**Final Exam**

- The final is on Monday, Dec. 10, from 10:30 am to 12:30 pm.
- The final is cumulative and covers both linear algebra and vector calculus, and there will be at least one problem on the divergence theorem.
Here are some practice problems:
- Linear Algebra Practice Problems and Solutions
- Vector Calculus Practice Problems (Revised to fix a typo in problem 9) and Solutions (Revised to fix calculuation errors in problems 1, 11, and 14.)

- On the final exam, you may use either 4 index cards (with size 4 by 6) OR 1 sheet of paper (8.5 by 11) with notes on both sides. You may also use a basic scientific calculator.
- Office hours:
- Saturday, 3-5pm in Blocker 156
- Sunday, 1-5pm in Blocker 156

**Divergence Theorem Examples:** Here are some examples of the divergence theorem.

**Solutions to Tests 2 and 4**: Solutions to Tests 2 and 4 had not been previously posted. Here they are:

**Recommended Problems on the Divergence Theorem**: Here are the recommended problems on the divergence theorem.

- TAMU: Section 7.3, # 7, 9
- Schaum's Outline: Chapter 6, #17, 18, 52, 53, 55

**Stokes's Theorem Examples:** Here are some examples of Stokes's Theorem.

**Test 4**

- Test 4 was on Friday, November 30.
- Solutions to Test 4
- Practice Problems and Solutions: Here are the practice problems for Test 4. Some revisions: The Practice Problems were revised on Tuesday to make the parametric equations in Problem 6 agree with the picture. The solution to Problem 1 was corrected on Wed.

**Homework**

- Homework 11 and Solutions to Homework 11 (The solution to problem 4(b) was revised on Thursday, Nov. 29, and the solution to problem 3(d) was revised on Wed, Nov. 28.)
- Problems on Surface Integrals: Here are some problems (with solutions) on parameterized surfaces and surface integrals.
- Homework 10 and Solutions to Homework 10
- Solutions to Homework 9
- Solutions to Homework 8
- Online Resource: The Calculus of Functions of Several Variables is a free online multivariable calculus textbook by Dan Sloughter. Section 3.6 and Section 3.7 of the book cover multiple integrals.
*Schaum's Outline of Vector Analysis*has very few problems on multiple integrals. If you are using*Schaum's Outline*, here are some recommended problems from the online textbook listed above (The Calculus of Functions of Several Variables):- Section 3.6: # 4-10, 12-14
- Section 3.7: # 1, 3, 8, 9, 17, 18, 19, 25

- Answers to the above recommended problems: Here are the answers to the recommended problems from Sections 3.6 and 3.7 of the online textbook.

**Online Resource:**
Notes on multivariable Calculus
by Paul Dawkins. These notes
cover most of the material that we have covered. For Test 4, I recommend looking at the following chapters:

- Line Integrals
- Surface Integrals: You can skip the section on the Divergence Theorem.

- Equations of Lines
- Equations of Planes
- Chain Rule
- Multiple Integrals: This is a chapter of the book. You can skip the section on surface integrals.
- Vector Fields

**Test 3**

- Test 3 was on Wednesday, November 7.
- Solutions to Test 3: Here are solutions to Test 3.
- Practice Problems and Solutions: Here are the practice problems for Test 3.

**Test 2 Info**: Test 2 was on Oct. 15. The median was 85 and the average was 84. Here are
solutions to Test 2.
More information on Test 2 (including statistics and the practice test)
can be found near the bottom of this page.

**Test 1 Info**: Test 1 was on Sept. 19. The median was 88 and the average was 85. Here are
Solutions to Test 1. More information on Test 1 (including statistics and the practice test)
can be found near the bottom of this page.

**Tentative Syllabus**: Here is a tentative syllabus for the course.

**Texts:** Any of the following will work as a textbook for the course:

*Linear Algebra and Vector Calculus at Texas A&M*by S. Leon and S. Colley. This book is available at the campus bookstore.- The two books
*Linear Algebra with Applications*, Sixth Edition by Steven J. Leon, and*Vector Calculus, Second Edition*by Susan Jane Colley. *Schaum's Outlines*of*Beginning Linear Algebra*and*Vector Analysis*.

**Instructor:**
Maria Belk (mbelk@math.tamu.edu)

**Time and Room:** MWF 12:40 pm - 1:30 pm in Blocker 156

**Office Hours**

My office hours are:

- Wednesday 3-4 pm
- Thursday 2-4 pm

**Online Resources:**

- The Calculus of Functions of Several Variables is a free online multivariable calculus textbook by Dan Sloughter. Section 3.6 and Section 3.7 of the book cover multiple integrals.
- Notes on multivariable Calculus by Paul Dawkins. These notes cover most of the topics we will be covering from multivariable Calculus.

