**Instructor**: Tim Chumley

**Office**: Clapp 423

**Phone**: 413-538-2525

**e-mail**: tchumley

**Office Hours**: Mondays 4:00-5:00; Wednesdays 4:30-5:30; Thursdays 1:00-2:00; additional availability by appointment

**Textbook**: *APEX Calculus 3* by Gregory Hartman, ISBN: 1719263663;

available as a free pdf or interactive html site.

Announcements will be posted on the Course Announcements Moodle forum throughout the semester, but essentially all other materials will be posted on this page.

Check the syllabus for all the important class policies (grades, attendance, etc.).

There will be weekly homework assignments throughout the semester to be turned in. Please read these guidelines for writing in our class.

**General information**. A selection of problems will be assigned to be written up individually and turned in each week.- These problems will be due
**Fridays at 5 pm**. - You may work with others but the writing should be done on your own.

- These problems will be due
**Gradescope**. Homework will be turned in through Gradescope.- You should be enrolled automatically if you have an account with your MHC email address. Please let me know if you have any issues logging in.
- Gradescope has made a short tutorial on submitting homework.

**Collaboration**. I want you to work together on the homework! The process of explaining your ideas to one another really helps in learning math. However, you must write up the assignments on your own and avoid copying others’ work directly. Also, please only write what you understand so that I know where to help, and please avoid using online forums like Math StackExchange, solutions manuals, or similar resources. A huge part of learning in this class comes from figuring out how to get unstuck by spending time thinking on your own or talking to me and classmates; these other resources can end up being counter-productive in the long term.**Rewrites**. Homework is for practice, and you are not expected to write perfect answers from the start!- You will be allowed to submit revisions of most problems for full credit each week.
- Your revisions will be due on
**Fridays at 5 pm**. - Please resubmit (only the problems you’re revising) on Gradescope.

Assignment | Due |
---|---|

Homework 0 | Jan 27 |

Homework 1 | Feb 3 |

Homework 2 | Feb 10 |

Homework 3 | Feb 17 |

Homework 4 | Mar 3 |

Homework 5 | Mar 10 |

Homework 6 | Mar 24 |

Homework 7 | Apr 7 |

Homework 8 | Apr 14 |

Homework 9 | Apr 21 |

Homework 10 | Apr 28 |

There will be quizzes most weeks that will be given on **Wednesdays**. The purpose of these is to check in to see that you’re comfortable with fundamental material and homework problems. Problems will always be related to the previous homework and class topics.

Quiz | Date | Material |
---|---|---|

Quiz 1 | Feb 1 | Homework 0 |

Quiz 2 | Feb 8 | Homework 1 |

Quiz 3 | Feb 15 | Homework 2 |

Quiz 4 | Feb 22 | Homework 3 |

Quiz 5 | Mar 8 | Homework 4 |

Quiz 6 | Mar 22 | Homework 5 |

Quiz 7 | Mar 29 | Homework 6 |

Quiz 8 | Apr 12 | Homework 7 |

Quiz 9 | Apr 19 | Homework 8 |

Quiz 10 | Apr 26 | Homework 9 |

There will be two midterms and a final exam. The dates for the exams are subject to change slightly.

Exam | Date | Format | Material |
---|---|---|---|

Exam 1 | Feb 24 | in-class | Homework 0-3 |

Exam 2 | Mar 31 | in-class | Homework 4-6 |

Exam 3 | May 4-8 | self-scheduled | Homework 7-10 |

Our plan is to cover parts of chapters 10, 11, 12, 13, and 14 of the textbook. Below is a rough outline of what is to be covered week by week through the semester. Please check back regularly for precise details on what is covered, as well as postings of class materials like lecture notes.

**Topic**: Section 10.1: Introduction to Cartesian Coordinates in Space. We discuss plotting points in three dimensions, the distance formula, and equations of some common surfaces (spheres, planes, and cylinders).**Class materials**: Lecture notes, worksheet**After class**: Work on Homework 0.

