Sunday, September 21, 2014

lecture 9, vector functions and space curves

Section 13.1 is a difficult read, but try to get a sense for what it covers and for the questions you'll be asked.

We've seen lines and curves in space that lie at the intersection of two surfaces, and we know how to write parametric equations for lines in space. Now, we'll parametrize a variety of space curves by starting with the Cartesian equations that describe those curves. We'll also create parametrizations out of thin air and then try to decide which surfaces the space curves lie on. 

Saturday, September 20, 2014

exam one draws near

Exam one is scheduled for Thursday, September 25 from 5:15 to 7 pm. The exam covers only chapter 12.

We will provide you with an equation sheet. You can find the equation sheet on the last page of each mock exam. Some facts are not on the equation sheet. You need to know how to measure distance between points in $\mathbb{R}^3$, the sphere equation, and the definitions of the dot and cross products.

Your exam room depends on your discussion section. You will share the room with Calculus II students. Be sure to leave an empty seat, or a Calc II student, between you and the next Calc III student.

  • Section 20, Business Auditorium
  • Section 21, CR 129
  • Section 22, CR 133
  • Section 23, Business Auditorium
  • Section 24, CR 129
  • Section 25, CR 133

  • To practice for the exam, use your WebAssign, discussion, and mock exam problems. Solutions are available for these problems. If you don't understand something ask questions at your discussion section and during our office hours. Spaced practice and self-examination may produce better results than cramming. 

    Friday, September 19, 2014

    uh oh!

    I forgot about the A&S honors convocation. My office hours end at 3:50 this afternoon.

    lecture 8, cylinders and quadric surfaces

    Read through section 12.6 before class and look at a few problems at the end of the section.

    As you look at the images of the surfaces, pay attention to the curves and lines that are drawn on the surfaces. These space curves are called traces, and they represent intersections between planes and the surfaces themselves. The traces help our brains interpret the images as curvy 2-dim objects living in $\mathbb{R}^3$.

    Our main objective is to describe the traces with Cartesian equations and then interpret them as lines, circles, parabolas or hyperbolas. This is how we match a particular quadric equation with it's graph. This is how we visualize surfaces.

    Thursday, September 18, 2014

    mock exams updated

    I just finished expanding the answers to the mock exams into full solutions. Have at them.

    solutions to week 3 discussion problems

    Here are my solutions to the discussion problems. Let me know if you find mistakes or solutions that are unclear.

    Wednesday, September 17, 2014

    pressed for time?

    Our schedule is a bit tight due to the semester starting on a Wednesday. If you need more time with assignments 3 and 4 hit the automatic extension button after the due dates have passed. And, if you need it, ask me, Curtis, or Andrew for an additional extension. We want you to understand this material. 

    Tuesday, September 16, 2014

    lecture 7, lines and planes

    The problems in section 12.5 can be daunting. One way to understand them is to build models that illustrate the object you are solving for, and the objects you have been provided with. The model should be specific about the objects required to solve the problem. For instance, to define a line you need a point and a tangent vector. Also, if the problem provides you with lines and planes, deconstruct those objects. You can always extract a normal vector and point from a plane equation. If you are given points connect them with vectors. We'll practice these skills.

    discussion, week 3

    This week's quiz problems involve simple dot and cross products. The discussion problems investigate some differences between real number algebra and vector algebra, the projection formula, and lines and planes in space. Consider these problems to be practice exam problems.

    Monday, September 15, 2014

    lecture 6, lines and planes, in space!

    Apologies for being so slow on this, but please scan section 12.5 before class.

    We'll start by computing the upward flux of an electric field through the parallelogram we played with on Friday. The computation involves a scalar triple product, or a cross product followed by a dot product. You'll see flux integrals in chapter 16.

    We'll spend most of the hour considering how to define lines and planes in space. We use tangent vectors to define lines and normal vectors to define planes. I'll save most of the examples for Wednesday and your discussion section.

    Thursday, September 11, 2014

    lecture 5, the cross product

    Skim section 12.4 before class. Do the usual thing. Look at the examples and figures, and a few problems from the end of the section.