- Homework 11 and Solutions
- Homework 10 and Solutions
- Homework 9 and Solutions
- Homework 8 and Solutions
- Homework 7 and Solutions
- Homework 6 and Solutions
- Homework 5 and Solutions
- Homework 4 and Solutions
- Homework 3 and Solutions
- Homework 2 and Solutions
- Homework 1 and Solutions

**Other materials:** This is a place for extraneous postings that were in the Announcements, but are no longer relevant:

- Fourier Series Notes (Revised on Oct. 11 to correct typos): Here are some notes on Fourier Series. They should be helpful for working problems 3 and 4 in Homework 7.
- Gram-Schmidt Examples: The Gram-Schmidt example in class on Monday, Oct. 8, was more computationally complicated than I had intended. This contains two examples. The first is the example from class; the second is the example I had intended to do (the only difference between the two problems is a negative sign).
- Examples of Solving Systems of Linear Differential Equations:
The book
*Schaum's Outline*of*Beginning Linear Algebra*does not cover this topic, so these are some example problems and solutions. These problems come from the recommended problems from Section 6.2 of the book*Linear Algebra and Vector Calculus at Texas A&M*. - More example problems on solving differential equations (Revised on Oct. 8 to correct an error in Example 5): These are some notes written by my husband Jim Belk; these notes include some examples of solving systems of differential equations and some examples of computing eigenvalues and eigenvectors when the characteristic polynomial has a multiple root. In Example 4(b) of these notes, the answer is left with complex numbers. We would want to obtain just the real solutions by using the method we used in class.
- Answers to recommended problems from Section 6.3:
The book
*Linear Algebra and Vector Calculus at Texas A&M*does not contain answers to recommended problems from Section 6.3. - In class on Wednesday, Sept. 5, I had intended to work one more example, but we ran out of time. Here is the example from class.

- Linear Algebra Toolkit: This is a great reference! It contains a bunch of modules that explain various linear algebra computations. Additionally, it will perform a lot of linear algebra computations for you. You may use it to compute any row reductions that you need for your homework.
- Here are some useful Wikipedia webpages:
- Row Space, Column Space, and Kernel (matrix): These are good articles on the material we covered on Friday, Sept. 7.
- Euclidean Subspace: This Wikipedia entry explains the material we covered in Week 2.
- System of Linear Equations: This Wikipedia entry is about solving systems of linear equations, the material we covered during the first week of class.

- A free Linear Algebra textbook by Jim Hefferon: This is a free online Linear Algebra textbook. You may find it useful as a secondary resource.

**Linear algebra lectures:**
These are the lectures for an MIT linear algebra course by Gilbert Strang, hosted by Google Video. (You can download the original versions from the MIT OpenCourseWare website, but they require RealPlayer.)

- Lecture 1: Geometry of Linear Equations
- Lecture 2: Elimination with Matrices
- Lecture 3: Multiplication and Inverse Matrices
- Lecture 4: Factorization into A = LU (Note: We are not covering this topic.)
- Lecture 5: Transposes, Permutations, Spaces R^n
- Lecture 6: Column Space and Nullspace
- Lecture 7: Solving Ax = 0: Pivot Variables, Special Solutions
- Lecture 8: Solving Ax = b: Row Reduced Form R
- Lecture 9: Independence, Basis, and Dimension
- Lecture 10: The Four Fundamental Subspaces
- Lecture 18: Properties of Determinants
- Lecture 19: Determinant Formulas and Cofactors
- Lecture 20: Cramer's Rule, Inverse Matrix, and Volume
- Lecture 21:Eigenvalues and Eigenvectors
- Lecture 22: Diagonalization and Powers of
*A*

Test 1 was on Monday, Oct. 15. The median on the test was 85 and the average was 84. Here are more statistics:

Grade | A | B | C or lower |

Score | 90-100 | 80-89 | < 80 |

Number of Students | 9 | 14 | 10 |

Here are the solutions:

Here are the Practice Problems for Test 2:

Test 1 was on Wednesday, Sept. 19. The median on the test was 88 and the average was 85. Here are more statistics:

Grade | A | B | C or lower |

Score | 90-100 | 80-89 | < 80 |

Number of Students | 14 | 11 | 8 |

Here are the solutions:

Here is the Practice Test: