Hello, Soomi. When you post a message to a group, you should get notifications whenever any new messages are posted to that group. This Daily Study Group ended about a month ago; most of the discussion from those daily sessions has ended. The Multivariable Calculus course taught during that DSG has now been released as an interactive course and is available for anyone to take and earn completion certificates: https://www.wolframcloud.com/obj/online-courses/introduction-to-multivariable-calculus/what-is-multivariable-calculus.html

If you have any questions during that course, this would be an excellent place to ask them. There also may be older entries in this discussion that may be helpful in learning the material. Good luck!

--phil (Not a Wolfram staffer; just a fellow student.)

Hi Mitch—try using CubeRoot rather than ^(1/ 3). (You may see why this is, now that I mention...)

Hi Mitch—if you could remind me of this next Wednesday, I would be happy to take a thorough look. I'm off for the next ~week, so can't check in the meantime. What I can at least say re: your comment is that CubeRoot and ^1/3 shouldn't return the same thing for either == or ===. The former explicitly gives the real cube root, whereas the latter refers to the more general cube root... at least when it's applied to a variable. For example, think about the cube root of 27—there's 3, obviously, but 3 e^(i 2 Pi /3) and 3 e^(i 4 Pi /3) both also give 3 if you cube them. When dealing with equations that have a ^(1/3), these latter types of roots are kept as possibilities whereas CubeRoot only keeps the real-valued one.

I downloaded the final exam yesterday and I don't think that the proposed answers to Question 11 have been fixed in response to the prior posting: 'On the Final Exam Lesson Eleven the domain of Log[Log[(x+y+x)^2 + 1]] does not include the endpoints so answers should only use "<" or ">".'

Also, it appears that the second listed answer to that question does not actually offer a choice: x+y+z<=-1 or x+y+z<=-1. Is that intentional? Thanks

On Quiz 3 problem 4, where f(x,y)= e^(-x-y), I think the first three answers are all correct. Taking a partial with respect to x or y just multiplies the function by -1. So all combinations of third partial derivatives will be the same; i.e. the negative of the function.

I didn't notice anyone else pointing this out. A minor typo in Practice Problem Set 2 Problem 13, The coordinate for A should be (0,2) instead of (0,-2).

Hi Arben;
Is the following answer correct? The processing took forever to complete and created several errors - see below. Additionally, when I replaced //N with //Simplify, I really got a strange looking answer.

Thanks,
Mitch Sandlin

Attachments:

StrangeAnswer.nb

(I think you may mean Lesson 31...)

Are you seeing a 1? In both my source notebooks and the deployed version I'm seeing this:

Thank you Juan! We'll look into this.

In the Framework, Lines and Planes, on the lessons view pane,

under Equation of a Plane, would it be worth appending " , represents an equation of the plane." to the first paragraph.

under the example of Equation of a Plane, the Dot Product symbol is missing between the two vectors ie it should read "therefore , <...> . <-3,1,4> = ... "

In the Framework under Cross Product, in the lesson view pane, all the Graphics have their WL code visible.

I must say the explanation of Magnus Effect and the referenced You Tube video is a high point in course applications.

In the framework, under Dot Product, in the lesson view pane under section Vector Projection, there is the WL statement SetOptions[ EvaulateNotebook[], ShowCellBracked->True ]

Perhaps already caught by others, In the framework for Dot Product, in the lesson view pane, the image under "Angles between Vectors" is blank.

In the lesson on Vectors I noticed that the description of what a vector is might be improved. Vectors are introduced as "Each vector contains two important pieces of information: its length (or magnitude) and the direction in which it points.". Then under the heading "Algebraic Representation of Vectors" we see that a vector can be written as <3,2> where in both entries are orthogonal distances that encode both the magnitude and direction. This leaves the student asking "what is the magnitude and direction then?". Later in the "Application" section we see a vector written as 50<cos(30),sin(30)> which is a magnitude and unit direction algebraic definition.

My question is whether it would be helpful to the student to see right away the magnitude * unit vector definition first followed up by an explanation that this is equivalent to the <distance, distance> Algebraic form ie. what the student sees in the application example ... 50 <cos(30),sin(30)> = 50 < sqrt(3)/2, 1/2> = <25 sqrt(3), 25>

I understand that it also requires the introduction of a unit vector, which is well done in the lesson as a decomposition of the <distance,distance> form.

I bring it up as something to think about.

Regards John

Your welcome Arben,
This framework is fundamental to many fields, so I'm willing to assist your call for comment. We're all tolerant to the most reasonable result within the time budget and the present condition of this framework is indeed quite good. I myself can only devote so much time to the call for comment.

