# Tagged Questions

Mathematical intuition is the instinctive impression regarding mathematical ideas which originate naturally without regard to formal mathematical proofs. It may or may not stem from a cognitive rational process.

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### What really is an indeterminate form?

We can apply l’Hôpital’s Rule to the indeterminate quotients $\dfrac{0}{0}$ and $\dfrac{\infty}{\infty}$, but why can’t we directly apply it to the indeterminate difference $\infty - \infty$ or ...
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### Intuitive explanation for formula of maximum length of a pipe moving around a corner?

For one of my homework problems, we had to try and find the maximum possible length $L$ of a pipe (indicated in red) such that it can be moved around a corner with corridor lengths $A$ and $B$ (...
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### Examples for proof of geometric vs. algebraic multiplicity

Here you see a supposedly easy proof of a well-known theorem in linear algebra: Although I know I should understand this, I don't :-( Obviously there are too many indices and stuff, so I don't see ...
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### Geometric interpretation of primitive element theorem?

The primitive element theorem is a basic result about field extensions. I was wondering whether there are nice geometric ways to visualize it or think about it. Since field spectra are singletons, it ...
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### Why does gradient descent work?

On Wikipedia, this is the following description of gradient descent: Gradient descent is based on the observation that if the multivariable function $F(\mathbf{x})$ is defined and differentiable ...
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### Intuitive Understanding of the constant “$e$”

Potentially related-questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate. I know there are many ways of calculating (or should I say "...
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### Paradox of the trumpet shape

This is a question I had for long time now, when you rotate the function $y=1/x$, $x>0$ (say $x$ and $y$ both measure meters) about the $x$ axes by $2\pi$ you get a shape which has infinite surface ...
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### What did Alan Turing mean when he said he didn't fully understand dy/dx?

Alan Turing's notebook has recently been sold at an auction house in London. In it he says this: Written out: The Leibniz notation $\frac{\mathrm{d}y}{\mathrm{d}x}$ I find extremely difficult ...
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### What is the intuitive meaning if multiplying by fractional 1?

first post ever on stack exchange in years of using it. Can anyone provide a historical or logical deduction of the reasoning behind multiplication by 1 via a fraction? For instance, in finance ...
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### Confusion about geometric interpretation of proof that $\mathbb R[X,Y,Z]/ \left\langle X^2+Y^2+Z^2 -1 \right\rangle$ is a UFD

I'm working through a proof that $R=\mathbb R[X,Y,Z]/ \left\langle X^2+Y^2+Z^2 -1 \right\rangle$ is a UFD. The idea is to localize at $1-x$ and show the result is a UFD. Since $R$ is atomic as a ...
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### Why is the area under a curve the integral?

I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself. However when it comes to the area ...
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### Chain Rule Intuition

We know that the chain rule is used to differentiate a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or ...
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### Least amount of repetitions s.t. probability greater than 1/2

Assume that for a formula $F$ over $n$ variables, there are exactly $k$ allocations that satisfy it. How many random samples from the set $\{0,1\}^n$ are necessary to find an allocation satisfying the ...
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### Intuition about the second isomorphism theorem

In group theory we have the second isomorphism theorem which can be stated as follows: Let $G$ be a group and let $S$ be a subgroup of $G$ and $N$ a normal subgroup of $G$, then: The ...
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### an outline of “intuitive mathematics”?

This question is related to the third answer in this post. There seems to be a difference between the intuitive idea of a thing (such as a function) and "models" of that thing in mathematics (such ...
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### Characteristic function of union of two sets formula and intuition

From http://topologicalmusings.wordpress.com/2008/03/20/inclusion-exclusion-principle-counting-all-the-objects-outside-the-oval-regions-2/ Is there an easier proof or way to calculate $1[A \cup B]$?...
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### cycloid of a unit-speed circle

In one of the lectures of the MIT OCW Multivariable Calculus course, the professor introduces the parametric equation of a cycloid in the plane, where $a$ is the radius of the circle that creates it, ...
I was looking at the symmetries of the Riemann tensor, and tried to prove a couple of properties, namely If $\nabla$ is torsion-free, then: (i) $R^a_{\,[bcd]}=0$, and (ii) $R^a_{\,b[cd;e]}=0$. ...