    We'll review the dot product and, maybe, do another projection example. The cross product is another useful way to multiply two vectors together. The cross product creates a new vector that is perpendicular to the original vectors. The magnitude of the cross product depends on the sine of the angle between the original vectors. The cross product, has a geometric definition and a component definition.

    We'll use the cross product to calculate the area of a parallelogram.

    solutions to week 2 discussion problems

    Here are solutions to the discussion problems. Problems 1 through 5 would be fair game for an exam.

    Tuesday, September 9, 2014

    lecture 4, vectors and the dot product

    Please flip through section 12.3 before class. Look at the figures, examples, a few problems.

    We'll talk about a few more problems from 12.2 and then look at the dot product. We use the dot product to project one vector onto another, and to measure the angle between two vectors. 

    Monday, September 8, 2014

    lecture 3, vector arithmetic

    Vectors are useful objects that have two properties, magnitude (length, norm) and direction. In this course we'll represent position, velocity, acceleration and force as vectors.

    Today we'll define two operations, vector addition and multiplication by scalars, that produce new vectors (12.2). Then we'll work on representing vectors by components.

    discussion, week 2

    Discussion sections meet this week. The quiz consists of three true-false questions about 12.1. The discussion problems involve topics from 12.1 and 12.2. In preparation for discussion, watch this MIT video on finding the centroid of a triangle.

    Wednesday, September 3, 2014

    lecture 2, review and vectors

    Before class, try to get a sense for what 12.2 covers and for the questions you'll be asked.

    We'll start by reviewing what we did Wednesday and then finish 12.1 by talking about how to measure distance between 2 points and about the equation for a sphere. Then we'll dig into vectors and vector arithmetic.

    Tuesday, September 2, 2014

    lecture 1, objects of our desire

    Before class, skim section 12.1 in the book. Look at the figures and examples and read four or five problems at the end of the section. Try to get a sense for what 12.1 is about.

    We'll talk about some common objects (points, curves and surfaces) and the ambient spaces ($\mathbb{R}$, $\mathbb{R}^2$, and $\mathbb{R}^3$) they live in. Our goal is to start writing equations for these objects. We'll also talk about how to measure distances between points in the various spaces

    Sunday, August 31, 2014

    syllabus and course schedule

    The syllabus for fall 2014 is mainly useful because it contains my contact info. My blog posts for August (scroll down) provide more details about topics covered in the syllabus. Some elements of the fall 2014 course schedule (updated 9/12) may change, but the exam schedule is set in stone. Notice that the schedule contains reading and homework assignments.

    Saturday, August 30, 2014

    need help?

    The difficulty level of this course is somewhere between hard and trivial. Anyone who understands single variable calculus can learn multivariable calculus, but it may take considerable effort. Asking for help is not a sign of weakness.

    One of the smartest things you can do is collaborate with other students on the homework. You will learn more if you collaborate, and it will take less time compared to working alone. But, and this is important, always write up detailed solutions to the homework problems on your own. Writing up your own solutions is the single best way to test your understanding of the material and to embed the material in your brain.

    Ask questions if you don’t understand something, especially during lecture and discussion. Your classmates will be grateful; they have the same question. And, please, come ask questions during our office hours (or make an appointment with me, Andrew or Curtis).

    You can also get help at the Mathematics Assistance Center, with the engineering honor society Tau Beta Pi, and at the new STEP Tutoring Center at Coe Library.

    Thursday, August 28, 2014

    how to learn, how not to learn

    There are a bunch of books out recently directed at those of us who forget what we've read as soon as we turn the page. All these books have similar prescriptions and are backed, to some extent, by observations of actual humans. This particular list of ten things to do and ten things not to do is by an engineering professor who was once a mathphobe. If you are curious, she teaches a MOOC on how to learn things.

    It's easy to fool yourself into thinking you understand the course material. Test yourself by working random homework problems. Don't just look at current problems, look at problems from past chapters also. Can you work these problems correctly without referring to notes or books? It takes time to understand the material at this level; you won't get there by doing homework problems at the last minute or by cramming before an exam.