Linear algebra introduces an arbitrary system of j linear equations in i unknowns xi as the sum over i and j of [aij xi] = bi, which is decomposed to A X = B, where a matrix is defined as A = [a11, ... a1j, a21, .... a2j, ... , ai,1, ... aij], and vectors defined as X=[x1, ... xj] and B = [b1, ... bj]. So a vector is defined simply as a row of matrix, a column of a matrix or more simply as a list of two or more scalars.

Applying vectors to Euclidean space whether (n= 2, 3) or to higher dimension, the concept of orthogonality is introduced, wherein the it understood that a change in any unknown, say xi, does not affect other unknowns, ie, in Euclidean 3D a change delta x does not change y or z. This is simply understood for 2D and 3D Euclidean space as we humans can visualize the orthogonal directions and associated unit vectors, but we need help to understand any applications of this beyond n=3.

That help is the concept of an orthonormal bases. For instance say we want to decompose a list of words describing colour. In literature all the colours are given a word or word combination (orange, white, bluish green, ...) where-in the word combinations are unique. Fields of study such as artificial intelligence, AI, might chose to model these assuming words and word combinations although unique as spelled are unique in meaning, that is can be modelled mathematically as orthogonal, ie that each colour word or word combination is orthogonal, however physicists study of light reveals colour is truly orthogonal as vector representation in the primary colours of <Red,Green,Blue>, ie. RGB as an n=3 orthonormal bases of colour. Fortunately the Gram-Schmidt process helps us determine an orthonormal basis when the xi are themselves not mutually orthogonal. I think this is an important thought as AI applications deepen beyond our comprehension of uniqueness.

In physics and engineering where the techniques of linear algebra, calculus, are well developed for conserved fields (read conservation of mass, momentum and energy) where the physics can be described in Euclidean space as orthogonal, it is preferred to keep separate the force vector functions and displacement vector functions, so the "physics" is more readable in the equations. An example of this would be the application example I liked and supplied for which I supplied a reworked notebook in a previous post, and should have separated the forces from the distances in all the equations. That way we would have seen an equation of the form f . d = mg <0,h> where mg is a force and the distance vector is <0,h> instead of f.d = mgh

Keep going Arben and team, You're doing well.
John

I left it running overnight and in the morning had some answers. It evaluated the inner integral but left a horrific mess for the integrand of the outer integral. However, it did get a numeric integral for the area. Arben, how can I ask it to show how long it took for each result?

Dear Arben, in Practice Problem Set 1, Problem 12, I guess that the slope of a line with eq. y = -5x + 3 is -5 and not -3. Therefore, the vector parallel to the line should be v = <1,-5> and not <1,-3>. The acute angle that results from the problem is 37.875 degrees and not 45 degrees. Best regards, Ruben.

The method of Lagrange multipliers was skipped and the physical example of the lesson notebook is not covered in the video.

Liking this example, but finding it terse in explanation I've reworked in in the attached file. That so it might be considered for inclusion in the course framework.

Regards, John

Attachments:

Lesson 18 Physic...nb

Very carefully.

Very interesting! If you'd like to recreate this in a notebook—perhaps with some of the functionality we'll learn about later this week, even—here's a start on the visualization aspect at least!

Any time! Tim's visualizations in the lessons are generally much nicer than mine in the exercises, but they do require correspondingly more work to make. We could say that this disparity is intentional in that it offers guidance to people at multiple levels of programming skill ;).

Arben - Thanks; I initially couldn't find the option ShowCellBracket in the Option Inspector. I knew it had to be somewhere in there. After you sent the one line of code, I finally found it.

Hi Juan—let me check.

• For Problem 4, I can confirm that the correct answer is among the choices but was not marked as the correct answer in the key. I have updated this.
• For Problem 10, I can confirm that the inequality should be ≤ rather than ≥ in order for the solution and graphic to match the given region.

Thanks for pointing these out!

Yes, that is just what I had in mind.

For those interested in more input on unstable rotation about the axis with the mean moment of inertia, here are some links:

• Wolfram Blog: Simulating Zero Gravity to Demonstrate the Dzhanibekov Effect and Other Surprising Physics Models
• Wolfram System Modeler (simulation starts at around 14:15)
• Video from "reality"
• Video from "science"

Still, it's impressive that you achieved the effect with a simple deck of cards under the influence of gravity. :-)

There is a typo in final solution to Lesson 2 Exercise 4 (equation of the sphere); ...(x-3)^2... should be ...(y-3)^2...

In a lecture long past, lesson 4 on the dot product, in the section "dot product of 2 different vectors", I guess that the difference vector u-v points in the wrong direction in the 3D image, it points from u to v, and should point from v to u. In lesson 3 on vectors, in the section "vector subtraction", in the 2D image the vector difference u-v does point in the correct direction, from v to u.

I also was preparing to post as well.

Hi Arben!

I have refreshed the page hoping this would be fixed but, no. The scratch notebook obscures the middle panel. It happens in Firefox, Safari, and Chrome. It does not happen with quizzes, only Practice Problems.

Examples: Chrome Safari Firefox

Please let me know if you need more information! Thank you!

Thank you, Arben, I will! :-D

Hello! Congratulations for this course! I'd like to know if the notebooks of classes are disponible.

Thank you! Yes, you should be able to download them from the framework we shared or directly from the amoeba link that we pin/sticky every day.

Adding a bit to the file ...

Attachments:

Average Values o...pdf

Thanks for the eagle-eyed catch, Juan—I've updated the notebook on our side. I'll look into your question about the quiz after today's session as well.

Hi Phil—in order:

1. Unlike the simple exercises in EIWL, there is—unfortunate as it may be—a 0% chance that we (or anybody, I think) could write code that would appropriately grade solutions to this type/level of exercise. In some cases, there is a strict numeric answer that could be entered and checked, but I don't think it makes sense to have an "enter your input here" only for some portion of the exercises. You can still work on the exercises in the scratch cloud notebook that you have available in the interface simultaneously with the exercises. I do, however, think that the lesson and exercise notebooks are best interacted with locally—we've made serious improvements to cloud such that it's even possible to host and interact with lots of the graphics and interfaces here, but when it comes down to it these are generally heavy notebooks and a local interface is always going to handle them noticeably better than a cloud interface.

2. They are indeed being worked on, but are not yet ready. The transcripts are actually done by a person!

3. All of the lesson and exercise notebooks are deployed to the beta course and should be in working order. The review notebooks are written as we go through the DSG in response to the questions people submit at the end-of-session surveys and are "DSG-exclusive", so to speak, on account of needing that context.

I wouldn't say much of anything is particularly easy to do in the framework :'). But I see what you mean and will ask the person who handles our cloud deployment whether this is possible.

You noted: that the difficulty in imitating what the EIWL course did was auto-grading. Don't auto-grade. Simply providing an interpreter in the exercises window should be easy.

Right, I said that doing auto-grading for something like this was impossible—but that's not to say that tasks simpler than that are trivial or even easy to accomplish; they're two orthogonal statements, so to speak. It's just that the auto-grading in particular is rather impossible; other notebook deployment possibilities which are sub-impossible can still span the range between "we'll trivially do that ASAP" and "that's a lot of work." Adding interactive portions to complex notebooks which auto-save, live, per-account and per-lesson, then are shown to each respective user each time seems like it falls closer to the latter to me—it would require modifying the framework such that the exercises aren't static notebooks, but auto-saving notebooks which display on a per-account basis.

...and that's excluding any responsiveness issues that might arise from having users not only typing into notebooks with complex 3D graphics and Manipulates and so on, but trying to generate their own such objects and save them to their own cloud copies of the notebooks. Then we have to worry about users making these objects in these notebooks but not making them in a way that runs well on cloud, and the support tickets that would be generated from people who have barely-functional things in their notebooks... I just mean to say that there are a lot of moving parts here that may not be fully visible to the end user. (Which is as it should be, but it's worth keeping in mind when imagining that something might be easy to implement!)

Hi Phil,

The problem with the separate scratch notebook is that it provides no way to store the student's work in context.

I agree with you. My solution is to copy/paste the exercises into notebooks on my desktop, and then program my solutions/answers into these. Keyboard shortcuts in the desktop version make programming so much faster and these are unavailable in the online framework. In the end, it's easier to retrieve my work on my device, organized to my liking.

For those without access to the desktop version, there is always the Wolfram Cloud: use a free account to create notebooks, save to the desktop, and access with the free Wolfram Player later. It wouldn't take much to have two browser tabs open in a vertically split screen, one with exercises or quizzes, and another with the Wolfram Cloud notebook.

Maybe one of these solutions could work for you. The second option is especially good because it makes every part of these study groups cost nothing to participants. It's so hard to beat.

