## Thursday, July 20, 2017

## 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).

## 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

## 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:

- $\int_I 1 \, dx = length(I)$ where $I$ is an interval on the $x$-axis.
- $\iint_D 1 \, dA = area(D)$ where $D$ is a region in $\mathbb{R}^2$.
- $\iiint_E 1 \, dV = volume(E)$ 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

### 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.

## Friday, March 31, 2017

### lecture 26: double integrals with polar coordinates

I'll compute the center of mass for a half disk and set up a double integral that gives the polar moment of inertia for a disk that is offset from the origin.

## Wednesday, March 29, 2017

### exam two results (updated)

Last Thursday, 157 brave calcunauts took the second exam. The average is now 77.6 and the quartile scores are 70, 79, and 87.7. This outcome is nearly identical to exam one!

If you are unhappy with your score, talk to your discussion leader, or me. Make an appointment if you can't attend our office hours.

If you are unhappy with your score, talk to your discussion leader, or me. Make an appointment if you can't attend our office hours.

### lecture 25: a trig substitution

We'll use double integrals to calculate some volumes where the region of integration is a sector of a circle. A smart way to evaluate these integrals is to make an inverse substitution using the polar coordinate system (13.3). The Jacobian determinant will make an appearance.

## Monday, March 27, 2017

### lecture 24; double integrals on non-rectangular regions

We'll work on visualizing regions of integration and reversing the order of integration (13.2). We may even talk about rewriting double integrals using trig substitutions (13.3).

## Thursday, March 23, 2017

### exam 2 solutions

- Krishna Sai Chemudupati found a major mistake in my solution to problem 9. The mistake is fixed (4/24 at 9:25 pm).

### effective learning techniques

An engineering professor answers the question "Which is the most effective learning technique you have experienced so far?"

... 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.

... 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.### exam two approaches!

Exam two begins this afternoon (Thursday, March 22) at 5:15 pm. This exam covers section

We will provide you with this equation sheet. Be careful, some facts are not on the equation sheet. You need to know how to compute the dot and cross products, how to integrate and differentiate, and understand the properties of the gradient vector. 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.

This exam is your opportunity to demonstrate to us that you understand the material. Be sure to read each question carefully, and draw sketches where appropriate. We expect complete solutions and correct notation. Be careful with the T/F questions; think, don't react.

Aaaa through Hanc, go to CR 302
Hans through Pont, go to CR 306
Post through Zzzz, go to CR 310

**11.6, and all of chapter 12 (except for 12.3)**.**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 compute the dot and cross products, how to integrate and differentiate, and understand the properties of the gradient vector. 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.

This exam is your opportunity to demonstrate to us that you understand the material. Be sure to read each question carefully, and draw sketches where appropriate. We expect complete solutions and correct notation. Be careful with the T/F questions; think, don't react.

**Your exam room is a function of the first four letters of your last name**.
We are sharing the rooms with Calculus I and II students. Make sure you are not sitting next to another Calculus III student.

To practice for the exam, use the problems from MyMathLab, discussion, and the mock exams and examples from the text. If you don't understand something ask questions at your discussion section and during our office hours.

To practice for the exam, use the problems from MyMathLab, discussion, and the mock exams and examples from the text. If you don't understand something ask questions at your discussion section and during our office hours.

## Wednesday, March 22, 2017

### uncertain office hours for march 22

I have an appointment this afternoon that interferes with my office hours. I'll try to get back as soon as possible.

### lecture 22: volumes by double integrals

We'll use double integrals to calculate the volume that lies above a rectangle (13.1) in the $xy$-plane and beneath a surface $z=f(x,y)$. If there is time, we'll also create a double integral that gives the volume above a triangle (13.2) and below the surface.

## Monday, March 20, 2017

### lecture 21: lagrange multiplier method refresher

I'll work two or three examples where we look for the extreme values of functions on curves or surfaces.

## Friday, March 10, 2017

### lecture 20: lagrange multiplier method

In calc I a function that is continuous on a closed interval is guaranteed to have an absolute maximum and an absolute minimum value on the interval. We'll chat about analogs to closed intervals in $\mathbb{R}^2$ and $\mathbb{R}^3$. And, we'll solve some problems (12.9) using gradient vectors to find the extreme value(s) of various functions on sets of points in both $\mathbb{R}^2$ and $\mathbb{R}^3$. A tiny amount calculus will occur.

