Monday, September 29, 2014

discussion, week 5

No quiz this week; next week's quiz will cover level curves and partial differentiation. 

Exams will be returned in discussion section. If you are in a Thursday section you can pick up your exam at my office on Tuesday or Wednesday afternoon. Solutions to the exam are available. Talk to Curtis or Andrew if you don't understand the solutions.

This week's problems are a mishmash of topics: parametrizing circles and ellipses in the plane; showing that a space curve lies on a particular surface; differentiating vector functions; plotting contour or level maps for simple functions; interpreting slopes from contour or level maps.

Sunday, September 28, 2014

lecture 12, level curves and partial derivatives

We skip section 14.2, so go ahead and read about partial derivatives in section 14.3.

I'll start by reviewing level curves. Level curves illustrate the shape of a function in the same way contour maps illustrate the topography of a region. This time we'll construct level curves for a function that is linear in $x$ and $y$. The graph of such a function is a plane. We'll see that the plane has different slopes in different directions. This observation will take us to partial differentiation.

Friday, September 26, 2014

exam 1 solutions

The person (or group) finding the most mistakes in my solutions to exam 1 will win a valuable prize!

lecture 11, vector calculus and functions of several variables

This is a late posting, but please skim through section 14.1 over the weekend.

I'll finish the computation of $\vec{r}(t)$, $\vec{v}(t)$, and $\vec{a}(t)$ for the space curve that marks the intersection of the spheres (section 13.2). Then we'll solve an initial value problem where integrate our way from an acceleration vector back to a position vector.

I'll take a brief look at section 14.1. We'll create some functions of several variables and then figure out how to describe their domains. We'll also sketch level curves for those functions. Our level curves are traces of the surface drawn in the $xy$-plane. Level curves are called contour curves by most people.

Thursday, September 25, 2014

exam one is over!

Exam one starts this afternoon at 5:15 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.

    Calculators and computers are not allowed at the exam.

    Tuesday, September 23, 2014

    lecture 10, space curves and calculus

    If you haven't looked at section 13.1 do it today. Take a peek at the first part of 13.2 also (the part about differentiation of vector functions).

    We'll try visualizing some other space curves and then differentiate the vector function $\vec{r}(t)$ to obtain velocity and and acceleration vectors. Finally, a little calculus.

    Monday, September 22, 2014

    discussion, week 4

    There is no quiz this week due to the Thursday exam. The discussion problems (revised Tuesday morning) focus on traces. Take this opportunity to discuss any problems you may have encountered while preparing for the exam. Let me know if you find mistakes in these solutions.

    exams and calculators

    I mentioned this in the syllabus, in class, and in the instructions on the mock exams, but let me say it again. Calculators are not allowed at the exams.

    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. 

    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 its 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.