  # stevem

Members

23

10 Neutral

• Rank
Quark

## Profile Information

• Location
UK
1. It would be easier to say $\frac{x^2 + 4x}{2y^2 + y} = 6 \implies x^2+4x=6\left(2y^2+y \right)$ Hence $2x+4=6\left(4y\frac{dy}{dx}+\frac{dy}{dx}\right)=6\frac{dy}{dx}(4y+1)$ So $\frac{dy}{dx}=\frac{2x+4}{6(4y+1)}$
2. And if you want a free equation editor to generate the LaTeX code for you, have a look at TeXAide
3. I agree with Dave that the most sensible thing to do is install LaTeX on your system. But if you really don't want to do that you'll need a LaTeX emulator like mimeTeX. Assuming that you are using Windows, then download mimetex.exe from there. You can then call mimetex from the DOS command line, but the simplest thing to do is write a batch file: @echo off mimetex -f mimetemp.tex -e mimetemp.gif Save your latex code in mimetemp.tex, run the batch file and the image will appear as mimetemp.gif.
4. ## I'd be interested in seeing various proofs that 0^0=1, if anyone has any

5. ## I'd be interested in seeing various proofs that 0^0=1, if anyone has any

! wasn't worried about that because everyone happily agrees that 0!=1 because, for example, it makes a nice formula for $e^x$. The problem that really concerns me is that you have used $x^0$ for any x. But you need to define $x^0$ for x=0 before you can use it in the formula for $e^x$ and that is where you get the circular argument. What you can do is say: I like the formula $e^x = \frac{x^0}{0!}+\frac{x^1}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^n}{n!}+...$ so I will define $0^0=1$ and $0!=1$ and then I can use this version rather than the one that starts 1 + x. The only snag is that not everyone would agree with this definition.
6. ## I'd be interested in seeing various proofs that 0^0=1, if anyone has any

Unfortunately, your argument is circular because of the way you've defined $e^x$. If you define it as $e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^n}{n!}+...$ then your argument no longer works.
7. ## I'd be interested in seeing various proofs that 0^0=1, if anyone has any

How do you get that? I repeat what I said - there is no proof that $0^0=1$. If you wish to use that definition then feel free to do so; it's just that not everyone likes that definition, whereas 0!=1 is, as Matt says, used by everyone, though, again you are free to leave it undefined.
8. ## I'd be interested in seeing various proofs that 0^0=1, if anyone has any

There is no proof. You can define $0^0$ to be whatever proves convenient. I prefer to say it is indeterminate because $\lim_{x \to 0^+}x^0=1$ but $\lim_{x \to 0^+}0^x=0$ but others disagree. See http://mathforum.org/dr.math/faq/faq.0.to.0.power.html for a discussion of this topic.
9. If you want arrows you can have them! $\ce{-> ->[\alpha] <- <=> <=>> <-> v ^}$ and even $\ce{Zn^2+ <=>[\ce{+ 2OH-}][\ce{+ 2H+}] {\underset{\text{amphoteres Hydroxid}}{\ce{Zn(OH)2 v}}} <=>[\ce{+ 2OH-}][\ce{+ 2H+}]{\underset{\text{Hydroxozikat}}{\cf{[Zn(OH)4]^2-}}}}$ There's more in the mhchem documentation (mhchem provides the Chemistry $\LaTeX$ here)
10. As Matt said and you will find that the entries will be aligned - in this case in the centre because of the c in \begin{array}{cccc} Unfortunately there's a bug in the $\LaTeX$ implementation on the forum so it shows wrongly in $A \times B = \left| \begin{array}{ccc} \hat i & \hat j & \hat k \\ A_x & A_y & A_z \\ B_x & B_y & B_z \\ \end{array} \right|$ but in normal $\LaTeX$ use it will do what you wanted.
11. It's not possible to solve this algebraically, if by that you mean using the usual 'elementary' functions. You can solve it using a computer package because it reduces to a non-elementary function called the LambertW function. This function is useful precisely because it allows one to solve these sorts of equations. It is a function that is one step beyond logs in the same way as logs are one step beyond polynomials. If we start with $3x+lnx=3$ then $e^{3-3x}=x$ which can be rewritten as $3e^3=3xe^{3x}$ or $3e^3=we^w$ where $w=3x$ w is the value of the LambertW function, which, as I've said isn't on a calculator, but is in most computer maths packages and you find $x=\frac{1}{3}\text{LambertW}(3e^3)$ which gives 1. Don't understand or (hopefully) want to learn more? Then there's an excellent beginners guide to LambertW (including why it's called that) in the American Scientist at Why W?
12. Not sure which you can't prove - that $\lambda$ is an injection or that it is a homomorphism or both so here is how to start for both these steps: 1. Let $a,b \in G$ and $\lambda_a=\lambda_b$. Apply these 2 functions $\lambda_a,\lambda_b$ to the identity of G and what can you deduce? 2. The binary operation in Sym(G) is composition of functions so for any g in G $\left(\lambda_a \circ \lambda_b\right)(g)=\lambda_a\left(\lambda_b(g) \right)=\lambda_a\left(bg\right) = \dots$ Then use the fact that 2 functions $\alpha,\beta$ in Sym(G) are equal if and only if $\alpha(g)=\beta(g)$ for all $g \in G$ Finally, apply what you have shown to give results about $\lambda$ using $\lambda(g)=\lambda_g$ Hope that helps.
13. If n=2 then $\left(T^{-1}AT\right)^2=T^{-1}ATT^{-1}AT=T^{-1}AIAT=T^{-1}A^{2}T$ This should help you to construct a proof by induction.
14. Just realised that you can get an anthology of Manifold in: Seven Years of Manifold: 1968-80 Ian Stewart (Editor), John Jaworski (Editor) Publisher: Shiva Pub, Nantwich ISBN: 0906812070
15. He did indeed. One must be more careful in stating results; I should have said that Theorem n! is even for all integers n > 1 Oh and, this is for Dave's benefit, the proof was thought up in the old Maths Institute at Warwick before Ian Stewart showed us how to do maths humour properly in a magazine called Manifold. If there are any still lying around somewhere at Warwick they are worth reading. My favourite cartoon in that magazine had a picture like this
×