Spivak Definition and 161 Threads

Please forgive any stupid mistakes I've made. On p.85, 4-5: If c: [0,1] \rightarrow (R^n)^n is continuous and each (c^1(t),c^2(t),...,c^n(t)) is a basis for R^n , prove that |c^1(0),...,c^n(0)| = |c^1(1),...,c^n(1)| . Maybe I'm missing something obvious, but doesn't c(t) =...

Prove the following: x^n - y^n = (x-y)(x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1}) Hmm ... I factored x then -y and came out with: x^{n}+...x^{2}y^{n-2}-x^{n-2}y^{2}...-y^{n} Argh. What's up with the middle part? I'm not sure where to go from here.

I just got Spivak's book a couple days ago, I've flipped through it and was looking through the Index. Does Spivak cover Related Rates or Optimization? I did not find it at all, so I am curious and wondering if it's under some other name. Thanks!

Must be something missing from my repertoire-- Spivak got me in Chapter 1! :-) Trying to find all x that satisfy: x + 3^x < 4 I've tried everything I can think of. Here are a few lines I've run down, to no avail: x + 3^x < 4 \Rightarrow e^{x+3^x} < e^4 \Rightarrow e^x \cdot...

I honestly enjoyed using James Stewart (yes heathen) when I did calc I, II, III and now these forums are buzzing with Spivak - as if he's the new Giancoli/Feynman of math textbooks whats the big deal? Oh - and i did check out the Amazon.com reviews - surprisingly the one bad review I read...

Hi. I have a question on proof of proposition 2 in chater 7 Spivak volume2. In the proof, he says that the n-dimensional distribution \Delta_{p}=\bigcap^{n}_{i,j=1}ker\Lambda^{i}_{j}(p) in R^(n+n^2) is integrable. Could anyone explain why it is an n dimensional distribution? Thanks.

I am reading a Vol2 of geometry book by Spivak. On page 220-221 he says that: "Notice that the possibility of defining covariant derivatives depends only on the equation..." The equation is some equality involving Christoffel Symbol. If anyone has this book, could you explain why what he...

I own both volumes of Apostol, but I must confess that when I bought them I had never heard of Spivak's book. I think it's pretty much a toss up for me now. What about you guys?

Suppose f(a) = g(a) = h(a) and f(x) <= g(x) <= (x) for all x Prove g(x) is differentible and that f'(a) = g'(a) = h'(a). So.. I need to prove that the following limit exists: lim h -->0 (g(x+h) - g(x)) / h but how can i use the fact that f(x) <= g(x) <= (x) for all x? Thanks

Does anyone know if there's worked out solution to the problems in spivak's calculus on manifolds? It's awfully easy to get stuck in the problems and for some of them I don't even know where to start... Also, if there isn't any, any good problem and 'SOLUTION' source for analysis on manifolds...

1) Find a function, f(x) which is discontinuous at 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4} ..., but continuous at any other points. Solution (I have come across, probably wrong and a half): f(x) = { 1 for all real x; 0 for 1/x where x is natural numbers. Can anyone tell me the answer to...