score update (after corrections)

Apologies & correction

I went home yesterday afternoon after classes, seminars and meetings and got pukey-sick, which is hopefully the only cause of my stupidity on Friday going over the quiz, for which I apologize and which I'm going to correct here.  If you still are confused on this issue, I'll be in my office tomorrow 11:30-12:30pm and Tuesday 12:30-1:30pm to discuss with you

The second problem on the quiz was to sketch out the regions of positive and negative divergence, which are as follows (blue is positive divergence and red is negative divergence)--I have imposed a coordinate system on the graph to give a language to describe parts of the graph:

Here's the correct reasoning. Calling the vector field F=Pi +Qj , the divergence is ∂P/∂x+∂Q/∂y; we can see that the Q is positive above the line y=1 and also below the line y=-1, and increasing as |y|>1 is more positive; you can see this in the way the vectors point upward-ish and are increasingly upward on each vertical line for x big enough or small enough.   On the other hand Q is negative between y=1 and y=-1, as you can see from the fact that the vectors point slightly downward.  Likewise, for P is positive for x bigger than 1 and also for x less than -1, and is more positive the more |x| is greater than 1 and is negative while x is  between x=-1 and x=1; you can see this from the way that the vectors point somewhat to the left when |x|<1 and to the right when |x|>1, and point increasingly to the right along horizontal lines as |x| is large enough.  

Now, since div(F)=∂P/∂x+∂Q/∂y, div(F) will be positive if the sum of ∂P/∂x and ∂Q/∂y is positive, that is if the rate of increase of P in the positive x-direction is greater than the rate of decrease of Q in the positive y-direction OR vice versa.  In the first quadrant, we can see that both ∂P/∂x and ∂Q/∂y are positive, in that they both P and Q increase from negative to positive values going from left to right (P) and from down to up (Q): this means div(F) must be positive.  The reverse is true in the third quadrant, so  div(F) must be negative.  In the half of the second quadrant above the line y=-x, Q increases a lot with increasing y while P decreases slightly with increasing x so ∂P/∂x is less negative than ∂Q/∂y is positive so div(F) must be positive; the opposite is true in the second quadrant below the line.  In the same manner, in the fourth quadrant above the line P is increasing quite a bit moving from left to right, while Q decreases not so much--it follows that ∂P/∂x is more positive than ∂Q/∂y is negative so div(F) must be positive, and again in the same manner the opposite is true in the fourth quadrant below the line.

Office Hours Monday and Tuesday

I announced this in class, but here it is anyway:
M 11:30AM-12:30PM
T 12:30-1:30PM

Finals Cram and Finals Workshop

From Brian England: I have decided to hold another problem session outside of the normal time frames next week.  I get out of my own final exam <<on Tuesday>> at 2pm and will head to the engineering tutoring center and be there by 2:30.  They have agreed to keep the place open until 5pm.  As always, your students are welcome to come.

Also: There will be a MAT267 Finals workshop Tuesday May 3 at from Noon to 1:00pm in BA201.  "Finals Workshops requires the student to bring class notes, practice material and questions to the workshop. Students should be prepared to collaborate with peers and tutors. This learning environment should foster recall and understanding of information for an upcoming exam."

  See <https://tutoring.asu.edu/calendar/>, Sign up: http://bit.ly/MAT267Final

13.7no7

I don't know why its coming out wrong!

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A few reasons.  First the vector element of area dA here is what our book calls dS, and not what the book calls dA--that is dA=(r_u x r_v )dudv, or since your parameters are called  r, θ  
dA=(r_r x r_θ )drdθ.  Second,  r, θ here are just parameters, not polar coordinates: you only have drdθ for your area element in the parameter plane, not rdrdθ (in fact I guess that the whole point of this problem is to fool you into making the mistake you did make, so that you can learn not to make it).  Finally, in your integral ∫∫_S F.dA it looks like you dropped the 9 from your F and also did the dθ integral incorrectly: the answer is not 2π because the F . (r_r x r_θ ) actually not independent of θ.

13.7no2

Can you help me with Let S be the part of the plane 3x+2y+z=4 which lies in the first octant, oriented upward. Find the flux of the vector field F=1i+1j+4k across the surface S.

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OK, here's an answer in general terms, so I don't do your homework problem for you.  You can write the plane as the graph z=f(x,y) of the function f(x,y)=4-3x-2y.  You can then use the usual parameterization of the graph of a function: r(u,v)=<u,v,4-3u-4v>, and then compute that
  r_u x r_v = ai+bj+ck, 
for whatever numbers a,b,c you get from computing the cross product.  Since the orientation is upward, you have to be a little careful: c has to be positive, so if   r_u x r_v  has c negative you have to use the -(r_u x r_v) instead to get the upward orientation.  The shadow of S in the xy-plane is the triangle enclosed by the x-axis, the y-axis, and the line 4-3x-2y=0, and the same triangle is the domain D of the parameterization. Then ∫∫_S F.dS=∫∫_D ∫∫_S F.(r_u x r_v) dA

13.6 no 3

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OK, the problem here is a practical one.  The "flow outward" this problem wants is in the units kg/sec. Since the density is kg/m^3,  and since the vector field v is in m/sec, and since dS has units m^2, the product density*v.dS has units of kg/sec.  Thus the rate of flow outward will be

5∫v.dS=5x(8/3)pi

 or (40/3)Pi.