**Topic**: Section 10.2: An Introduction to Vectors. We talk about a new mathematical object, the vector in Euclidean space, that encodes geometric notions of magnitude and direction. Our focus is on the interplay between algebra and geometry of vectors, and we discuss vector addition and scalar multiplication.**Class materials**: Lecture notes, worksheet**After class**: Complete today’s worksheet and finish Homework 0.

**Topic**: Section 10.3: The Dot Product. We introduce a new mathematical operation between vectors and discuss its relationship to the angle between vectors. We define the notion of orthogonal projection.**Class materials**: Lecture notes, worksheet**After class**: Read the introduction to Section 10.4 up through the first example.

**Topic**: Section 10.4: The Cross Product. We define another new operation between vectors. Unlike the dot product, the cross product yields a new vector, and we discuss its algebraic and geometric properties.**Class materials**: Lecture notes, worksheet**After class**: Read Section 10.5 up through Example 10.5.4.

**Topic**: Section 10.5: Lines. We discuss how to represent lines in three dimensional space using parametric equations and symmetric equations.**Class materials**: Lecture notes, worksheet**After class**: Finish Homework 1 and do today’s worksheet.

**Topic**: Section 10.6: Planes. We return to planes in three dimensional space, going beyond the coordinate planes, and discuss how to find their equations and their relationship to the dot product.**Class materials**: Lecture notes, worksheet**After class**: Read Section 11.1 up through Example 11.1.2.

**Topic**: Sections 11.1: Vector-valued functions. We discuss functions \(r: \mathbb R \to \mathbb R^n\) where \(n=2, 3\). Such functions are thought of as parametric curves that trace the trajectory of a particle moving in the plane or space.**Class materials**: Lecture notes, worksheet**After class**: Read the Derivatives subsection of Section 11.2 up through Example 11.2.5.

**Topic**: Sections 11.2: Calculus of vector-valued functions. We continue our discussion of vector-valued functions focusing on the idea of derivatives, tangent vectors, tangent lines, and velocity.**Class materials**: Lecture notes, worksheet**After class**: Read Section 12.1 up through Example 12.1.3.

**Topic**: Section 12.1: Introduction to Multivariable Functions. We introduce functions of two variables, the notions of cross-sections, level curves, and contour plots, and we begin our discussion of how to visualize the 3d graphs of some standard examples (the paraboloid, cone, and hyperbolic paraboloid).**Class materials**: Lecture notes, worksheet**After class**: Start on Homework 3. Find level curves of \(z=y^2-x^2\) and plot its graph in CalcPlot3D.

**Topic**: Section 12.3: Partial Derivatives. We introduce the idea of partial derivatives as a way of understanding rates of change of functions along cross section curves.**Class materials**: Lecture notes, worksheet**After class**: Finish today’s worksheet and review the ideas of concavity and second derivatives from Calculus I.

**Topic**: Section 12.3: Partial Derivatives, continued. We introduce second order partial derivatives and their geometric interpretations.**Class materials**: Lecture notes, worksheet**After class**: Read the first two pages of Section 12.6.

**Topic**: Section 12.6: Directional Derivatives. We discuss a generalization of the partial derivative where we consider the change in function values as we vary the inputs in arbitrary directions. We discuss a related notion called the gradient vector and its geometric properties.**Class materials**: Lecture notes, worksheet**After class**: Read the first page of Section 12.7 and Example 12.7.1 stopping at the directional derivative discussion.

**Topic**: Section 12.7: Tangent Planes. We discuss the derivation of the formula for the tangent plane of \(f(x,y)\) at a given point \((x_0,y_0)\).**Class materials**: Lecture notes, worksheet**After class**: Study for Exam 1.