## Wednesday, March 8, 2017

### lecture 19: critical points and the second derivatives test

We'll use the gradient vector to identify and classify critical points for a function of two variables: $f(x,y)=x^3+y^3-3xy$. Once we know how the function works we'll use algebra to make sure we located all the critical points, and the second derivatives test to check our interpretations of those critical points (12.8).

## Tuesday, March 7, 2017

### the big picture

In chapter 12 we describe surfaces in $\mathbb{R}^3$ with single Cartesian equations that depended on some combination of $x$, $y$, and $z$. If the equation of the surface is expressed as $g(x,y,z)=0$ (or any constant) then the gradient of the function $g(x,y,z)$ is normal to, or perpendicular to, the surface.

Both tangent and normal vectors are used in the final weeks of the semester, when we integrate the tangential component of some vector field along a curve, or the normal component of another vector field over a surface.

## Monday, March 6, 2017

### lecture 18: tangent planes and differentials

We'll use the gradient vector to attach tangent planes to surfaces that are described by implicit or explicit equations. The tangent plane lies very close to the surface at points near the point of attachment, so the tangent plane equation can be rearranged to give a linear approximation and a total differential. We'll work three examples.

## Friday, March 3, 2017

### lecture 17: properties of the gradient vector

We'll finish the example problem from Wednesday and then look at the properties of the gradient vector (12.6). They are:

- The range of the directional derivative is $-| \, \vec{\nabla}f \,| \le D_{\hat{u}}f \le | \, \vec{\nabla}f \,|$.
- $\vec{\nabla}f$ is the direction in which $f$ increases most rapidly, aka the direction of maximum increase.
- $-\vec{\nabla}f$ is the direction in which $f$ decreases most rapidly, aka the direction of maximum decrease.
- $\vec{\nabla}f$ is perpendicular to level curves of $f(x,y)$ in $\mathbb{R}^2$ or level surfaces of $f(x,y,z)$ in $\mathbb{R}^3$.

The fourth property gives us a spiffy way to create tangent planes to surfaces. And, tangent planes are a gateway to linear approximations.

## Wednesday, March 1, 2017

### lecture 16: chain rule and directional derivatives

I'll work two more chain rule examples. In one case, we'll rid the world of the scourge of implicit differentiation (12.5).

Then we'll find the rate of change of a function in an arbitrary direction in the function's domain. Dot products will appear as will an amazing vector, the gradient vector, that is constructed from the first derivatives of the function (12.6).

Then we'll find the rate of change of a function in an arbitrary direction in the function's domain. Dot products will appear as will an amazing vector, the gradient vector, that is constructed from the first derivatives of the function (12.6).

## Monday, February 27, 2017

### lecture 15: partial derivatives, and the chain rule

We'll talk more about partial derivatives (12.4) today; we'll look at implicit differentiation and we'll compute some second and third derivatives. If there is time, we'll look at the multivariable chain rule (12.5). It's a nice excuse to compute partial derivatives and, maybe, look at a total derivative.

## Friday, February 24, 2017

### lecture 14: partial derivatives

We'll talk more about traces and level curves and compute slopes on surfaces in various directions. And then we'll talk about partial derivatives (12.4).

## Wednesday, February 22, 2017

### lecture 13: functions of two variables

We'll use planes and quadric surfaces to create functions of two variables (12.2). We'll examine the domains of these functions, construct level curve maps, and compute slopes on the surfaces. We may even compute some partial derivatives (12.4).

## Monday, February 20, 2017

### exam one results (updated, again)

Last Thursday, 153 brave calcunauts took the first exam; four others took a make-up exam on March 2. The average is now 77.7 and the quartile scores are 70, 79, and 87. So, slightly more than 50% of you have scores of 79 or higher.

If you are unhappy with your score, talk to your discussion leader, or me. Make an appointment if you can't attend our office hours.

If you are unhappy with your score, talk to your discussion leader, or me. Make an appointment if you can't attend our office hours.

### lecture 12: quadric surfaces

We'll review planes (12.1) by working several examples. Then we'll turn to the more interesting quadric surfaces. We've played with several of these, including spheres, circular cylinders and cones.

As you read through section 12.1 look at the images of the surfaces. Pay attention to the curves drawn on those 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 those traces as lines, circles, parabolas or hyperbolas. This is how we match a particular quadric equation with its graph. This is how we visualize surfaces.

