# Tag Info

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Yes, there is a way to profitably read mathematical proofs, but it takes time. Here is an excerpt from the "note to the reader" in an excellent topology book: "It is a basic principle in the study of mathematics, and one too seldom emphasized, that a proof is not really understood until the stage is reached at which one can grasp it as a whole and see it ...

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You want to find $P(X > 3\: \text{or}\: Y > 3)$, which is the same as $1 - P(X \le 3\: \text{and}\: Y \le 3)$. Since $X$ and $Y$ have the same distribution and are independent, $P(X \le 3\: \text{and}\: Y \le 3) = P(X \le 3)P(Y\le 3) = P(X \le 3)^2$. So you compute 1 - P(X \le 3)^2 = 1 - \left(\frac{1}{2} + \frac{1}{4} + \frac{1}{8}\right)^2 = 1 - ...

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One reason for defining things that way is that then the Fundamental Theorem of Calculus for piecewise continuously differentiable functions is immediate from FTC for continuously differentiable functions, without needing to mess around with limits. Like so: Say $f:[0,2]\to\mathbb R$ is continuous. Say $f$ is continuously differentiable except at $1$. Say ...

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For functions of a single real variable, it is very easy to deal with endpoints: just define the derivative as one-side limits. For functions of several real variables, everything get more difficult, since boundary points cannot always be described in a single way. It is possible to defined derivatives on closed subsets, but we need a less intuitive ...

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Perhaps one of the best pieces of advice I've heard regarding this type of dilemma (not only in math, but for all practices), is that if it's not fun then you need to change it up. Your math interests sound like they've become more of an obsession than fun. Math clearly fits the way you think, and you strive for satisfaction in understanding it, but you've ...

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I would say that one of the best texts for advanced calculus is "Vector Calculus, Linear Algebra, and Differential Forms" by J. Hubbard and B. Hubbard. click here You may find this as an additional and helpful resource to "Advanced Calculus" by Buck.

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