I'm currently working through Schutz's "A first course in general relativity" as a preparation for a graduate course in General Relativity based on Carroll's notes. I'm a little confused about vectors, one-forms and gradients.
Schutz says the gradient is not a vector but a one-form, because...
Let n \ \epsilon \ \mathbb{N}, \ n \geq 2 and p, \ q \ \epsilon \ \mathbb{R}. Consider f: \ \mathbb{R} \ \rightarrow \ \mathbb{R} defined by f(x)=x^{n}+px+q.
Suppose n is odd, prove that f has at least one and at most three real roots.
I thought about the intermediate value theorem for...
I knew the definition of a Cauchy sequence, but I still can't find the solution.
I think I can't, given a certain n and f_n I can find f_{n+1} and I see that once n approaches infinity f_n becomes either -1 or 1, but I don't know how to work from there. In fact this is all quite new to me.
We consider the space C^0 ([-1,1]) of continuous functions from [-1,1] to \mathbb{R} supplied with the following norm:
||f||_1 = \int_{-1}^{1} |f(x)| dx
a. Show that ||.||_1 defines indeed a norm.
b. Show that the sequence of functions (f_n), where
\begin{align*}
f_n(x) &= -1...
When do you consider time being wasted?
As for me, I consider spending 8 years proving one of the most challenging mathematical problems ever not as much a waste of time as spending 8 years watching television.
Actually, I think the only one you can judge wether someone wasted his time or...
There isn't one objective answer to the question "is it worth studying mathematics", it depends on your definition of value. If you aim at making much money or serving your country/the human race in the best possible way, I don't think mathematics is the way to go. However, if you're looking for...
\frac {dy} {dx} = x y^2 - y
I used Mathematica's DSolve function and found the correct answer:
y(x) = \frac {1} {1 + x + C e^{x}}
However, I don't have any idea what method to use to solve it with pencil and paper...