Hi Phil—I recall Feynman using that, yes :). I can only say that we will look into trying to implement this functionality, but can't offer any promises. As for testers, it's basically us on the Wolfram U team testing all of this stuff out. Admittedly, I'm not sure that we considered the perspective of someone new with no local installation that wanted everything in one editable auto-saved notebook per exercise set. It might be that the difficulties I mentioned—and others that I did not—precluded that consideration, or it might be that I personally think that these lessons and exercises all work best locally. (It's hard to say after the fact.)

If or when this fictitious student/man (person) form Mars has a problem, a) the clever student, say a student from, oohhh, let’s say from MIT [...]

"Man from Mars" refers to someone who is a complete beginner with something -- not someone who has gone through several Wolfram U courses (like me). Why do you presume that someone with essentially zero background would guess there was some way to save a scratch notebook?

the less nerdy but willing-to-ask learner would be helped by the very friendly, capable, and ever willing-to-help Wolfram folks

Except that most are not willing-to-ask. Someone won't ask how to do something if they don't even know that it's possible. I "SAVE NOTEBOOK" button in the framework would be a Captain Obvious way to inform students that it was possible to save a notebook. Without an explicit bread crumb like that, you will only have a tiny percentage of students figuring out something like that. They wouldn't even know to ask the question.

Hi Juan—yes, there's clearly an issue with this question. I'll modify it such that it makes sense and have it redeployed.

That is indeed correct, James—thanks for the catch!

Hi Juan—thank you for your kind comments. In that exercise, the idea is that the function is a product of one function of x and y and another function of z. This means that you can find where the whole function is continuous in the x-y plane by looking at that first function, find out where it's continuous "out of the plane" (i.e. where z != 0) by looking at the second function, and then find the full 3D region where the whole function is continuous by taking the intersection of these two regions. ("Join" is a little imprecise there, I agree. I will update the notebook now to specify "intersection".)

For example, imagine a situation where f(x,y,z) = sqrt(x^2 + y^2 - 1) ln(1-z).

• You can say that g(x,y) = sqrt(x^2 + y^2 - 1) and h(z) = ln(1-z), so that f(x,y,z) = g(x,y) h(z).
• g(x,y) is continuous as long as its argument is non-negative, so you need that x^2 + y^2 - 1 ≥ 0, or x^2 + y^2 ≥ 1. This corresponds to the whole plane with the unit circle centered at the origin cut out.
• h(z) is continous as long as its argument is positive, so you need 1 - z > 0, or z < 1.
• In total, f(x,y,z) must be continuous only where x^2 + y^2 ≥ 1 and z < 1 at the same time. That looks like:

You're welcome, of course! Maybe that's how I had intended it, but your comment made me realize that it was a bit unclear, so I'm glad that I could update it.

Thanks John—this is also fixed now.

If you have the link, you should be able to access it. You should get the link in your reminder emails each day, and it's of course put up in the chat during each course. If the link isn't working for you, please email us (with the link) at wolfram-u at wolfram dot com and we'll figure out what's going on.

That's surprising to me—I've entered the preview room for tomorrow's lecture and the link is already pre-populated. I recall seeing a sticky link there in today's class, though I didn't need to click it (for obvious reasons). That link should be pre-populated for the entire study group series, so I don't see any reason that it shouldn't have been there on Friday or today. Also, that link works when I'm fully logged out and in incognito mode, so there should be no permissions issues.

I'll keep an eye out tomorrow, but something seems fishy!

@Arben Kalziqi , here's a screenshot 3 minutes before the course is scheduled to start today, September 19, 2023:

The community forum link is noted, but there is no link for the download blob. Something is unhappy in the workflow....

Here's the long answer: "We used to explicitly post the URL for the download materials and the community forum URL in the Chat panel. We have changed this: while we still explicitly note the URL of the community forum, we now place the DOWNLOAD link behind a label. There are good reasons for this change! Sorry for any confusion this change may have caused." :)

Hi Tom—it's in the same place for all DSGs, at the top of the chat panel in a light cyan color. You can see it explicitly circled in red in the screenshot within this thread.

@tom pearce , e-mails from the "Wolfram U Team" should have the URL of the code blob. I just verified: the e-mail sent at 11AM EDT this (Thursday) morning definitely has the link the code blob. I recommend keeping all e-mails from the "Wolfram U Team" in your e-mail archive.

The e-mail sent out at 5:34 PM EDT today (Thursday) noting that today's video was available for viewing has the URL for the beta version of the interactive framework for the course. All videos and exercises are accessible from that.

I believe that each e-mail from "Wolfram U Team" should have both of these URLs, but each e-mail today was missing one of the links. The links do not change; once you have both of them, you should be fine.

