# Tagged Questions

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### Intuitively, how do you explain the concept of Flux?

Lately in my physics and mathematics classes, I've come across the concept of Flux. And although I've been able to define them mathematically and figure out how to use them. I'm still not entirely ...
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### What is defined by rate of change at a single point?

Rate of change measures how fast a process is going when it moves from one point to another. It measures the change of, say, $Y$ when $X$ moves from $X$ to $X + \Delta X$. But my problem arises when ...
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### Borsukâ€“Ulam theorem for $n=2$

How one can intuitively prove the following statement: At any moment there is always a pair of antipodal points on the Earth's surface with equal temperatures. What about a rigorous proof?
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### Why don't taylor series represent the entire function?

Say, I have a continuos function that is infinitely differentiate on the interval $I$. It can then be written as a taylor series. However, taylor series aren't always completely equal to the function ...
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### Length of a curve by integration: why won't flat segments do?

Maybe my question is a duplicate, but I guess I don't know the right terminology to find it elsewhere. I would be happy to delete it if someone can point out a duplicate. From elementary calculus, ...
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### Surface area of a Hypersphere

Hypersphere in 4 dimensions, I am having problem with finding the surface area of it. please help. I know that surface area will have 3 dimensions in 4 dimensional space, I am having trouble to ...
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### Evaluating $\int \frac{1}{\sqrt{x^2 + a^2}}\, dx$ without resorting to trigonometric $u$-substitution

I am looking for a quick and intuitive way to evaluate this indefinite integral without resorting to any trigonometric functions. I'm not sure if it is at all possible to do so, but I was just ...
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### Why is the derivative a limit?

Start by assuming that function curves are made of an infinite amount of lines (i.e. look at the image above but instead of approximating it using a finite number of lines, use infinite lines). This ...
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### Why are derivatives lines?

If you look at a function "infinitely close", the difference between two points is a line: __ __/ __/ Where each "__" is a point, and "/" is the ...
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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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### Guide to sketching graphs of basic functions

I'm taking a test soon, where I would be asked to sketch graphs. I wonder if there is any kind of guide or general tutorial on the net how to carry out sketching. For instance I would much like to ...
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### What really is an indeterminate form?

We can apply lâ€™HĂ´pitalâ€™s Rule to the indeterminate quotients $\dfrac{0}{0}$ and $\dfrac{\infty}{\infty}$, but why canâ€™t we directly apply it to the indeterminate difference $\infty - \infty$ or ...
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### Intuition behind $\vec{e_i} \times \vec{e_j}=\epsilon_{ijk} \vec{e_k}$ (Levi Civita)

Let $\vec{e_i}$ denote a unit vector. Then we can write: $\vec{e_i} \times \vec{e_j}=\epsilon_{ijk} \vec{e_k}$, where $\epsilon_{ijk}$ is the Levi Civita symbol. Can someone intuitively explain me ...
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### Explanation for $\lim_{x\to\infty}\sqrt{x^2-4x}-x=-2$ and not $0$

I am trying to intuitively understand why the solution to the following problem is $-2$. $$\lim_{x\to\infty}\sqrt{x^2-4x}-x$$ ...
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### non-archimedean in lay terms

I've dabbled with studying infinitesimals off and on for years ... Robinson, Keisler, Bell ("Smooth Worlds"), etc., even a bit of category theory. But I'm not a mathematician and tend to jump in way ...
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### Volume of a hypersphere

We know that the area of a circle (2-D) =$\pi r^{2}$ and the volume of a sphere (3-D)= $\dfrac{4}{3}\pi r^{3}$. Question:What is the "volume"(or whatever that is called) of a n-dimensional sphere? ...
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### The thought process of derivatives explained (intermediate calculus) “derivatives with respect to what”

My intention here is to contribute, if there is a problem with my solution or explanation--if it is wrong--please add a comment and don't just down vote. My answer represents my understanding and I ...
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