**Topic**: Exam 1.**After class**: Read the first two pages of Section 12.8.

**Topic**: Section 12.8: Extreme Values. We use the gradient to discuss the ideas of local extrema, saddle points, and critical points.**Class materials**: Lecture notes, worksheet**After class**: Read Theorem 12.8.3 and Example 12.8.6.

**Topic**: Section 12.8: Extreme Values. We introduce the notion of constrained optimization and discuss how to use techniques from Calculus I to find global extrema of functions of two variables.**Class materials**: Lecture notes, worksheet**After class**: Read about Lagrange Multipliers in Open Stax Calculus.

**Topic**: Section 12.8: Extreme Values. We discuss the method of Lagrange multipliers for solving constrained optimization problems. For a textbook resource on Lagrange Multipliers see Open Stax Calculus 3 Section 4.8.**Class materials**: Lecture notes, worksheet**After class**: Read the*Area of a plane region*portion of Section 13.1.

**Topic**: Section 13.1: Double Integrals and Area. We begin our discussion of multivariable integration and show how it can be used to find areas of planar regions.**Class materials**: Lecture notes, worksheet**After class**: Read the start of Section 13.2 up through Example 13.2.2.

**Topic**: Section 13.2: Double Integrals and Volume. We continue our discussion of iterated integrals, now with a focus on integrals where the integrand is not a constant, but rather an arbitrary function of \(x\) and \(y\). Such integrals are viewed as producing a signed volume and we will work on examples of changing order of integration and how this how a change of order of integration can make it possible to compute an integral by hand.**Class materials**: Lecture notes, worksheet**After class**: Read Section 9.4 up through Example 9.4.3.

**Topic**: Sections 9.4: Polar coordinates. We discuss the polar coordinate system in the \(xy\)-plane, an alternative to Cartesian \(xy\)-coordinates, which provides a convenient way to describe planar regions that are bounded by curves like arcs of circles.**Class materials**: Lecture notes, worksheet**After class**: Read Section 13.3 up through Example 13.3.1. Enjoy your break!

**Topic**: Spring break, no class.

**Topic**: Spring break, no class.

**Topic**: Spring break, no class.

**Topic**: Section 13.3: Polar Integrals. We work on double integrals in polar coordinates.**Class materials**: Lecture notes, worksheet**After class**: Read Example 13.6.2.

**Topic**: Section 13.6: Triple Integrals. We introduce the concept of a triple integral and how to view it as computing the mass of a solid region in three-dimensional space. We begin to see examples of setting up limits of integration.**Class materials**: Lecture notes, worksheet**After class**: Read Example 13.6.4.

**Topic**: Section 13.6: Triple Integrals. We work on more examples of setting up limits of integration for triple integrals.**Class materials**: Lecture notes, worksheet**After class**: Read Examples 13.7.1 and 13.7.2.

**Topic**: Section 13.7: Cylindrical Integrals. We introduce the cylindrical coordinate system for points \((x,y,z)\) in three dimensional space. The \(x\) and \(y\) coordinates are replaced by \(r\) and \(\theta\) as in polar coordinates and \(z\) is left alone.**Class materials**: Lecture notes, worksheet**After class**: Study old homework problems, worksheets, and lecture examples!

**Topic**: Review.**Class materials**: worksheet, solutions**After class**: Continue your review and study for the exam.

**Topic**: Exam 2.**After class**: Read Example 13.7.5 through Theorem 13.7.2.

**Topic**: Section 13.7: Spherical Integrals. We introduce the spherical coordinate system for points in three dimensional space. Such points are determined by the point’s distance \(\rho\) from the origin, the polar angle \(\theta\) and an angle \(\phi\) with respect to the positive \(z\)-axis.**Class materials**: Lecture notes, worksheet**After class**: Read the first three pages of Section 14.1.

**Topic**: Section 14.1: Line integrals of scalar fields. We introduce the notion of integrating a function \(f: \mathbb R^2 \to \mathbb R\), \(n=2,3\), over a parametrized curve. These integrals, denoted \(\int_C f\,ds\), are called line integrals of scalar fields.**Class materials**: Lecture notes, worksheet**After class**: Read Section 14.2 up through Example 14.2.1.

