• Mathematician Chat v. 3.999...
    1,232 replies, posted
For any natural number k, k! is the product of all natural numbers less than or equal to k. The gamma function extends this idea in a nice way so that expressions like (1/2)! and (i-1)! have meaning. What's important is that the gamma function has no closed form, so it can only be written as an infinite product, infinite series or nonelementary improper integral. These result in really complicated ways to write integers of the form (k-1)! like 1, 2, 6, 24, 120... In the example that 41 posted, he used the infinite product definition to write 6. The large Π means that you multiply each of the infinitely many terms, in the same vein as a Σ. So, he's saying that (1/4)*(16/5)*(81/48)*... = 6
What's the flow through the surface which is connected by the points (1,0,0), (0,1,0) and (0,0,1) if the field is F=(x-y+xy, -2x+y, xz)?
I'm a little confused by the terminology of the question. You're looking for the flux through the convex hull of those points?
I think that's what I want, not really familiar with the terminology. [editline]2nd March 2015[/editline] After doing some research, yes. [editline]2nd March 2015[/editline] Nevermind, I solved it.
How do you properly calculate volumes of a region bounded by functions and in which a solid is created so that the region is its base and perpendicular to the x or y axis are shapes (squares, semicircles)?
[QUOTE=Niven;47240825]How do you properly calculate volumes of a region bounded by functions and in which a solid is created so that the region is its base and perpendicular to the x or y axis are shapes (squares, semicircles)?[/QUOTE] By...a double integral over the (presumably) planar region of the base?
So I just read a mathoverflow post that pointed out that you can find lim x->0 sin(x)/x without using L'Hopital's rule. It's just the derivative of sin(x) at x = 0. My grade in calc 1 was over 100% but now I'm convinced I should never have passed.
[QUOTE=JohnnyMo1;47246844]So I just read a mathoverflow post that pointed out that you can find lim x->0 sin(x)/x without using L'Hopital's rule. It's just the derivative of sin(x) at x = 0.[/QUOTE] But uhh, isn't that the same thing? Because when you use L'Hopital's rule on sin(x)/x, you just happen to be left with only the derivative of sin(x). As in : [IMG]http://i.imgur.com/TZ3WKaR.png[/IMG]
[QUOTE=Block;47248513]But uhh, isn't that the same thing? Because when you use L'Hopital's rule on sin(x)/x, you just happen to be left with only the derivative of sin(x). As in : [IMG]http://i.imgur.com/TZ3WKaR.png[/IMG][/QUOTE] Sure, but it can be made much easier than that. lim x->0 sin(x)/x is just lim h->0 sin(0 + h)/h. [editline]3rd March 2015[/editline] Of course you'll arrive at the same answer (or something is very wrong), I just never noticed you could do it straight from the definition with nothing but a relabeling (and even that's not really necessary).
[QUOTE=JohnnyMo1;47246844]So I just read a mathoverflow post that pointed out that you can find lim x->0 sin(x)/x without using L'Hopital's rule. It's just the derivative of sin(x) at x = 0. My grade in calc 1 was over 100% but now I'm convinced I should never have passed.[/QUOTE] Wouldn't you already need to know the derivative of sin in order to actually find a value for the limit?
[QUOTE=Block;47248513]But uhh, isn't that the same thing? Because when you use L'Hopital's rule on sin(x)/x, you just happen to be left with only the derivative of sin(x). As in : [IMG]http://i.imgur.com/TZ3WKaR.png[/IMG][/QUOTE] can anyone tell me what lim stands for? (I do have an idea what sin(x) and cos(x) are)
limit [code] lim f(x) x->0 [/code] Means you imagine taking points closer and closer to 0, and for every point you choose, you evaluate the function, f, that's written to the right of lim. Sometimes, as you get closer to 0, the corresponding values of the function will approach some number. This number is denoted as above.
[QUOTE=Krinkels;47251824]Wouldn't you already need to know the derivative of sin in order to actually find a value for the limit?[/QUOTE] Sure, but what's the problem with that?
Wouldn't that be circular reasoning? [editline]4th March 2015[/editline] I mean it works so you can use it to remember it but would it mathematically sound? (might be way out of line here, but i think my teacher said something like this)
Why is that circular? Nothing precludes you from knowing the derivative of sin(x) before you try to evaluate lim x->0 sin(x)/x. In fact, you have to if you want to do it by L'Hopital's rule too! [editline]3rd March 2015[/editline] Like, is it cheating if I say I can evaluate x*x^2 without the product rule because I know that x*x^2 is x^3 and I already know the rule for polynomials?
[QUOTE=JohnnyMo1;47246844]So I just read a mathoverflow post that pointed out that you can find lim x->0 sin(x)/x without using L'Hopital's rule. It's just the derivative of sin(x) at x = 0. My grade in calc 1 was over 100% but now I'm convinced I should never have passed.[/QUOTE] The first principles derivation was the way it was presented to me in high school. which ghetto-ass school did you go to?
That's not really common based on the discussion I saw on mathoverflow.
