Scott’s digital self has moved.

I still don’t know what I was waiting for.

To get greater control over the blog organization and appearance, I have moved to scottastrong.com Nothing new will appear on this site but it will stay up since there are roughly 200 posts which will take a long time to port to the new blog.

But I can’t trace time.

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2014 in review

The WordPress.com stats helper monkeys prepared a 2014 annual report for this blog.

Here's an excerpt:

The concert hall at the Sydney Opera House holds 2,700 people. This blog was viewed about 30,000 times in 2014. If it were a concert at Sydney Opera House, it would take about 11 sold-out performances for that many people to see it.

Click here to see the complete report.

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Gradesheet

Yeah I’ve never known nothing quite like this, don’t it feel like tonight might never be again.

Here you will find my long overdue gradesheet for the course (pdf here). A couple of notes:

  1. Group Assignment #3 has not yet been graded.
  2. Homework #5 has not been graded.
  3. The FG column is your grade if you were to not take the final exam, which is worth 30% of your final grade, according to our syllabus.
  4. To help, I’ve made two more columns which is a statement of what you need on the final to secure an A or a B.
  5. 10.31 Extra credit was not added in an will add 1% to your homework average.
  6. Note: The midterm grades currently do not reflect the points added to those students who did problem #4 the hard way.
  7. Note: Discretionary = 40% (30% for HW and 10% for Group Assignments), MT=30% and final exam =30%

Lastly, there will appear a checkbox on the final exam where you can decide whether you would like your final grade to be on a +/- grade system. For those of you who are trying to make grade changes, this could be helpful. If you are at an 85% and are trying to go for an A in the class, and you miss, then your hardwork may net an A- or a B+. However, if you are trying to make a grade move and the test did not feel super great and you stand to drop to a B- then you can opt-out and ride that B into the break. Your choice but caveat emptor.

I’ll be your bleedin’ heart, I’ll be your cryin’ fool. Don’t let this go too far. Don’t let it get to you.

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Homework 5 – Master list and submission zone

Do you see the way that tree bends? Does it inspire? Leaning out to catch the sun’s rays, a lesson to be applied.

Here you will find the homework 5 submission zone.

Homework 5 is comprised of the following blog posts:

  1. The original 11.13.2014 posting of surface integral problems.
  2. The first addendum.
  3. The final addendum.

A full list of problems is as follows:

  1. Evaluate \displaystyle \int\!\!\int_S 4\, dS, where \displaystyle S is the surface with parametric equations \displaystyle x=uv, \displaystyle y=u+v, and \displaystyle z=u-v, for \displaystyle u^2+v^2\leq 1.
  2. Evaluate \displaystyle \int\!\!\int_S xz\,dS, where \displaystyle S is the part of the plane \displaystyle z=1-x-y in the first octant.
  3. Find the flux of \displaystyle {\bf F} across \displaystyle S, where \displaystyle {\bf F}(x,y,z)= x^2\hat{\textbf{i}}+xy\hat{\textbf{j}}+z\hat{\textbf{k}} and \displaystyle S is the part of the paraboloid \displaystyle z=x^2+y^2 below the plane \displaystyle z=1 with upward orientation.
  4. Set up \displaystyle \int\!\!\int_S {\bf F}\cdot d{\bf S}, where \displaystyle  {\bf F}(x,y,z)=2\hat{\textbf{i}}+3\hat{\textbf{j}}+z\hat{\textbf{k}} and \displaystyle S is the sphere \displaystyle x^2+y^2+z^2=4. (Parameterize with spherical coordinates and set up the integral only, do not solve. Do not use the Divergence Theorem.)
  5. If {\bf{F}}=(-y \hat{\textbf{i}}+ x \hat{\textbf{j}})/(x^{2}+y^{2}), show that \displaystyle \int_{C}{\bf F}\cdot d{\bf r}=2\pi for every positively oriented simple closed path that encloses the origin.
  6. Use Stokes’ Theorem to evaluate \displaystyle \oint_C {\bf F}\cdot d{\bf r} where \displaystyle {\bf F}(x,y,z)=2z\hat{\textbf{i}}+(8x-3y)\hat{\textbf{j}}+(3x+y)\hat{\textbf{k}} and \displaystyle C is the boundary of the part of the plane \displaystyle 2x+2y+z=2 in the first octant, oriented counterclockwise as viewed from above.
  7. Replicate the following work found here. You will not be asked to submit your work only on your comfort level with its replication. [11/15/2014 POSTED!.]
  8. Assuming that this were the domain of integration associated with a surface integral, how does the integrand change the physical interpretation of the surface integral?
  9. Use the transformation \displaystyle  u=x+2y and \displaystyle  v=y-x to evaluate \displaystyle  \int_0^{\frac{2}{3}}\!\!\int_y^{2-2y} (x+2y)(y-x)\,dx\,dy.
    • Hint! To figure out the bounds in the uv-plane, consider transforming the triangular xy-region defined by the three lines into a the uv-plane. Specifically, what does y=x imply about v? Also, what does x=2-2y imply about u? Lastly, what’s the third curve from the xy-plane?
  10. Use the transformation \displaystyle  u=x+y and \displaystyle  v=x-y to evaluate \displaystyle  \int\!\!\int_R (x+y)e^{x^2-y^2}\,dA, where R is the rectangle enclosed by the lines \displaystyle  x-y=0, \displaystyle  x-y=2, \displaystyle x+y=0, and \displaystyle  x+y=3.
    • Hint! Same logic is the previous problem, the region in uv-plane should be simpler.
  11. In class we considered the region left behind after we excised a cone from a cylinder. In both classes we considered an order of integration given by dV = r \, d\theta dr dz. Do the problem again where the order of integration is given by dV = r \, d\theta \, dz \, dr.
  12. For the same situation, what happens if the cylinder is wider than the cone and vice versa? For instance, suppose that the cylinder is defined by r_{1}^{2} = x^{2} + y^{2} and the cone is defined, by \displaystyle \left(\frac{r_{0}z}{h}\right)^{2} = x^{2}+y^{2}, where 0 \leq z \leq h, what is the volume left behind when r_{0}<r_{1} and vice versa?
  13. In class, on Tuesday, we consider the problem of the region shared by the intersection of the two solid bodies defined by the surfaces \rho=1 and \rho = 2 \cos(\phi). Using triple integrals, find the volume of the first solid body that is not contained within the second and vice versa.
  14. Use the Divergence Theorem to calculate \displaystyle \int\!\!\int_S {\bf F}           \cdot d{\bf S}, where \displaystyle {\bf F}=x^3 \hat{\textbf{i}}+xz^2 \hat{\textbf{j}}+3y^2z \hat{\textbf{k}} and \displaystyle S is the surface of the solid bounded by the paraboloid \displaystyle z=4-x^2-y^2 and the \displaystyle xy-plane.
  15. [Not required but I wanted to make 15 problems. ] Assume that the temperature of a room is proportional to the square of the distance to the center and this distribution causes a heat flux vector field that is proportional to the negative of the gradient of the temperature. We showed in class through flux integrals that the amount of heat energy flowing through the sphere of radius R centered at the origin is -8\pi D \alpha R^{3}. Verify this result via the divergence theorem.

a truant finds home…and a wish to hold on…
but there’s a trapdoor in the sun…immortality…

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Homework 5 – Final addendum

People who work together will win, whether it be against complex football defenses, or the problems of modern society.

Here you will find a growing repository of notes and here you will find a set of notes about divergence.

For your final question I would like you to:

  1. Use the Divergence Theorem to calculate \displaystyle \int\!\!\int_S {\bf F}           \cdot d{\bf S}, where \displaystyle {\bf F}=x^3 \hat{\textbf{i}}+xz^2 \hat{\textbf{j}}+3y^2z \hat{\textbf{k}} and \displaystyle S is the surface of the solid bounded by the paraboloid \displaystyle z=4-x^2-y^2 and the \displaystyle xy-plane.

If you want to, but you don’t have to, then try this problem as well.

  1. Assume that the temperature of a room is proportional to the square of the distance to the center and this distribution causes a heat flux vector field that is proportional to the negative of the gradient of the temperature. We showed in class through flux integrals that the amount of heat energy flowing through the sphere of radius R centered at the origin is -8\pi D \alpha R^{3}. Verify this result via the divergence theorem.

