# math 2210

what one fool can do, another can

## Wednesday, May 17, 2017

### status report

I will be uploading an adjusted midterm exam average for you in a little bit. Then, Andrew, Bang and I will make a decision about where to place grade boundaries. If your score is just below one of the current cutoffs you'll have to wait to see where your grade ends up.

## Monday, May 15, 2017

### exam 4 results (updated)

Last week, ~~149~~ 150 brave calcunauts took exam 4. The mean is 66.9 and the quartiles are 56, 67.5, and 80.25. This was a difficult exam for many people but, overall, results are comparable to those from recent semesters:

Fall 2016: mean 70.3, quartiles 61, 74, and 84.

Spring 2016: mean 63.8, quartiles 54, 65, 76.

## Saturday, May 13, 2017

## Friday, May 12, 2017

## Thursday, May 11, 2017

### exam 4 is history!

Exam 4 started at 3:30 pm, Thursday, May 11. The exam covers sections 14.1-14.8. Calculators and phones are not allowed at the exam. We will provide you with this equation sheet. Confused by the integral theorems? Here is an incomplete guide to using the theorems to compute line and surface integrals.

The equation sheet does not contain the transformation equations for cylindrical/polar coordinates, but you should know them by now, along with the formulas for $dA$ and $dV$.

Practice integration and partial differentiation. Know how to change variables in double and triple integrals. Know how to compute dot and cross products. Know how to compute the divergence and curl of a vector function. Know how to parametrize circles and lines. Know the cosine and sine of common angles like $0$, $\pi/6$, $\pi/4$, $\pi/3$, $\pi/2$, $\pi$, and $2\pi$ radians.

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

To practice for the exam, use your MyMathLab, discussion, and mock exam problems. Solutions are available for these problems. If you don't understand something ask questions in class, at your discussion section, or during our office hours.

How to prepare for this exam?

The equation sheet does not contain the transformation equations for cylindrical/polar coordinates, but you should know them by now, along with the formulas for $dA$ and $dV$.

Practice integration and partial differentiation. Know how to change variables in double and triple integrals. Know how to compute dot and cross products. Know how to compute the divergence and curl of a vector function. Know how to parametrize circles and lines. 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**.
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 your MyMathLab, discussion, and mock exam problems. Solutions are available for these problems. If you don't understand something ask questions in class, at your discussion section, or 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.

## Friday, May 5, 2017

### weekend challenge

We know that curl $\vec{F}$ is $$\vec{\nabla} \times \vec{F} = \langle \frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\rangle.$$ Is it true that $$ \textrm{div}(\textrm{curl} \, \vec{F}) = \vec{\nabla} \cdot (\vec{\nabla} \times \vec{F})= 0?$$ Assume that the scalar functions $P$, $Q$, and $R$ are so smooth that their mixed second partial derivatives are equal, like $$\frac{\partial^2 P}{\partial x \partial y}=\frac{\partial^2 P}{\partial y \partial x}.$$

### lecture 41: divergence theorem

I'll give a quick review of Stokes' theorem and then move on to our final extension of the fundamental theorem, the divergence theorem (14.8). You can find an explanation for why the divergence theorem involves $\vec{\nabla} \cdot \vec{F}$ in my notes.

You shouldn't leave the course thinking these theorems are only useful for avoiding a difficult line or flux integral. In my experience they are mainly useful for turning conservation laws into partial differential equations.

You shouldn't leave the course thinking these theorems are only useful for avoiding a difficult line or flux integral. In my experience they are mainly useful for turning conservation laws into partial differential equations.

### calculate your total score

A high score on the fourth exam can markedly improve your course grade. When we calculate your total score the lowest of your first three exam scores will be replaced by your fourth exam score (unless your fourth exam score is lower than all your scores on the first three exams).

To see how this works, transfer your three exam scores, average discussion score, and average MyMathLab score to this Excel file (hit the download button at the top of the page). Then enter your best guess for exam four.

To see how this works, transfer your three exam scores, average discussion score, and average MyMathLab score to this Excel file (hit the download button at the top of the page). Then enter your best guess for exam four.

**If the DropBox link is too annoying, you can download the Excel file from WyoCourses. Look for Files/WhatsMyGrade.xlsx**.## Wednesday, May 3, 2017

### closed and open surfaces

From a Wikipedia article on surfaces. Closed surfaces (left) have no boundaries. Open surfaces (right) have closed boundary curves (in red).

### lecture 40: stokes' theorem

Today we'll look at Stokes' theorem (14.7). Stokes is another extension of the fundamental theorem. It relates the flux across a surface to a line integral around the boundary of the surface. My notes give some details on how the integrals are related. We'll verify Stokes' theorem by computing a surface integral and a line integral. Cool beans!

## Monday, May 1, 2017

### lecture 39: surface integrals (part 2)

After finishing any leftovers from Friday, we'll calculate the flux of a vector function $\vec{F}$ across various surfaces (14.6).

In this drawing, the flux of $\vec{F}$ across the oriented surface element $d\vec{S}$ is represented by the volume of the gray boxes. You can see that the flux is largest when $\vec{F}$ and $d\vec{S}$ are parallel, but decreases as the angle between the vectors increases. In detail, the differential of flux is equal to the dot product between $\vec{F}$ and $d\vec{S}$ and the flux through an entire surface is $$\iint \vec{F} \cdot d\vec{S} = \iint \vec{F} \cdot \hat{n} \, dS.$$

In this drawing, the flux of $\vec{F}$ across the oriented surface element $d\vec{S}$ is represented by the volume of the gray boxes. You can see that the flux is largest when $\vec{F}$ and $d\vec{S}$ are parallel, but decreases as the angle between the vectors increases. In detail, the differential of flux is equal to the dot product between $\vec{F}$ and $d\vec{S}$ and the flux through an entire surface is $$\iint \vec{F} \cdot d\vec{S} = \iint \vec{F} \cdot \hat{n} \, dS.$$

## 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.

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 three is over!

Exam three is totally over. The exam covers sections

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.

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

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?

**13.1**through**13.7**.**No 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**.
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).

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