2020年4月28日 · Let me provide some context. I was specifically looking at the application of linear algebra to the stress tensor $\sigma$ and how we get the transformed stress state in some rotated coordinate system by the following multiplication: $\sigma'=Q\sigma Q^{-1}(Q$ is a rotation matrix defined by the angle we want to rotate the axes).

2010年8月30日 · There are infinitely many possible values for $1^i$, corresponding to different branches of the complex logarithm. The confusing point here is that the formula $1^x = 1$ is not part of the definition of complex exponentiation, although it is an immediate consequence of the definition of natural number exponentiation.

The Law of Signs $\rm\: (-x)(-y) = xy\:$ isn't normally assumed as an axiom. Rather, it is derived as a consequence of more fundamental Ring axioms $ $ [esp. the distributive law $\rm\,x(y+z) = xy + xz\,$], laws which abstract the common algebraic structure shared by …

2020年3月30日 · A-1 A means that first we apply A transformation then we apply A-1 transformation. When we apply A transformation we reach some plane having some different basis vectors but after apply A-1 we again reach to the plane have basis i ^ (0,1) and j ^ (1,0). It means that after applying A-1 we reach to the transformation which does nothing.

Summing to infinity cannot truly be evaluated as a sum as it is a process of addition that never ends, rather than a final result. What we can say fairly categorically is that the only acceptable numerical answers are $1, 0,$ and $\frac{1}{2}$.

2015年5月11日 · There are multiple ways of writing out a given complex number, or a number in general. Usually we reduce things to the "simplest" terms for display -- saying $0$ is a lot cleaner than saying $1-1$ for example.

Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

2013年1月15日 · The main reason that it takes so long to get to $1+1=2$ is that Principia Mathematica starts from almost nothing, and works its way up in very tiny, incremental steps. The work of G. Peano shows that it's not hard to produce a useful set of axioms that can prove 1+1=2 much more easily than Whitehead and Russell do.

$$\ln(n+1)\le\sum_{i=1}^n\frac1i\le\ln(n)+1$$ This is a rather tight upper limit and lower limit you can use to approximate your answer. One could also note that $$\sum_{i=1}^n\frac1i=\int_0^1\sum_{i=0}^{n-1}x^i\ dx=\int_0^1\frac{1-x^n}{1-x}\ dx$$ We also have the Euler-Maclaurin expansion: