Wednesday, May 11, 2016

exam 4 approaches

Exam 4 begins at 1:15 pm on Wednesday, May 11. The exam covers sections 13.7, and 14.1-14.8.

Calculators and computers are not allowed at the exam. We will provide you with an equation sheet. You can find the equation sheet on the last page of the new mock exam.

The equation sheet does not contain the transformation equations for cylindrical coordinates, but you should know them by now.

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.

Your exam room is a function of the first three letters of your last name.
• Aar through Kla, go to CR 133
• Kle through Pow, go to Berry Center 138
• Put through Zzz, go to Geology 216
Leave an empty seat between you and the next calcunaut.

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, during our office hours, or at the review session.

Wednesday, May 4, 2016

divergence theorem, section 14.8

I'll give a quick review of Stokes' theorem and finish the flux problem.

And, we'll look at one final version 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.

Monday, May 2, 2016

homework twenty-two

Pearson's MyMathLab website is sick, very sick. Check its status at status.pearson.com. I was able to extend the deadline for homework twenty-two to Wednesday night.

stokes' theorem, section 16.8

Today we'll look at Stokes' theorem (16.8). 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. We'll also look at some ramifications of the theorem.

Friday, April 29, 2016

surface integrals (part 2), section 14.6

After finishing any leftovers from Wednesday, we'll compute the flux of a vector field $\vec{F}$ across open or closed surfaces. The flux across the surface element $d\vec{S}$ is represented by 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 flux is equal to the dot product between $\vec{F}$ and $d\vec{S}$.

Wednesday, April 27, 2016

surface integrals (part 1), section 14.6

First, a word on section 14.5. You want to be able to compute the curl and divergence of a vector function. You also want to know that if $\vec{F}$ is conservative then $\vec{\nabla} \times \vec{F} = \vec{0}$. There are examples in the book, on YouTube and in this week's discussion problems. We'll do some additional work with the curl and divergence next week, after we have discussed Stokes's theorem and the divergence theorem.

Today we'll work on integrating scalar functions over common surfaces. To save time I'll let you read about the vector and scalar surface area elements on your own.

The WyoCast recording system failed again.

This table contains some common surface area elements:

Monday, April 25, 2016

green's theorem and vector operators, sections 14.4 and 14.5

We'll verify Green's theorem, use it to play some tricks, and get to know some friends of the gradient vector.

The WyoCast recording system failed.

Friday, April 22, 2016

conservative vector functions, section 14.3 and 14.4

We'll will finish Wednesday's project and see why a vector function of the form $\vec{F}(x,y,z)=\vec{\nabla}f(x,y,z)$ is said to be conservative. The short story is that the fundamental theorem of calculus requires that energy (kinetic plus potential) be conserved if the force is conservative. And, this last statement explains why math rulz!

If there is time, we'll start work on the next version of the fundamental theorem, Green's Theorem in section 14.4. Green's Theorem links double integrals to line integrals.

Wednesday, April 20, 2016

the fundamental theorem, section 14.3

We will use the fundamental theorem from calculus I to construct a fundamental theorem for line integrals involving $\vec{F} \cdot d\vec{r}$. 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.

hi res video from section 01

Tuesday, April 19, 2016

exam three results

Last Thursday, 146 brave calcunauts took exam three. The results are similar to those for exams one and two. The mean is 73.4 and the quartile scores are 66, 74.5, and 83. One surprise is how many fewer people scored in the 90s. No perfect scores this time, but one person did score 99. As usual, a number of people wrote beautiful solutions.

For comparison, last semester the mean was 74.4 and the quartile scores were 63.8, 76, and 87.

People had problems with the T/F questions (slow down and think?) and the Lagrange multiplier question (too much time has passed?). The median scores on the individual problems are:
14/20, 6/8, 8/8, 8/8, 12/12, 9/12, 6/8, 4/8, and 12/12.

Monday, April 18, 2016

line integrals, sections 14.2 and 14.3

I'll review the work and flux integrals and finish the problems from Friday. Then, we'll look at alternate ways to express the work and flux integrals and compute some work integrals using conservative force fields.

low resolution video and audio from section 01

Friday, April 15, 2016

preliminary solutions for exam 3

4/17 11 AM: I fixed my error in the answer to problem 4 and a Bang found a typo in the table in the answer to problem 7.

4/18 8 PM Bryan fixed an error in problem 8.

I need to dress these solutions up, but won't have chance until next week.

line integrals, section 14.2

We will integrate the product $f(x,y,z) \, ds$, where $ds = | d\vec{r}|$, along a space curve. We'll solve the problem by using the parametric equations for the curve to make a change of variables so that $f(x,y,z) \, ds = f(t) | \vec{r}^\prime(t) | dt$. Integrals of this type arise when we need to calculate the mass and center of mass of a wire.

More importantly, we will think about replacing $f(x,y,z)$ with a component 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. Another is to compute the flux of $\vec{F}$ across a curve in $\mathbb{R}^2$.

no control of camera or microphone!

Thursday, April 14, 2016

exam three approaches

Exam three begins at 5:15 pm today (Thursday, April 14). The exam covers sections 12.8, 12.9, and 13.1 through 13.5, but you still need to understand how to work with vectors. No electronic devices are allowed at the exam.

We will provide you with an equation sheet. You can find the equation sheet on the last page of the mock exam. Some facts are not on the equation sheet. You need to know how to differentiate, find limits of integration and integrate. 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.

• Aar through Kla, go to CR 133
• Kle through Pow, go to Berry Center 138
• Put through Zzz, go to Geology 216

• Leave an empty seat between you and the next calcunaut.

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.

An engineering professor offers alternatives to cramming:

Wednesday, April 13, 2016

line integrals, section 14.2

Today 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 $ds=\sqrt{(dx)^2+(dy)^2+(dz)^2}$. The trick is to use the parametric equations of the curve to make a change of variables. This type of integral is used to compute physical properties of wires (think length, mass, and center of mass).

We'll look at section 14.1 after we talk about a second type of line integral on Friday.

low res video from section 01 (lip readers only?)

Monday, April 11, 2016

change of variables, section 13.7

We'll solve several double integrals with complicated regions of integration by applying either a forward or inverse substitution.

low res video from section 01

Friday, April 8, 2016

mass calculations, section 13.6

We'll do center of mass calculations for 1d, 2d and 3d mass distributions.

No wyocast video today?

Wednesday, April 6, 2016

triple integrals, section 13.5

We'll work another example using cylindrical coordinates and then look at the spherical coordinate system. The spherical coordinate system is most useful when one or more of our boundaries is a sphere.

high res video from section 1