The links that Arben mentioned in the BigMarker presentation are only viewable (and accessible) during the live presentation of the course. AFAICT, they are not available in the replay of the course. But you should be good if you kept the e-mails from "Wolfram U Team" today. That should get you running tonight.

Hello,

I notice on Exercise 1 in the notebook Lesson 8 Exercises.nb the first entry is 1/(x-1) instead of 1/(t-1)

Regards, John Burgers

Arben, As I suggested in the Q & A, I would like to see an osculating sphere added to the 3D graph under the Unit Binormal Vector section. Does the osculating sphere contain the osculating circle? Do they have the same radius?

Online References: (1) https://mathworld.wolfram.com/OsculatingSphere.html (2) Osculating Spheres of Space Curves - Brown University → https://www.math.brown.edu/tbanchof/balt/ma106/lab4.html?dtext46.html

DG Textbook References: (3) E. Kreyszig, “Differential Geometry”, Dover (1991) pp. 54-55; (4) R.S. Millman, G.D. Parker, "Elements of differential geometry", Prentice-Hall (1979) pp. 39; (5) D.J. Struik, "Lectures on classical differential geometry", Dover (1988) pp. 25

Video: (6) Osculating Plane, Circle, and Sphere for a Space Curve → https://www.youtube.com/watch?v=08C4U8x8F04

One of my wife’s favorite shrubs is a spiral cypress or boxwood topiary. I found pictures of a couple and used them to estimate curves that would approximate their shapes. I think a curve something like this would make a more satisfying animation of an osculating sphere moving along its curve than would a helix where the osculating sphere would not change size.

Arben, It is Fabulous! Thank you so much for your efforts on this!

This isn't a problem, I just thought it was hilarious. Before I evaluated the code in one of the lesson notebooks, this was the error:

It would be funny if "graphicsStuff" actually became a thing!

Coming in Wolfram Language v14.0: graphicsStuff!

Are you testing us? That would have to be big-g GraphicsStuff. :)

Cool, thanks, Arben! [star eyes emoji]

(this message board does not like emojis)

Hi Carl—it should be there now. Not sure why it was late; there can be a little variability with BigMarker on occasion.

Hi Lori—I actually have been getting your first problem myself and had assumed that it was just something in my installation as I have a custom magnification set! Interesting; I'll look into this. And yes, the solution cells for the exercises are meant to be open. You are of course free to try them yourself first.

A little secret useful thing is that if you hold—on a Mac, at least—Option while clicking a cell bracket, it'll select all cells of that type. So you can hold opt, click the cell bracket for any Solution cell, then right click the cell and click "close all subgroups". It will then collapse all of the Solution cells.

Thanks for your feedback, Lori. I'll talk to the team about perhaps uploading and deploying these notebooks with the Solutions cells collapsed by default.

Indeed :). I believe the exercise notebooks which will be deployed to the full course framework will have this change made when the course goes live.

Thanks Thomas; this is fixed in the latest version of the exercises.

Hi Michael. Could you give me an example of what you're seeing? I think that most things should work well enough in 12, but I don't have a copy installed currently.

Holy moly! I'll ask internally about this.

Thanks for the follow-up. I tested out a bunch of the lessons and they all open that way---the words are vertical. I wonder if there's a simple setting I can adjust to view them as intended....

By the way, I tried on both Windows 10 and MacOs 11.7.8 (two different computers) and had the same result.

Yes--I thought about the magnification too. Changing that did not make a difference.

Thanks Carl. I believe we've found the issue and will be pushing out an update.

Hi, Michael. I'm running Mathematica 12.2, and I'm having no problem using the notebooks of this course. I've never having any issue downloading/running notebooks from Wolfram Research. I suspect your problem is not related to the version of Mathematica you're running.

I do get "Why the bleep" warnings -- a warning that I'm running an earlier version of Mathematica that the notebooks were saved in. I just ignore that error and use the notebook:

No problem, Charles. I think as a fellow physicist, you'll find a lot to appreciate here.

Trivial, but there is a typo in Lesson 3, Exercise 3:

vec1 = {4, 4 - 1};

should be

vec1 = {4, 4, -1};

which requires one change further down:

Graphics[{Red, Arrow[{{0, 0}, vec1}], Blue, 
  Arrow[{{0, 0}, unitVec1}]}]

should be:

Graphics3D[{Red, Arrow[{{0, 0, 0}, vec1}], Blue, 
  Arrow[{{0, 0, 0}, unitVec1}]}]