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this post was submitted on 13 Sep 2026
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TechTakes
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Thanks. So far, the questions I missed on the first exam were mainly due to sign errors when calculating determinants for cross-products, so that's a simple enough fix I suppose. I've seen some of the material before, as halfway through the last time I took Calc 3 was when the Covid lockdowns hit. Even though classes are back in-person, this current instructor is still presenting most of the material in static notes and lecture videos online, so it's still a tough adaptation for me. We're using the James Stewart text that is pretty common everywhere, though thankfully the instructor has set it up so I don't have to pay for ebook access yet again.
I picked up the famous Spivak text a few years ago, and am only just now working up the courage to crack it open. I've had a few years' gap in between classes, so reviewing the earlier material with more proofs involved is probably a worthwhile exercise in any event. I'm given to understand that particular text is something of an on-ramp to real analysis. I've been considering trying for a math minor once I transfer for engineering, and it seems like real and/or numerical analysis would be two of the prime upper-division classes for that.
Yeah, there's not much to be done about sign errors, other than practicing problems until you reach the point where you always make an even number of them.
You might not need it for this course, but cross products got easier for me once I learned my way around the Levi-Civita_symbol and the Einstein summation convention. This is covered in chapter 10 of Boas' book; she cites Evett (1966).
Real analysis is definitely the kind of course that can benefit from an on-ramp. It's often a student's first encounter with proofs that involve a lot of machinery: theorem after theorem of epsilon-delta business to establish something that you didn't doubt in the first place. I think it's easy to get lost in the details there, particularly if care isn't taken to explain what confusions these methods were devised to avoid.
The OpenStax textbook covers the development of the general equations of the various conic sections. If the Stewart book covers that in such detail, it must be jammed into the appendices or exercises someplace. I'll have to find the time to go over that, at least; I still don't have a great intuition for hyperbolas.
Yes, this looks like what I need to understand the negative j term in the cross product formula. Cambridge looks to have the 3rd edition of Boas' book up for download, so I have a copy now. We'll see how well I comprehend it, I feel like I'm missing the concept of index permutation to really understand the Wikipedia article.
I don't know how much traffic we'd get if we tried to cultivate a math discussion here. I did make an account on Mathstodon too and perhaps I should spend more time there. Tutoring at this level is sparse, at least at a community college. There's one tutor for my class available via Zoom, and the instructor himself is apparently teaching 5 classes when he's only under contract for three. If I signed up for 5 classes as a student, I'm sure I'd be setting off all sorts of alarm bells in the registration department, if they even let me go through with it.
The Levi-Civita symbol is, in programming lingo, a function that takes three arguments:
epsilon(i,j,k)wherei,jandkare integers that label the coordinate axes. Let 0 stand for the x axis, 1 for the y and 2 for the z. Thenepsilon(0,1,2)equals 1. If we swap the values of any two arguments, we flip the sign: e.g.,epsilon(1,0,2)evaluates to -1. Now, swap again:epsilon(1,2,0)must change sign again, so it evaluates to 1. But the sequence 1,2,0 is just the original sequence 0,1,2 read cyclically from a different starting point. In this way, we can work out what the value of theepsilonfunction is for any input. When the inputs are all different, the output is either 1 or -1, depending on whether the input sequence is "in order" or "in reverse order". Write the numbers 0, 1 and 2 around a circle. Read clockwise, and you get one output; read counterclockwise, and you get the other. If two inputs are the same, thenepsilonmust evaluate to zero, because, for example,epsilon(0,0,1)with the first two inputs swapped isepsilon(0,0,1)again, meaning that the output must be minus itself.