May 24th
May 24th
Syllabus for Calculus
Calculus, Part III
| Linear algebra : |
matrices and matrix algebra, systems
of linear equations , eigenvalues and
eigenvectors, orthogonal matrices and diagonalization. |
| Vector calculus: |
vector fields, derivatives, line and
surface integrals, Green’s, Stokes' and
divergence theorems. |
| Ordinary differential equations: |
ordinary differential equations and
systems of ordinary differential
equations; Laplace transform methods; power series solutions, Bessel’s
and Legendre’s equations; applications. |
| |
| Use of symbolic manipulation and
graphics software . |
| Text: |
Zill, Dennis and Cullen, Michael:
Advanced Engineering Mathematics,
Second Edition (Sudbury, MA: Jones & Bartlett Publishers, ©1999) |
| |
| Syllabus: |
Chapter Section & Topic
Vectors, Matrices and Vector Calculus |
Core
Problems |
Maple
Problems |
| 7 |
Vectors 297 |
|
|
| 7.6 |
Vector Spaces 331 |
1, 5, 6, 9, 11, 16, 19, 23, 26, 29 |
|
| |
| 8 |
Matrices 341 |
|
|
| 8.1 |
Matrix Algebra 342 |
9, 10, 13, 15, 17, 21, 27, 29, 33,
37, 45 |
|
| 8.2 |
Systems of Linear Algebraic Equations
351 |
3, 5, 11, 21, 27 21, 43 |
|
| 8.3 |
Rank of a Matrix 363 |
3, 11, 15 |
20 |
| 8.4 |
Determinants 368 |
11, 13, 17, 19, 25, 27, 29 |
|
| 8.5 |
Properties of Determinants 374 |
11, 13, 17, 21, 27, 31, 39, 40 |
|
| 8.6 |
Inverse of a Matrix 381 |
5, 13, 21, 25, 27, 31, 35, 45, 49,
55, 58 |
|
| 8.7 |
Cramer' s Rule 391 |
1, 5, 9, 11, 13 |
|
| 8.8 |
The Eigenvalue Problem 395 |
1, 5, 11, 13, 17, 23 |
19, 27 |
| 8.9 |
Powers of Matrices 400 (Optional) |
1, 5, 7, 11, 15 |
|
| 8.10 |
Orthogonal Matrices 404 |
3, 9, 15, 21 |
|
| 8.12 |
Diagonalization 418 |
3, 11, 15, 27, 33, 35,39 |
19, 30 |
| 8.15 |
Method of Least Squares 436 |
1, 3 |
6, 7 |
| |
| 9 |
Vector Calculus 447 |
|
|
| 9.1 |
Vector Functions 448 |
1, 5, 13, 15, 19, 25, 29, 35, 37, 43 |
3, 7, 23 |
| 9.2 |
Motion on a Curve 454 |
5, 9, 11, 17 |
|
| 9.3 |
Curvature and Components of
Acceleration 459 |
1, 3, 13, 23 |
|
| 9.7 |
Divergence and Curl 480 |
11, 15, 21, 27, 35, 39 |
1, 5 |
| 9.8 |
Line Integrals 486 |
3, 5, 13, 17, 23, 27, 35 |
|
| 9.9 |
Line Integrals Independent of the
Path 495 |
3, 9, 13, 17, 23, 27 |
|
| 9.12 |
Green's Theorem 516 |
3, 11, 17, 23, 25, 27 |
|
| 9.13 |
Surface Integrals 521 |
5, 13, 15, 25, 33, 37 |
|
| 9.14 |
Stokes' Theorem 529 |
3, 9, 13, 15 |
|
| 9.16 |
Divergence Theorem 546 |
1, 9, 13, 19 |
|
| 9.17 |
Change of Variables in Multiple
Integrals 552 |
|
|
| |
Chapter Section & Topic
Ordinary Differential Equations* |
Core Problems
|
Maple
Problems
|
| 2 |
First- Order Differential Equations
35 |
|
|
| 2.5 |
Solutions by Substitutions 64 |
5, 13, 17, 25, 29 |
|
| 2.6 |
A Numerical Solution 68 |
1, 3, 5, 9, 13 |
|
| 2.7 |
Linear Models 73 |
3, 11, 17, 23, 32 |
33 |
| |
| 3 |
Higher-Order Differential
Equations 101 |
|
|
| 3.6 |
Cauchy-Euler Equation 136 |
7, 13, 21, 28, 37 |
|
| 3.8 |
Linear Models: Initial-Value Problems
147 |
5, 9, 21, 25, 29, 33 |