As you read through section 12.1 look at the images of the surfaces. Pay attention to the curves drawn on those 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 those traces as lines, circles, parabolas or hyperbolas. This is how we match a particular quadric equation with its graph. This is how we visualize surfaces.

## Sunday, February 19, 2017

## Friday, February 17, 2017

### lecture 11: planes and surfaces

We'll review the main points about differentiating and integrating vector functions by working a few examples (11.6, 11.7).

But then we have to jump into chapter 12. Before long you will be finding partial derivatives and working with the multivariable version of the chain rule. We'll start by talking about planes (12.1). I'll show how to assemble a plane equation from a point and a normal vector. Some of our examples will involve cross products and lines.

But then we have to jump into chapter 12. Before long you will be finding partial derivatives and working with the multivariable version of the chain rule. We'll start by talking about planes (12.1). I'll show how to assemble a plane equation from a point and a normal vector. Some of our examples will involve cross products and lines.

## Thursday, February 16, 2017

### exam one solutions

Here are my solutions to exam one. Please let me know if you find any sketchy math. We'll try to have the exams graded, and scores uploaded, by Tuesday morning.

### exam one is history

Exam one began at 5:15 pm on Thursday, February 16. The exam covers only sections 11.1-11.5.

We will provide you with this equation sheet. 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 how to compute the dot and cross products. 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.

This exam is your opportunity to demonstrate to us that you understand the material. Be sure to read each question carefully, and draw sketches where appropriate. We expect complete solutions and correct notation. Be careful with the T/F questions; think, don't react.

Aaaa through Hanc, go to CR 302
Hans through Pont, go to CR 306
Post through Zzzz, go to CR 310

**No electronic devices are allowed at the exam**.We will provide you with this equation sheet. 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 how to compute the dot and cross products. 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.

This exam is your opportunity to demonstrate to us that you understand the material. Be sure to read each question carefully, and draw sketches where appropriate. We expect complete solutions and correct notation. Be careful with the T/F questions; think, don't react.

**Your exam room is a function of the first four letters of your last name**.
We are sharing the rooms with Calculus I and II students. Make sure you are not sitting next to another Calculus III student.

To practice for the exam, use the problems from MyMathLab, discussion, and the mock exams and examples from the text. If you don't understand something ask questions at your discussion section and during our office hours.

To practice for the exam, use the problems from MyMathLab, discussion, and the mock exams and examples from the text. If you don't understand something ask questions at your discussion section and during our office hours.

## Tuesday, February 14, 2017

### homework review

To prepare for the exam, rework the homework problems. From the MyLab page, click on MyLab Course Home, and then GradeBook.

## Monday, February 13, 2017

### lecture 10: calculus of vector functions

We'll differentiate and integrate vector functions (11.6 and 11.7) and review a bit by writing parametric equations for tangent lines to space curves.

### week four discussion

You'll be reviewing for exam one this week. Bring your toughest questions. Look on the discussion tab for this week's problems.

## Friday, February 10, 2017

### lecture 9: calculus of vector functions

We'll spend more time talking about lines and curves in $\mathbb{R}^3$ (11.5). Then we will figure out how to differentiate vector functions (11.6).

## Wednesday, February 8, 2017

### lecture 8: lines and curves in $\mathbb{R}^3$

We'll decide if two lines are parallel, intersecting, or skew (not parallel, not intersecting). Then we'll parametrize a curve or two (11.5).

## Monday, February 6, 2017

### lecture 7: lines in $\mathbb{R}^3$

After doing one more cross product example, we'll write parametrized equations for lines in $\mathbb{R}^3$ (11.5). We'll figure out how to tell if a particular point is on the line, and we'll project the line onto one of the coordinate planes and figure out how to describe the projected line with parametric and Cartesian equations. If there is time, we'll try to decide if two specified lines are parallel, intersecting, or skew (not parallel, not intersecting).

## Sunday, February 5, 2017

## Friday, February 3, 2017

### lecture 6: cross products in $\mathbb{R}^3$

We'll wrap up the dot product and look at the cross product (11.4). We'll use cross products to determine the area of parallelograms and triangles in $\mathbb{R}^3$. You'll play with additional examples in discussion next week.

## Wednesday, February 1, 2017

### lecture 5: dot product, cross product

I'll do two examples involving the dot product (11.3). We'll calculate the work done by a constant force acting on a mass that is moving through $\mathbb{R}^3$, and we'll compute the flux of a force across a line segment in $\mathbb{R}^2$. Then, we'll define the cross product (11.4) and look at some of its properties.

## Monday, January 30, 2017

### lecture 4: vector arithmetic, dot product

We'll do a little vector arithmetic using basis vectors (11.1, 11.2) and, also, look at some applications of the dot product (11.3). This video shows how to use vector triangles to locate the centroid of a triangle.

## Friday, January 27, 2017

### lecture 3: vector arithmetic in $\mathbb{R}^2$ and $\mathbb{R}^3$

We'll do the vector arithmetic I promised on Wednesday.

## Wednesday, January 25, 2017

### lecture 2: vector arithmetic in $\mathbb{R}^2$ and $\mathbb{R}^3$

We'll finish Monday's discussion by looking at simple surfaces (planes, cylinders and spheres) in $\mathbb{R}^3$. Then we will move on to vectors and vector arithmetic in sections 11.1 and 11.2. Basic quantities such as position, velocity, acceleration, and force are represented by vectors. We will add vectors and also scale vectors. We'll talk about basis vectors and see how to express a general vector in terms of the basis vectors.

## Monday, January 23, 2017

### no discussion sections this week

Due to bad weather, the discussion sections will not meet this week (January 23-27).

### lecture 1: points, curves and surfaces

We'll talk about some common objects (points, curves and surfaces) and the spaces ($\mathbb{R}$, $\mathbb{R}^2$, and $\mathbb{R}^3$) they live in. Our goal is to figure out how many Cartesian equations are required to describe the different objects in the different spaces. Also, we'll talk about how to measure distances between pairs of points in the various spaces.

Believe it or not, this stuff will be useful throughout the semester.

Believe it or not, this stuff will be useful throughout the semester.

## Sunday, January 22, 2017

### office hours

Office hours are an important part of the course; if you are not using them you are not fully participating in the course.

My official office hours are M, W, and F from 3:10 - 5:00 pm, or by appointment. I'll post office hours for the discussion leaders (Bang and Andrew) as soon as possible. Keep checking the link at the top of this page.

My official office hours are M, W, and F from 3:10 - 5:00 pm, or by appointment. I'll post office hours for the discussion leaders (Bang and Andrew) as soon as possible. Keep checking the link at the top of this page.

### the etext is full of examples and problems ...

if you can find it. To make your way from the WyoCourses course page to the HTML version of the textbook, look for these links:

- MyLab and Mastering
**MyMathLab with Pearson eText Course Home****Course Tools****HTML eBook****alternate version of your eBook****HTML version of your textbook**

### text

*Multivariable Calculus*by Briggs, Cochran, and Gillett (2nd Edition, ISBN 978-0-321-96516-5). Don't freak out, the chapters in this book might be included in the text you used for Calculus I and II. If not, you can access the electronic version of the book through MyMathLab.

The textbook (either ebook or paper) is an integral part of the course. If you are not using it you are missing out on a large number of worked examples that will help you solve homework problems. Some of these examples may appear on the exam.

### mymathlab

You can reach MyMathLab through WyoCourses. And, there is a permanent link to MyMathLab in the list of links on the right side of this page.

If you paid for access to MyMathLab in Math 2200 or 2205 you will not have to pay for access this semester. If you haven't paid for access, don't worry; you can signup for temporary access and then purchase a license when the trial period ends.

If you see an error message when you click on MyLab and Mastering, try enabling popup windows in your browser.

If you paid for access to MyMathLab in Math 2200 or 2205 you will not have to pay for access this semester. If you haven't paid for access, don't worry; you can signup for temporary access and then purchase a license when the trial period ends.

If you see an error message when you click on MyLab and Mastering, try enabling popup windows in your browser.

### a shiny new semester

This course is all about differentiating and integrating functions of two and three variables. Along the way, we'll play with a variety of curves and surfaces that live in three dimensional space. We'll figure out how to compute slopes on these objects and, eventually, how to integrate vector and scalar functions over the objects. In the final weeks of the semester we'll work with generalizations of the fundamental theorem of calculus that tie together different types of integrals. It is a fascinating and useful combination of geometry and calculus that never fails to excite me.

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