Friday, April 28, 2017

lecture 38: surface integrals (part 1)

Today we'll work on integrating scalar functions over common surfaces (14.6). To save time I'll let you read about the vector and scalar surface area elements on your own. This table contains area elements for some common surfaces:

Thursday, April 27, 2017

exam 3 results

Last Thursday, 151 brave calcunauts took exam 3; two more will take a makeup exam Sunday. The average is 74.8 and the quartile scores are 64, 76, and 85. Looking at the median, 50% of the scores are higher than 76 and 50% are lower. I will upload scores to WyoCourses on Thursday afternoon, after exams are returned to discussion section 25.

For comparison, the median scores in Spring and Fall 2016 were 75.5 and 75.

Wednesday, April 26, 2017

lecture 37: green's theorem

After wrapping up some details from the fundamental theorem of calculus for line integrals (14.3), we'll look at another version of the fundamental theorem (called Green's Theorem, section 14.4) that connects double integrals to line integrals over closed curves in $\mathbb{R}^2$. 

Monday, April 24, 2017

lecture 36: conservative vector functions

On Friday we used the fundamental theorem from calculus I to construct a fundamental theorem for line integrals involving $\vec{F} \cdot d\vec{r}$ (14.3). The fundamental theorem only works if the vector function is conservative: $\vec{F} = \vec{\nabla}f$. We'll create a test to tell whether a generic vector function is conservative, and we'll figure out how to construct the potential function $f(x,y,z)$ for a conservative vector function.

Friday, April 21, 2017

lecture 35: line integrals and the fundamental theorem

I'll review the line integrals from last week and talk about what happens when we reverse the direction of integration. Then, we'll look at alternate ways to express work and flux integrals (14.2) and compute some work integrals using conservative force fields (14.3).

Thursday, April 20, 2017

exam 3 solutions

Here are my solutions to exam 3. Please let me know if you find any mistakes.

exam three is over!

Exam three is totally over. The exam covers sections 13.1 through 13.7No electronic devices are allowed at the exam.

We will provide you with this equation sheet. Be careful, some facts are not on the equation sheet. You need to know how to find limits of integration and how to integrate. You need to know how to calculate areas using double integrals and how to calculate volumes using double or triple integrals. You should also know the cosine and sine of common angles like $0$, $\pi/6$, $\pi/4$, $\pi/3$, $\pi/2$, $\pi$, and $2\pi$ radians.
Your exam room is a function of the first three letters of your last name.

  • Aaa through Iri, go to CR 302
  • Jag through Rac, go to CR 306
  • Rai through Zzz, go to CR 310

  • Once again, we are sharing the rooms with Calculus II (or I) students. Don't sit next to another Calculus III student.

    To practice for the exam, use the problems from MyMathLab, your discussion section, and the mock exams (problems from 13.7 appear on mock exams 4a, 4b, and 3c) and examples from the text. If you don't understand something ask questions at your discussion section and during our office hours.

    How to prepare for this exam?
    ... I didn’t just do a math homework problem and turn it in. Instead, particularly if it was an important homework problem, I would work it and rework it fresh, spacing the practice out over several days. I wouldn’t peek at the answer unless I absolutely had to. That ensured I really could solve the problem myself—that I wasn’t just fooling myself that I knew it. After I was comfortable that I could really solve the problem by myself on paper, I then “went mental,” practicing the steps in my mind until the solution could flow like a sort of mental song. I could perform this kind of mental practice at times people often don’t think to use for studying—like in the shower, or when I was walking to class. I found that this attention to chunking eventually gave me sort of magic powers—I could glance at many problems, even ones I’d never seen before, and know virtually instantly how to solve them. ... Sometimes people think they suffer from test anxiety when they perform poorly on [a] test, but surprisingly often, they don’t. They’re simply experiencing panic as they suddenly realize they don’t know the material as well as they thought they did. They haven’t created neural chunks.

    Wednesday, April 19, 2017

    lecture 34: line integrals (work and flux)

    We will continue working with line integrals (14.2), replacing $f(x,y,z)$ with components of a vector function $\vec{F}(x,y,z)$. One application is to compute the work done by $\vec{F}$ on a mass that moves along a curve: $$\textrm{work = }\int \vec{F} \cdot \hat{T} \, ds $$ where $\hat{T}$ is a unit tangent vector to the curve. Another application is to compute the flux of $\vec{F}$ across a curve in $\mathbb{R}^2$: $$\textrm{flux = }\int \vec{F} \cdot \hat{n} \, ds $$ where $\hat{n}$ is a unit normal vector to the curve. To evaluate these line integrals we'll use the parametric equations for the curve to make a change of variables (like last time).

    Monday, April 17, 2017

    lecture 33: line integrals

    I'll talk about how to integrate a scalar function $f(x,y,z)$ along a curve in $\mathbb{R}^3$ using the scalar arc length element $ds=| \, d\vec{r} \, |$. The trick (14.2) is to make a change of variables using the parametric equations of the curve: $$\int f(x,y,z) \, ds = \int f(t) \, | \, \vec{r}^{\, \prime}(t) \, | \, dt. $$ This type of integral can be used to compute physical properties of wires (think length, mass, and center of mass).

    Friday, April 14, 2017

    lecture 32: change of variables

    We'll wrap up any undone triple integrals and then rewrite several double integrals with complicated regions of integration by applying either a forward or inverse substitution (13.7).

    Wednesday, April 12, 2017

    lecture 31: spherical coordinates

    I'll remind you of the details of the spherical coordinate system and finish working examples (13.5). We may even talk about general changes of variables (13.7).

    Monday, April 10, 2017

    lecture 30: cylindrical and spherical coordinates

    We'll work another example (13.5) using cylindrical coordinates and then look at the spherical coordinate system. The spherical coordinate system is most useful when one or more of the boundary surfaces is a sphere centered at the origin.

    Friday, April 7, 2017

    mass integrals

    lecture 29: cylindrical coordinates

    We'll turn our attention to regions of integration that have cylindrical symmetry (13.5), symmetry about one axis. We'll use a mash up of polar coordinates and the $z$-axis (cylindrical coordinates) to make quick work of these cases.

    Wednesday, April 5, 2017

    wednesday office hour delayed

    I have to attend a practice talk at 3 today. I hope to start my office hour at 4 pm.

    lecture 28: triple integrals

    I'll be working examples from section 13.4. You should be aware of this interesting pattern:
    1. $\int_I 1 \, dx = length(I)$ where $I$ is an interval on the $x$-axis.
    2. $\iint_D 1 \, dA = area(D)$ where $D$ is a region in $\mathbb{R}^2$.
    3. $\iiint_E 1 \, dV = volume(E)$ where $E$ is a solid region in $\mathbb{R}^3$.
    If there is time we may look at the cylindrical coordinate system (13.5).

    Monday, April 3, 2017

    monday office hours

    I have a meeting at 3, so my office hours are delayed until at least 3:30.

    lecture 27: triple integrals

    We'll wrap up the polar coordinate examples (13.3) and then mosey along to section 13.4 to look at triple integrals. Today we'll focus on using triple integrals (in Cartesian coordinates) to compute volumes.