Questions tagged [geometric-intuition]
The geometric-intuition tag has no usage guidance.
6
questions
48
votes
2
answers
7k
views
Geometric interpretation of the half-derivative?
For $f(x)=x$, the half-derivative of $f$ is
$$\frac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}} x = 2 \sqrt{\frac{x}{\pi}} \;.$$
Is there some geometric interpretation of (Q1) this specific derivative, and, (...
67
votes
11
answers
10k
views
How should one think about non-Hausdorff topologies?
In most basic courses on general topology, one studies mainly Hausdorff spaces and finds that they fit quite well with our geometric intuition and generally, things work "as they should" (sequences/...
34
votes
6
answers
6k
views
How to explain the concentration-of-measure phenomenon intuitively?
One way to phrase the
"concentration-of-measure"
phenomenon is that,
for a Euclidean sphere $S^d$ in $d$ dimensions, for large $d$,
"most of the mass is close to the equator, for any equator."1
Q. ...
85
votes
12
answers
87k
views
Why is the gradient normal? [closed]
This is a somewhat long discussion so please bear with me. There is a theorem that I have always been curious about from an intuitive standpoint and that has been glossed over in most textbooks I ...
36
votes
5
answers
4k
views
What is the general geometric interpretation of modules in algebraic geometry?
Algebraic geometry is quite new for me, so this question may be too naive. therefore, I will also be happy to get answers explaining why this is a bad question.
I understand that the basic philosophy ...
24
votes
2
answers
12k
views
Geometric interpretation of Cartan's structure equations
Given a linear connection on a Riemmanian manifold $M$ and $\phi^1,...,\phi^n$ a local frame for $T^*M$ we can define the connection 1-forms $\omega^j_i$. We define the curvature 2-forms by $\Omega_i^...