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Re: MATH2601 Linear Algebra/Group Theory Questions \text{Prove that if }V\text{ is a finite dimensional vector space and }W\text{ is a subspace of }V,\text{ then }W\text{ is also finite dimensional.}

Re: Several Variable Calculus \text{Suppose that }f:\mathbb{R}^n\to \mathbb{R}\text{ is such that the sets}\\ \{\textbf{x}\in \mathbb{R}^n:f(x)>d \}\\ \{\textbf{x}\in \mathbb{R}^n:f(x)<d \}\\ \text{are open for every }d\in \mathbb{R} \text{Prove that }f\text{ is continuous}

Re: HSC 2017 MX2 Marathon Just solve the quadratic-in-disguise

Re: HSC 2017 MX2 Integration Marathon (1) - Odd and even functions (2) - u=tan(x) (3) - u l-> a-u (4) - Addition of equations \text{Lemma L:}\\ \tan (A+B) = \frac{\tan A + \tan B}{1-\tan A \tan B}\\ \text{Let }A=\tan^{-1}a, \, B=\tan^{-1}b\text{ then}\\ \tan^{-1}a +...

Re: Announcement from BOSTES - significant change to calculus courses Sure, but it doesn't even mean much in 2U anyway. Rarely (I don't recall of any instance myself) do 2U questions require you to be worried about the order in which things are implied. The questions are designed so that...

Re: Several Variable Calculus Wait, I can't see it - Can't tell how a region cut off by the rectangular hyperbola is a ball

Re: Several Variable Calculus ...? These are homework questions I'm stuck on. I'd be too worried about getting a 0 mark for the course if these were assignment questions.

Re: Several Variable Calculus That is one beautiful trick. _____________________________________ \text{RTP: }\Omega= \{(x,y)\in \mathbb{R}^2: \, x^2-y^2<1 \}\text{ is open.} I definitely don't want to use the Lagrange multiplier theorem to find the closest point (x,y) to work off, so any...

Re: Several Variable Calculus There was a typo; it was meant to be a less than or equals to. I think InteGrand just ignored or figured I had typo'd and just used the correct inequality. ________________________________________________________________ \text{Prove that }\mathbb{Q}\text{ is not...

Yep

Re: Several Variable Calculus \text{Prove that }\lim_{(x,y)\to (0,a)}\frac{\sin (xy)}{x}=a \text{This result may be assumed: }\left| \frac{\sin a}{a}-1\right|=|a|^2, \, a\in \mathbb{R}

Re: MATH2601 Linear Algebra/Group Theory Questions It's basically this \begin{align*}x^2&=e\\ \implies (x^{-1}x)x&=x^{-1}e\\ \implies ex&=x^{-1}\\ \implies x&=x^{-1}\end{align*} Having used the associativity axiom and the definition of the identity element. Right-operating also works

Re: MATH2601 Linear Algebra/Group Theory Questions Just out of curiosity, what's an example of a vector space and/or a field which is defined in a way, that does not use the conventional means of addition and (scalar) multiplication?

Re: Several Variable Calculus Appears so counterintuitive though. I can't visualise what's going on here

Re: Statistics Very true. So grateful.

Re: Several Variable Calculus \text{Bit confused. How is it possible for }\lim_{x\to 0}\lim_{y\to 0}f(x,y)\\ \text{and the reverse order of limiting to not exist} \text{but then }\lim_{(x,y)\to (0,0)}f(x,y)\text{ exists}\\ \text{for }f(x,y)=(x+y)\sin \frac1x \sin \frac1y

Re: Statistics I don't know; InteGrand can do as he wants. But why are you asking me this?

Re: Statistics Quick question regarding probability spaces: \text{Is the convention to take }\mathbb{P}(A) \ge 0\text{ or }\mathbb{P}(A) > 0?

Re: Announcement from BOSTES - significant change to calculus courses I have a feeling 2U would get confused by logic. But the proofs in 2U tend to not really feature problems with logic Also it was a bit hard for me to grasp at the start as well. Discrete maths saved me.

Re: Several Variable Calculus Is it basically just this? \begin{align*}\text{Let }\textbf{x} &\in B_\delta (\textbf{a},\epsilon)\\ \implies \delta (\textbf{x},\textbf{a}) &< \epsilon \\ \implies c_1 \rho (\textbf{x},\textbf{a}) &< \epsilon\text{ for some }c_1\in \mathbb{R}^+\\ \implies \rho...