano, Picard, Direction Fields (slides, pdf)
Start variables separable DE 1.4.
Variables separable method references:
Reference slides for separable DE.
Separable Equations 2007 slides, separability test, tests I and II (6 pages, pdf)
Separable Equations 2008 slides, separability test, tests I and II (9 pages, pdf)
Separable Equations manuscript, classification (pdf)
1.4 Page 40 Exercise slides (4 pages, 500k)
How to do a maple answer check for y'=y+2x (TEXT 1k)
15 Jan
Discussion of 1.3 problem. Distribution of maple lab 1.
Exam 1 review, problem 1.
[Meucci and Zhang]: Today in class: All you need to know about quadratics,
Start Maple Lab 1.
16 Jan: Collected in class, Page 26, 1.3-8,14.
The exercises on Page 41, 1.4: 6, 10 will be due next
week. Theory of separable
equations continued, section 1.4.
Tests for quadrature (f_y=0) and linear (f_y indep of y) types.
Separable equation test.
Examples for Midterm 1 problem 2.
Example 1: Show that y'=x+y is
not separable using the TEST
Separable Equations slides, separability test, tests I and II (6 pages, pdf)
Example 2: Find the factorization f=F(x)G(y) for y'=f(x,y), given
f(x,y)=2xy+4y+3x+6 [ans: F=x+2, G=2y+3].
Basic (but useless) theorem: y(x) = H^(-1)( C1 + int(F)),
H(u)=int(1/G,u0..u).
Separation test: F(x)=f(x,y0)/f(x0,y0),
G(y)=f(x0,y), then FG=f if and only if y'=f(x,y) is separable. Basic
theory discussed.
The solutions y=constant are called equilibrium solutions. Find
them using G(c)=0. Non-equilibrium solutions from y'/G(y)=F(x) and a
quadrature step. Implicit and explicit solutions.
Next time 1.5, theory of linear DE y'=-P(x)y+Q(x). Integrating factor,
fraction for replacement of y'+py.
Started topic of partial fractions, to be applied again in 2.1-2.2.
References:
Reference slides for separable DE.
Separable Equations 2007 slides, separability test, tests I and II (6 pages, pdf)
Separable Equations 2008 slides, separability test, tests I and II (9 pages, pdf)
Separable Equations manuscript, classification (pdf)
1.4 Page 40 Exercise slides (4 pages, 500k)
How to do a maple answer check for y'=y+2x (TEXT 1k)
Heaviside coverup method manuscript, 4 pages pdf
Click here
18 Jan:
Evaluation of integrals by the division algorithm. More on the variables
separable method. Solutions for 1.4-6,10.
Linear integrating factor method 1.5. Application to y'+y=e^x. Testing
linear DE y'=f(x,y) by f_y independent of y. Examples of linear equations
and non-linear equations. Picard's theorem implies a linear DE has a unique solution.
Main theorem on linear DE and explicit general solution.
Due next Page 41, 1.4: 6, 10, 18, 22, 26 and Page 54, 1.5: 8, 18, 20, 34.
References for linear DE:
Linear integrating factor method, Section 1.5, slides (pdf)
Linear DE method, Section 1.5 slides: 1.5-3,5,11,33+Brine mixing (9 pages, pdf)
Linear DE part I (Integrating Factor Method), (typeset, 8 pages, pdf)
Linear DE part II (Variation of Parameters, Undetermined Coefficients), (typeset, 7 pages, pdf)
How to do a maple answer check for y'=y+2x (TEXT 1k)
Linear first order slides, integrating factor method (2 pages, pdf)
Week 1, Jan 7,9,11: Sections 1.1,1.2,1.3.
07 Jan: Three Fundamental Examples
introduced: growth-decay, Newton Cooling, Verhulst population.
See Three Examples (pdf)
Fundamental theorem of calculus. Method of quadrature [integration method
in Edwards-Penney].
Sections 1.1, 1.2. Example for problem 1.2-1, similar to
1.2-2. Details about exams and dailies.
Panels 1 and 2 in the answer check for an initial value
problem like 1.2-2: y'=(x-2)^2, y(2)=1.
Answer checks. Proof that "0=1" and logic errors in presentations.
Maple tutorials start next week. Maple lab 1 is due soon, please
print it from the link
Maple Lab1.
Week 1 references (documents, slides)
Three Examples, Fundamental Theorem of Calculus, Method of quadrature, Decay law derivation, Background formulas. 6 slides, pdf.
Three Examples (pdf)
Three Examples, solved 1.2-1,2,5,8,10 by Tyson Black, Jennifer Lahti, GBG, 11 slides, pdf.
Log+exponential Background+Direction fields PDF Document (4 pages, 450k). Decay law derivation. Problem 1.2-2. Direction field examples.
08 Jan: Intro by Tuesday TA staff. Discuss submitted work format ideas,
examples and problem 1.2-2. Please submit 1.2-2 on Wednesday.
09 Jan:
Collected exercise 1.2-2. Exercises 1.2-4, 1.2-6, 1.2-10 discussed in
class. Slides projected: Tyson Black 1.2-1, Jennifer Lahti 1.2-2,10, Background,
3 Examples, Decay Equation Derivation.
Integration details and how to document them using handwritten
calculations like u-subst, parts, tabular. Maple and Matlab methods.
Integral table methods.
Euler's directional field
visualization, tools for using Euler's idea, reduction of an initial
value problem to infinitely many graphics, showing the behavior of all
solutions, without solving the differential equation.
Lecture on 1.2-8.
Direction field reference:
Direction fields manuscript, 11 pages, pdf.
Threading edge-to-edge solutions
is based upon two rules: (1) Solution curves don't cross, and (2)
Threaded solution curves must match tangents with nearby arrows of the
direction field. See the direction field document above for
explanations.
For problem 1.3-8, xerox at 200 percent the textbook page and paste the
figure. Draw threaded curves on this figure according to the rules in the
direction field document above.
11 Jan: Continue lecture on direction fields and existence-uniqueness 1.3. Discuss 1.3-8.
Collected in class Page 16, 1.2: 4, 6.
Picard-Lindelof Theorem, Peano Theorem, example
y'=(y-2)^(2/3), y(0)=2, like 1.3-14.
The Picard-Lindelof theorem and the Peano theorem are
found in this slide set:
Peano and Picard Theory (3 pages, pdf).
For problem 1.3-14, see
Picard-Lindelof and Peano Existence theory manuscript, 9 pages, pdf
Peano and Picard Theory, 3 slides, pdf
Picard-Lindelof and Peano Existence Example, similar to 1.3-14, 1 slide, pdf
Summary of Peano, Picard and direction Fields [Jan 2008]
Peano, Picard, Direction Fields (slides, pdf)