Attendance up to 4/20

(click on image to enlarge)

Exam 3 update glitch

Just at the end of my office hour (and a half) today I crashed my spreadsheet.  I was saving often, so this could only possibly affect the people who were in my office after 12:30PM.  I *think* I recovered all of the correct scores, but if you were in my office after 12:30PM today, and correcting your exam or quiz 9 scores, check the following list.  If there is a problem, bring your additional scores (the new points I gave you on your paper) to my office hour tomorrow or class on Friday and I'll fix the problem.

Course Evaluations (a.k.a. grade your instructor)

MyASU -> My Classes -> Course Evaluations

Current scores and projected final grade

(click on image to enlarge)








Notes:
1)"Estimated final score" is made from the current total scores combined according to the rule (Exam Scores/300)*75+Homework*15 +Quizzes*10
2) these scores do not include test corrections if you havent done them yet.

Extra credit projects

Here is your extra credit opportunity--20pts total.  The project will be three pages double spaced, 1 inch margins.  Projects will be delivered in pdf format to my inbox AND in my department mailbox (in Wexler Hall 231) before Wednesday May 4 at noon.  No credit will be given for projects that do not meet the format or deadline.  Each project will have statements clearly cited.  While I offer starting references in as Wikipedia pages, each project will require you to locate a minimum of two other published references; you may wish to speak to your engineering professors for suggestions.  Below I offer two topics, I'm open to adding other topics to the list if you can come up with one that has a compelling application of multivariable calculus.



Topics:

1) The use of divergence, curl and gradient in electrodynamics.
Some starting references:

A) https://en.wikipedia.org/wiki/Electromagnetism
B) https://en.wikipedia.org/wiki/Classical_electromagnetism
C) https://en.wikipedia.org/wiki/Maxwell's_equations

Specific questions to answer:  What does the divergence of an electric field tell you?  What does the curl of an electric field tell you?  What does the divergence of a magnetic field tell you?  What does the curl of a magnetic field tell you?  What does the gradient of an electric potential tell you? Suppose that there is an electron sitting at the origin. What is flux integral over the sphere of radius 1?


2) The use of divergence, curl and gradient in fluid dynamics.
Some starting references:

A) https://en.wikipedia.org/wiki/Fluid_mechanics
B) https://en.wikipedia.org/wiki/Fluid_dynamics
C) https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations

Specific questions to answer with regard to a velocity field of a fluid in motion: What does the divergence  tell you?  What does the curl tell you?  How are these related to pressure?  What are the properties of a potential flow?

Lecture Notes 4/18/16 through 4/22/16

Lecture Notes 4/18/16

Lecture Notes 4/20/16

Lecture Notes 4/22/16

13.5 number 2

I'm not sure what I am doing wrong? why do i keep getting it wrong, i've
reworked it a few times already.

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Well, what are you trying?  Let's see....you've done the divergences correctly, so that's not the problem.   Uh...wiki has this formula for the curl:

which is the same as ours...which for part A) gives curl(F)=-xi + (2y-(-y))j+(-2z)k, so it looks like you made an sign error leading to an arithmetic error in the j component.  In part B) you've done the curl correctly, so there was only the one error

13.5 number 9 (updated)

I do not know how to approach this at all.  Please give me some insight!  I
work the majority of the weekend, so an extension until Sunday would be
heaven-sent too.  If not, I'll do what I can before midnight.  Just
throwing that in there! ;-)

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First of all, the book talks about this in some detail. I recommend that everybody read it.  The *general* notion is that the curl measures both the amount and direction of circulation around an axis--for which counter-clockwise is positive--as well as the direction of the axis as a vector.  The important examples to keep in mind are the ones we worked in class last Friday: the vector fields

F=xi+yj+zk for which curl(F)=0

and the vector field F=yi-xj+0k, which looks like this:

for which curl(F)=-2k (and the negative is because the vector field is circulating clockwise around the z-axis).

The Final Exam

TUESDAY, May 3: 7:10-9:00pm in COOR 184

Homework Due This Friday

Hello Professor Taylor, 

I was hoping that we could extend the 13.6 homework to Wednesday or Friday next week considering we finished the notes today. Also it would be extremely hard for me to make it into the tutoring center before Friday and spend a decent amount of time on it, so any extension into next week would help. Thank you!

Regards,

AND

Hello Dr. Taylor,

I was planning to begin HW for section 13.6 this evening
(since I'm fairly certain it was open since Friday), but after walking away
from my computer for a bit and refreshing my browser I no longer have
access to 13.6.  Were we supposed to have access to it originally?  Or was
this recent change an error?

Thank you for your time.

*************************************

Oops, right you are.  OK, 13.6 is now due on the 29th along with 13.7. (which is the last day of classes, mind you.)  It's still open though, so you can work on it, even though it's not due until next week.

Lecture Notes 4/15/16

Lecture Notes 4/15/16

About negative volumes...

There aren't any.  If you get a negative answer for a volume you did something (very) wrong.  You should figure out what it is.  Then fix it.
(Of course, you could hope that you just made a sign mistake at the last minute and that the true answer is just the negative of your negative answer.  From what I've seen on a lot of exams, though, odds are that you made a *much* bigger mistake than that)

RE: taking notes

Article on NPR

<<Not that it's easy to take laptop notes in math class.>>