43 |
| 3.9 |
Linear Models: Boundary-Value
problems 163 |
11, 21, 25, 31 |
1, 33 |
| |
| 4 |
The Laplace Transform 189 |
|
|
| 4.1 |
Definition of the Laplace Transform
190 |
3, 7, 15, 23, 33, 39, 43 |
|
| 4.2 |
The Inverse Transform; Transforms of
Derivatives 195 |
7, 15, 21, 27, 35, 36, 43 |
|
| 4.3 |
Translation Theorems 204 |
5, 9, 15, 19, 23, 29, 33, 41, 45, 57, |
|
| |
|
60, 61,67, 75 |
|
| 4.4 |
Additional Ope rational Properties 215 |
5, 13, 19, 25, 31, 43 |
20, 51 |
| 4.5 |
Dirac Delta Function 224 |
3, 9, 13 |
|
| 4.6 |
Solving Systems of Linear Equations
227 |
3, 7, 11, 15 |
|
| |
| 5 |
Series Solutions of Linear
Equations 235 |
|
|
| 5.1 |
Solutions about Ordinary Points 236 |
3, 9, 13, 17, 27, 29 |
5, 7, 33 |
| 5.2 |
Solutions about Singular Points 246 |
3, 9, 13, 17, 23 27 |
|
| 5.3 |
Two Special Equations 256 |
3, 11, 21, 35, 38 |
36 |
| |
| 10 |
Systems of Linear Differential
Equations 567 |
|
|
| 10.1 |
Preliminary Theory 568 |
5, 9, 13, 19, 23 |
|
| 10.2 |
Homogeneous Linear Systems 575 |
3, 13, 21, 29, 35, 45 |
15, 17 |
| 10.3 |
Solution by Diagonalization 588 |
5, 9 |
|
| 10.4 |
Nonhomogeneous Linear Systems 590 |
7, 19, 23, 27 |
30 |
*A brief review of elementary methods of solution will be
given (see chapter/section listings below).
OLD EXAM QUESTIONS also form a part of the core.
The core problems indicate the kind of basic problems you
will need to be able to solve by hand. They also provide a guide to
the basic level of difficulty to be expected on the final exam.
Note: All sections of Math 240 have a COMMON FINAL EXAM
The fol lowing chapters and sections contain material
covered in previous courses. Students are encouraged to
review this material as needed.
| Vectors, Matrices and Vector
Calculus |
Ordinary Differential Equations |
| 7 |
Vectors 297 |
1 |
Introduction to Differential Equations 5 |
| 7.1 |
Vectors in 2-Space 298 |
1.1 |
Definitions and Terminology 6 |
| 7.2 |
Vectors in 3-Space 304 |
1.2 |
Initial-Value Problems 15 |
| 7.3 |
The Dot Product 309 |
1.3 |
Differential Equations as Mathematical Models 21 |
| 7.4 |
The Cross Product 317 |
|
| 7.5 |
Lines and Planes in 3-Space 323 |
2 |
First-Order Differential Equations 35 |
| |
2.1 |
Solution Curves Without the Solution 36 |
| 2.2 |
Separable Variables 44 |
| 9 |
Vector Calculus 447 |
2.3 |
Linear Equations 51 |
| 9.4 |
Functions of Several Variables 464 |
2.4 |
Exact Equations 59 |
| 9.5 |
The Directional Derivative 470 |
|
| 9.6 |
Planes and Normal Lines 477 |
3 |
Higher-Order Differential Equations 101 |
| 9.10 |
Review of Double Integrals 502 |
3.1 |
Preliminary Theory: Linear Equations 102 |
| 9.11 |
Double Integrals in Polar Coordinates 511 |
3.2 |
Reduction of Order 114 |
| 9.15 |
Review of Triple Integrals 534 |
3.3 |
Homogeneous Linear Equations with Constant
Coefficients 117 |
| |
|
3.4 |
Undetermined Coefficients 123 |
| |
|
3.5 |
Variation of Parameters 132 |
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