Mathematics 226 homework, Fall 2014

• Homework 1, due on Friday, September 12: PDF. The assignment covers Sections 10.1 and 10.2 of the textbook. You are responsible for all of the material in these sections, except that we will skip "Hanging cables and chains," pp. 574-576.
• Homework 1 Solutions: PDF.
  
  
• Homework 2, due on Wednesday, September 24:
  • Section 10.3: 13, 15, 27
  • Section 10.4: 5, 9, 17, 19
  • Section 10.5: 23
  The assignment covers Sections 10.3, 10.4, and 10.5 of the textbook. You are responsible for all of the material in these sections, except that we are skipping the examples with distances (pp. 592-593). Determinants (pp. 582-583) will be covered in more detail in the linear algebra co-requisite class. There will be no homework covering Section 10.6 at this time, but we will use cylindrical and spherical coordinates throughout the course, so you should do some of the exercises from this section for your own practice. Please note that the textbook provides the answers to the odd-numbered questions, so that you can check whether you are doing it correctly. By the same token, no credit will be given for the correct answer alone. You must demonstrate the reasoning that gets you this answer.
• Homework 2 Solutions: PDF.
  
  
• Homework 3, due on Wednesday, October 15:
  • Section 12.2: 19
  • Section 12.3: 23, 28
  • Section 12.4: 11, 16, 17
  • Section 12.5: 17, 19, 23
  The assignment covers Sections 12.2-12.5 of the textbook. You are responsible for all of the material in these sections, except we have skipped "Distance from a point to a surface", pp. 688-689 (we will study this type of problems more systematically later on), and "Homogeneous functions", pp. 700-701. You do not have to memorize the different types of partial differential equations in Section 12.4 or its exercises.
• Homework 3 Solutions: PDF.
  
  
• Homework 4, due on Friday, October 24:
  • Section 12.6: 7, 11, 19
  • Section 12.7: 13, 17, 27
  • Section 12.8: 5, 15
  The assignment covers Sections 12.6-12.8 of the textbook. You are responsible for all of the material in these sections, except as follows:
  • Section 12.6: We have skipped "Differential and Legendre Transformations," pp. 713-714. "Functions from n-space to m-space", pp. 709-711, were mentioned only briefly. You will need to know the definition of the Jacobian matrix (top of page 710) and the approximation formula dy = Df(x)dx from page 711.
  • Section 12.7: We have skipped "Rates Perceived by a Moving Observer," pp. 722-723.
  • Section 12.8: Theorem 8 is stated in the most general form possible, so that the same statement covers both of the main applications: the cases n=1 (function defined implicitly by an equation) and n=m (change of coordinate systems). In this class, we will focus on these two cases. No proof of Theorem 8 is given in the course textbook, but if you are interested, you can find it in more advanced textbooks, e.g. "Advanced Calculus" by G. Folland or "Calculus on Manifolds" by M. Spivak.
    "Choosing Dependent and Independent Variables," pp. 730-731, refers to applications to physics. We are skipping it here, but if you studied this material in your physics courses, you may want to read this, as it may clarify the mathematical interpretation of the formulas from physics.
• Homework 4 Solutions: PDF.
  
  
• Homework 5, due on Wednesday, November 12:
  • Section 13.1: 25, 27
  • Section 13.2: 11
  • Section 13.3: 7, 15, 25. (Hint for #25: If g(x,y)=g(a,b) defines y=h(x) implicitly near (a,b), what is the derivative of the 1-variable function f(x,h(x))?)
  The assignment covers Sections 13.1-13.3 of the textbook. The questions from Sections 13.1-13.2 can be solved using only the methods from those sections, but if you prefer to use the method of Lagrange multipliers from Section 13.3, feel free to do so. We have skipped "Linear programming", pp. 756-757.
• Homework 5 Solutions: PDF.
  
  
• Homework 6, due on Friday, November 21:
  • Section 13.5: 11
  • Section 14.1: 17, 19
  • Section 14.2: 15 (hint: change the order of integration), 21, 27, 29
  • Section 14.3: 5, 7
  The assignment covers Sections 13.5 and 14.1-14.3 of the textbook. In Section 13.5, we focused on "Linear regression" (pp. 777-779) and skipped "Applications to Integrals" (pp. 779=781).
• Homework 6 Solutions: PDF.
  
  
• Additional practice problems are posted here.

Guidelines for writing and submitting your homework:

• Show all work. Write clearly and legibly, in complete sentences. For full credit, your solutions need to be not only correct but also clearly explained.
• Do all problems in sequence if possible. For each solution, identify clearly the problem number (and the section number, if applicable). Staple your assignment.
• Please note that the textbook provides the answers to the odd-numbered questions, so that you can check whether you are doing it correctly. By the same token, no credit will be given for the correct answer alone. You must demonstrate the reasoning that gets you this answer.
• You may discuss the homework with other students, but the final write-up must be your own.
• If you cannot come to class, ask a friend to hand in your assignment or drop it off at your instructor's office before 12 noon on the due date. Late homework will not be accepted. Answers/solutions will be posted immediately after class.
  
  

  ---

  [Mathematics Department] [University of British Columbia]