**Topic**: Section 14.2: Vector fields. We discuss functions of the form \(F: \mathbb R^n \to \mathbb R^n\) where \(n = 2,3\). These are vector-valued functions but their domain is multidimensional. Our focus will be on understanding how to visualize them and on flow lines.**Class materials**: Lecture notes, worksheet**After class**: Read the first two examples of Section 14.3.

**Topic**: Section 14.3: Line integrals of vector fields. We introduce the notion of a line integral of a vector field over a parametrized curve.**Class materials**: Lecture notes, worksheet**After class**: Read Theorem 14.3.2 and the preceding discussion starting at the heading The Fundamental Theorem of Line Integrals.

**Topic**: Section 14.3: Line integrals of vector fields, continued. We discuss conservative vector fields and the Fundamental Theorem of Line Integrals.**Class materials**: Lecture notes, worksheet**After class**: Read about flow in the introduction to Section 14.4, skimming past the discussion of flux for the moment, and then read Theorem 14.4.1.

**Topic**: Section 14.4: Green’s Theorem. We discuss path-dependent line integrals along simple closed curves and how they are related to double integrals.**Class materials**: Lecture notes**After class**: Work on Homework 9.

**Topic**: Section 14.4: Green’s Theorem, continued. We discuss the proof of Green’s Theorem and get practice using it.**Class materials**: Lecture notes, worksheet**After class**: Read Example 14.4.1. Read the supplementary notes about the Curl Test.

**Topic**: Section 14.4: Flux. We discuss another kind of line integral: the flux of a vector field across a curve in the plane.**Class materials**: Lecture notes, worksheet**After class**: Read the last two pages of Section 14.4.

**Topic**: Section 14.4: The Divergence Theorem. We discuss computing flux integrals using the planar version of the Divergence Theorem.**Class materials**: Lecture notes, worksheet**After class**: Work on Homework 10.

**Topic**: Chapter 14 practice. We work on some problems to get more practice with line integrals, the Fundamental Theorem of Line Integrals, and Green’s Theorem.**Class materials**: Lecture notes, worksheet, 3blue1brown video**After class**: Work on Homework 9 revisions.

**Topic**: Review. We spend the day working on review problems for Exam 3 and take our final quiz of the semester.**Class materials**: worksheet, solutions**After class**: Work on Homework 10.

**Topic**: Office hour, no class.**After class**: Study for Exam 3. Enjoy your summer! Keep in touch!

Here are a few ways to get help:

**Office Hours**: Mondays 4:00-5:00; Wednesdays 4:30-5:30; Thursdays 1:00-2:00; additional availability by appointment**TA help**: We will have class TA’s who will be available for help during drop-in sessions. The schedule is as follows:- Sundays, 2:00 - 4:00 pm, in Clapp 401
- Wednesdays, 7:00 - 9:00 pm, in Clapp 422 with Cathy
- Thursdays, 7:00 - 9:00 pm, in Clapp 422 with Phuong

**Study groups**: Other students in the class are a wonderful resource. I want our class to feel like a community of people working together. Please get in touch if you’d like me to help you find study partners, and please reach out if you’d like others to join your group. You may work together on homework, explain and check answers, but make sure you write what*you*know on homework in order to get good feedback.**Piazza**: I’ve set up an online forum for asking and answering questions about class material, including homework problems. My plan is to check the forum every day and answer questions as they come up, but I hope everyone in the class will pitch in and answer others’ questions when possible.

- Wolfram Alpha: a useful way to check your answers on computations. It can do algebra and calculus, among other things, and it understands a mix of English and symbols.
- Desmos: a nice website for graphing functions.
- GeoGebra: a website similar to Desmos, but good for 3d plotting too.
- CalcPlot3D: a more sophisticated plotter with the ability to make contour plots.