[QUOTE=JohnnyMo1;47253855]Why is that circular? Nothing precludes you from knowing the derivative of sin(x) before you try to evaluate lim x->0 sin(x)/x. In fact, you have to if you want to do it by L'Hopital's rule too! [editline]3rd March 2015[/editline] Like, is it cheating if I say I can evaluate x*x^2 without the product rule because I know that x*x^2 is x^3 and I already know the rule for polynomials?[/QUOTE] I mean you use the sin h/h limit when finding the derivative of sin(x) so it already needed to be known in the first place? Please correct me if I'm wrong. [editline]4th March 2015[/editline] Okay, sure, you can find it in different ways, but that's one way.
[QUOTE=JohanGS;47256878]I mean you use the sin h/h limit when finding the derivative of sin(x) so it already needed to be known in the first place? Please correct me if I'm wrong. [editline]4th March 2015[/editline] Okay, sure, you can find it in different ways, but that's one way.[/QUOTE] You can, but as you said there are other ways. I'm not defining anything here, so it's really just a method to evaluate.
Hi, bit of a stupid question here, but I was doing a few inductive proofs for an assignment and I came across a basic algebra problem. [img]http://puu.sh/goeoW/87fc0cec57.jpg[/img] How is the second to last step simplified into the last step? I feel so dumb right now; I know that it [I]does[/I]​ simplify, but I forget the steps how to simplify into that.
Let A = (-7)^(k+1) Note that -7A = (-7)^(k+2) The last bit of the second to last step is 8A/4. Now the expression becomes: (1 + 8A - A)/4 = (1 + 7A)/4 = (1 - (-7A))/4
OOOOOOH. Of course! I knew everything up until the last step, but something in my mind told me "no, adding 8(-7)^k+1 won't give me the answer I want..." I should stop jumping to conclusions like that. Thank you!
I really find mathematics an interesting subject, but when I was 13 years old I had cancer and missed out on a lot of the foundational math. (which was taught in the 6th-7th grade curriculum) which is necessary for building on with more advanced courses. Right now I'm in college Trigonometry and I find it insanely interesting. The problem is that those foundational holes in my knowledge really hurt me in some aspects. I am currently running through the entirety of Khan Academy's math courses from pre-k to Calculus and beyond so that I can gain a better understand. But since I am a computer science major I am really going to need to be able to do the extremely advanced math courses eventually. I can learn anything but it just takes me time, Does anyone have any recommended resources or advice which I could use to help me flesh out my knowledge/understand/love for math?
[QUOTE=Llamalord;47267742]I really find mathematics an interesting subject, but when I was 13 years old I had cancer and missed out on a lot of the foundational math. (which was taught in the 6th-7th grade curriculum) which is necessary for building on with more advanced courses. Right now I'm in college Trigonometry and I find it insanely interesting. The problem is that those foundational holes in my knowledge really hurt me in some aspects. I am currently running through the entirety of Khan Academy's math courses from pre-k to Calculus and beyond so that I can gain a better understand. But since I am a computer science major I am really going to need to be able to do the extremely advanced math courses eventually. I can learn anything but it just takes me time, Does anyone have any recommended resources or advice which I could use to help me flesh out my knowledge/understand/love for math?[/QUOTE] This is something that I do, but if you ever find yourself wondering where you'll ever use something in mathematics, I just start brainstorming what applications that can be used for them to motivate myself to keep going
Hey, can anyone help me with this proof? [img]http://puu.sh/goYqK/86840fed1c.png[/img] I don't quite know what to do with the coefficients after simplifying the terms to the inductive hypothesis.
First off, that shouldn't be 5^(-2). It's 5^2. What's important is that 25 - 4 = 21. To be more specific, let 21i = 4^(k+1) + 5^(2k-1) for some integer i. RHS = 4(4^(k+1) + 5^(2k-1)) + 21(5^(2k-1)) = 4(21i) + 21(5^(2k-1)) = 21(4i + 5^(2k-1))
You're right, just me being stupid with numbers again. Thank you! It seems that the further I get into more advanced mathematics, I make even more algebraic mistakes. WHEN WILL IT STOP.
Can someone give a decent explanation to what an eigenvector and eigenvalue is? They're stated as characteristic roots of systems in my engineering classes, but other than that I don't know what they are and why they are be considered characteristic roots.
[QUOTE=gangstadiddle;47272124]Can someone give a decent explanation to what an eigenvector and eigenvalue is? They're stated as characteristic roots of systems in my engineering classes, but other than that I don't know what they are and why they are be considered characteristic roots.[/QUOTE] That's a pretty bad explanation from your class, imo. A square matrix is a concrete representation of a linear transformation, i.e. a function from a vector space to itself. The eigenvectors of a linear transformation (or matrix) are the vectors that are unchanged after the transformation except for possibly being scaled by a constant. The eigenvalue associated with an eigenvector is the amount the vector is scaled by. Wikipedia's picture is a good example: [url]http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#mediaviewer/File:Mona_Lisa_eigenvector_grid.png[/url] Blue is an eigenvector of the transformation with eigenvalue 1, red is not.
What does 'characteristic roots of systems' mean?
Sorry, you need to Log In to post a reply to this thread.