And is seemingly more intuitive than infinite decimals, I think.
At least for me.
Especially regards to the rest of analysis. Introduction by infinite decimals would result in setting up students thinking this is some sort of central topic to analysis.
It's not, though.
Whereas IVT has the benefit of being both intuitive, and a defining result in analysis.
@RyanUnger maybe that makes more sense as to why I would express skepticism about using infinite decimals?
You'd have to phrase it more in terms of the sequences, but then I am afraid it would lose the simplicity.
And would be more of a "here, follow me blindly down this trail"
High school kids are fine with infinite decimals. We're talking intuition, not rigor, here.
We're not trying to define/verify the field axioms using infinite decimals.
What's more, in high school AP courses, there's nothing resembling a justification of the IVP. To do anything rigorous, you have to introduce the LUB axiom or, equivalently, the monotone convergence axiom.
When I taught the Spivak rigorous calculus course 15 times or whatever it was, I finessed any sort of construction of the reals and let them keep their intuition, and just enunciated the LUB axiom for proofs.
Actually, no one ever asked me in all those years.
My biggest problem in the Spivak course with (mostly) freshmen was keeping them from dropping out of it too fast. I did my best to get to actual differentiation faster than the book does, because otherwise I knew I'd lose over half my class.
The main reason Apostol started his book with integrals was to contend with the know-it-all students who'd seen calculus in high school, I firmly believe. Of course, there's historical precedent, too.
I don't know where you went to school. There used to be Analysis 2 most places, but not enough student interest. That was true at UGA, because the potential Analysis 2 students had already had my multivariable course, where we did most of that stuff actually.
Sad fact: The preponderance of calculus books are written by people who are non-expert in geometry or other things where one truly needs to grok multivariable calculus.
That's certainly true of Thomas. It fell apart badly in that part.
But the calculus book was amusing: He proved the max/min test in 2 variables by restricting to lines (which doesn't work) and then later had an exercise which showed that wasn't valid. I believe it was that book.
I would hope that someone who writes specifically a multivariable book is better than people who write the huge tomes that have everything. Certainly Marsden/Tromba was written by people with a feel for the material.
I'm trying to remember the first course I saw Sobolev spaces. Probably seminar in grad school. Maybe the Hörmander SCV course I took as a senior. Yeah, probably that.
the difficult thing, as I said, is that there's a difference between the history of the formalization, i.e. when it first was called that in the literature
@RyanUnger the idea of a "continuum", I can sort of get behind. The problem is that I don't think it is in any way obvious what this ought to be.
You will want to be very careful with how you go about it. As it stands, the rationals are a "continuum" in a naive sense that you can find a rational between any two rationals.
And so your approach with an "it's like a continuum" approach would have to specifically approach it with regards to completeness.
And then I think it gets a little heavy then. As in, "why do we phrase it this way, why not stop?".
And whenever I see someone ask something like that, that's when the pedagogue brings out the IVT.
That even though the rationals and the reals both "look" the same with respect to the geometric "number line", there still remain gaps.
E.G., a graph like x^2 - 2 manages to slip by without actually touching the rational number line.
If for a surface $f_x = 0$ and $f_y = 0$, Is finding a curve such directional derivative along the tangent of the curve is $0$ all the time, sufficient to show that the given point is saddle point?
Oh, perhaps. But I have read in elementary multivariable texts that one have to check the sign of the determinant to check the nature of critical points.
I am currently reading this arxiv.org/pdf/0903.5400.pdf , and it defines in defintion 2.2 , that a point can be saddle is two paths (say $t_1$ and $t_2$) intersect tranversally and there is local maxima along the path of $t_1$ and local minima along the path of $t_2$
But it seem to me, there can be an alternative definition( I am open to accept it wrong) that if one is able to find a curve that the directional derivative along the tangent doesn't change, and $f_x =0$ and $f_y = 0$ then the point would be saddle
$f_x = 4x^3 - 4y^2 - 4x$ taking $y = 0$, $f_x = x^3 - x$, which is positive of negative value of x and negative for positive of x. $f_y = -4y^3 - 8xy$ for x = 0, $f_y = -4y^3$ which is positive of negative value of y and negative for positive of y.
I believe one can conclude but that that along those particular paths, there is relative maxima.
By the way, what are the possible cases when $ \nabla f = \vec 0$ minima, maxima, saddle point or something else? . Like in single variable it can be minima, maxima or nothing. Do that "nothingness" creep here ?
I am currently reading this https://arxiv.org/pdf/0903.5400.pdf , and it defines in defintion 2.2 , that a point can be saddle is two paths (say $t_1$ and $t_2$) intersect tranversally and there is local maxima along the path of $t_1$ and local minima along the path of $t_2$
But I don't know for which vector space it works as transformation, and what would be the meaning of the transformation. I have just taught myself to find the nature of critical points by looking the sign of determinant of the matrix.
Ich frag so "Okey letztes jahr bin ich schon akzeptiert worden und hatte gar kein Zeugnis dabei weil ich sie noch nicht hatte" und die Frau so "Ja das kann man nachreichen" und ich so "Okay also den ganzen Abschluss kann man nachreichen aber einen Stempel nicht"
