Lecture Notes Dump (more on the way)

October 12, 2016

October 14, 2016

October 17, 2016

October 24, 2016

October 26, 2016

October 28, 2016

October 31, 2016

November 2, 2016

November 9, 2016

November 14, 2016

November 21, 2016

November 28, 2016

November 30, 2016

How to learn math

People are asking me how to learn math again. I'm not an expert, but here are a couple of links that might be useful:

Stanford Math 53: How to study math.

Barbara Oakley's Learning How to Learn

All about the final exam

Hello Dr. Taylor, I just wanted to confirm where we are going to be taking the final for this class. I am in your MWF class. I assume we take the final exam in the lecture classroom?

Thank you, 

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Nope. Per the syllabus (remember that thing?), the final exam will take place in DISCVRY 250 on December 6, from 7:10PM-9:00PM. Plan to be well studied, well rested, wide awake and sober.

Class on the 23rd: Yes

Hi Professor, just wondering if you are holding class on Wednesday the 23rd of November. Thanks!

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Yes.  The great State of Arizona is paying me to the teach on the 23rd, and so teach I must.

A common point of confusion...

When a problems says to integrate ∫ F.dr along a path, or any path, from point A=(a, b, c) to point
B=(d, e, f),  that means that the direction is taken FROM A TO B.  For some reason lots of people want to do this in reverse, which means they get the negative of the right answer.  In the particular if
F=▽f, lots of people will write ∫ F.dr=f(a, b, c) -f(d, e, f), instead of the correct answer which is
f(d, e, f) -f(a, b, c)

13.3#11

Hello Professor! I'm a bit stumped here. I am not really sure where to go from this point. My understanding of a conservative vector field is that it is path (in!)dependent, and should turn up the same value for work for any path between two points.  (true!)

However, I don't see what I am doing wrong. I have been drawing out the fields and attempting to estimate paths, but I cannot get past these answers. Could you please enlighten me as to what
paths I am thinking of incorrectly?

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Trying to guess about wha their line integrals will be for various paths is *NOT* very helpful, because there are so many possible paths.  There are two keys required for these problems.  The first is remembering that being path independent/conservative is the same as (if the domain is reasonable) being a gradient.  The second is to have a gallery of examples of what gradient fields look like.   For example, the gradient of a linear function is always a constant vector field, that is at every point you have the same vector with the same magnitude pointing in the same direction. Do you have any of those?  I suggest you look at your favorite quadratic functions and sketch out their gradient fields, there are some of those in the collection above.  You should also be able to look at a vector field sketch above, and guess what are the functions F_1 and F_2, and from that understand if
∂F_1/∂y=∂F_2/∂x is  possible.

Section 13.4 due date

Hello Professor Taylor!

Quick question: Is section 13.4 meant to be due on Sunday? 

Thanks! 

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Now it's Friday

Extra Credit Opportunity

Here's what I propose: a three page, single spaced essay on the topic of "What Multivariable Calculus is Good for in My Chosen Major".

Here are the rules:
1) No plagiarism, i.e. you can not copy paste from wikipedia, or the text book or anywhere else. It must be in your own words. Your are allowed to quote from your chosen source, but it must be attributed in a Citations section.

2) If you make an assertion you should document the source/reason why that assertion is true, again in the citation section

3) The essay should be structured with a one paragraph introduction, in which you tell me what you are going to tell me, a one paragraph Conclusions section, in which you tell me what you told me, and in between those at least two pages worth of making your case, clearly and cogently.

This essay will be worth 10 midterm exam valued points.

To receive credit Essays must be submitted in both electronic text (not ms word, not pdf) and as hardcopy to my mailbox in the school of mathematics.  
THE DUE DATE: 11:59PM, December 7th   

13.3#1

Hi Professor, for problem one on section 13.3, I have the following answers:
a. N
b. N
c. 5xsin(y)+5y^2


I know they are all correct, but WebWork keeps saying that the variable "N" isn't allowed, even though it accepts it for Part b.


Please get back to me as soon as you can.
Thanks.*******************

The Webwork software has a random number generator that inputs slightly different parameters for each user; everybody gets to work a slightly different problem.  If you click the 'email instructor' button at the bottom of the page, webwork will send me a link to your specific problem and I can give you an answer specific to that problem.  That being said, I just worked the problem from my webwork account and there was just one "N", so I guess you may have computed the partial derivatives incorrectly when you decided that a. was not conservative.

Last minute worries

Dr. Taylor,

Currently, the homework for 13.4 is due 11/20, but considering that we never got to it, can we have an extension for that section?

Thank you,

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Yes.

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Hello, Dr. Taylor.

I recall you telling us in class that the method we learned to determine whether or not a vector field is conservative <true!> (and how to calculate the potential function of a conservative field <not true!>) did not apply to three dimensional functions; you mentioned that we would learn this after the Friday's test.

Accordingly, how should we calculate the potential function for this 3D vector field?

Thanks.

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In any case, you should start doing partial integrals to compute the function while enforcing constraints relating partial derivatives to the components of the vector field.   Just like in the two dimensional case, except that a partial derivative kills an arbitrary function of the two other variables. 

Practice Test3 #3

Using spherical coordinates, I have the limits 0<theta<2pi, 0<phi<pi/4, but I am not sure about rho. What would be the lower limit of rho? (I think the upper limit is 3)

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That's correct. Since you are in the cone, every straight line leaving the origin stays in the cone, and intersects the sphere.  This means that the lower limit for ρ has to be 0.

Practice Test 3 #7

For this one I set me integral up as:  Int[0,2pi] Int[0,4] Int[0,16-r^2] rdzdrdtheta

I am not getting the right answer can you please tell me what I am doing wrong?

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Yes, the inequality x^2+y^2≥1 means that the lower limit of integration for r  is 1. This means the domain has a hole in it, from above it looks like this:

 

13.2 #4

I am a bit confused about this question. I know that the middle one (the
one at the top, perpendicular to the vector field) has its integral, and
therefore work, equal to zero. However, the other two don't make sense. The
one traveling in the direction of the vector field should be positive,
right?

The vector field is in the same direction as the path, so that
equates to a positive line integral? If that is the case, why is it the
smallest? Am I just confused?

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The one traveling in the same direction of the vector field, C3, will have dr parallel to F, hence F.dr is positive, hence the integral will be positive (this is greater than the integral over C1, which is zero). On the other hand integrating over C2, dr will be in the opposite direction as F, hence F.dr is negative hence so is the integral over C2.

HW 13.2

Would it be alright to request a homework extension to section 13.2? Just to the weekend, so we can have more time to study up and work on other exams. And of course, go over the material in detail. Just to compensate for the interrupted schedule math has taken lately. 

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OK, 13.2 is now due on Sunday night.

How to remember formulas

This link talks about scientific results on how to study. Especially note the section on how to study formulas.

12.7n#13

I am very confused about this problem. Not only does it not state what form the variables should take for theta and rho, but the problem randomly resets upon different answers.

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I'm not sure what was going on for you but it looks like you did it correctly.

Current Estimated Grades 11/3/16