The result can be made physical and may help you think about the divergence theorem.

Notes: I will make a master list later today and a submission zone on Thursday.

Arrakis teaches the attitude of the knife — chopping off what’s incomplete and saying: “Now it’s complete because it’s ended here.”

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12.03.2014 – Journal Prompt (E+M)

I was born not knowing and have had only a little time to change that here and there.

The following blog posts were developed last year and are related to the concepts from electricity and magnetism that started and ended the course with this year. Last year there was more stress on wave motion, which I’ve moved to MATH235 since there just isn’t time to talk about this in MATH224.

  • 8.23.2013 blog post about mass-spring, string and drumhead motion.
  • 8.25.2013 blog post about electromagnetic phenomenon and Maxwell’s equations.
  • 12.4.2013 blog post about electromagnetic waves and signal transmission. In the post you will find this pdf which is a more detailed set of notes about how the fundamental theorem of calculus applied to Maxwell’s equation results in Gauss’, Faraday’s, Ampere’ and Maxwell’s laws.

Remember that the question on deck is how/why does/can a radio receive information and make sound originating elsewhere. This same question is applicable to all wireless devices and is, of course, complicated but steeped in applied science and engineering principles. Applied science says that it can be done and engineering figures out a way to do it so that your cell phone does not have a giant antenna sticking out of it. Well, at least not any more.

Oh the times you probably missed.

Next semester in MATH235 we will talk about:

  1. How a mass-spring system is remarkably similar to the circuit that is inside a radio and how the material properties of the circuit can affect which signal in space is received.
  2. The vibrational modes of a thin circular membrane.
  3. The existence of electromagnetic waves that transport the information from the radio station to the radio.
  4. The existence of traveling waves in pressure medium.

We will not talk about:

  1. How signals are processed.
  2. The signal/current on the wire and how this is coupled to an electromagnet to put shapes on the thin circular membrane. Here is a nice animation of a speaker working.
  3. Design of wave guides.
  4. All that goes into how we hear anything at all.

Good thing there are plenty of other courses which touch on these aspects of STEM.

Study hard what interests you the most in the most undisciplined, irreverent and original manner possible.

Posted in CalcIII, Journal Prompt, Partial Differential Equations, The Fundamental Theorem of Calculus | Tagged , , , , | Leave a comment

11.8.2013 Journal Prompt – (Hydrogen Atom Revisted)

Scott Strong:

Here is a discussion of first moments (center of mass) and second moments (moments of inertia) as they relate to the probability density function (electron cloud) of the ground state (1s1) hydrogen atom electron. Some quick and interesting facts:

  1. There is a 68% chance of finding the electron outside of the Bohr radius.
  2. The expected/average value, which is the most likely place you would find the electron, is at 3a_{0}/2.
  3. There is a 71.5% chance of finding the electron one standard deviation away from the average.

Originally posted on Scott Strong:

So, in class we talked about how the Schrödinger equation,

$latex displaystyle (1) , , , , i hbar psi_{t} = frac{hbar^{2}}{2m}triangle psi + V psi$

defines a wave function, $latex psi : mathbb{R}^{n+1} to mathbb{C}$, such that the quantum mechanical particle subjected to the potential field, $latex V:mathbb{R}^{n} to mathbb{R}$, has an associated probability density defined by $latex psi psi^{*}: mathbb{R}^{n+1}to mathbb{R}^{+}_{0}$. For a hydrogen atom $latex V(r) = -e^{2}/4pi epsilon_{0} r$ where $latex e$ is the charge of the nucleus and $latex epsilon_{0}$ is the permittivity of the vacuum between the atomic core and the electron. Using techniques from partial and ordinary differential equations, it can be shown that the ground state, $latex 1s^1$ orbital, wave function is given by

$latex displaystyle (2) , , , , psi(r) = frac{1}{sqrt{pi a_{0}^{3}}} e^{-r/a_{0}}$

where $latex a_{0}approx 5.27 times 10^{-11} m$ is the Bohr radius. There are other excited…

View original 1,346 more words

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