# Tagged Questions

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### Lecture notes ready for $\LaTeX$

Are there on the internet lecture notes in calculus in .tex or .txt format, that is, ready to be edited/modified/re-used and compiled using $\LaTeX$? EDIT: now I am specifically asking for calculus, ...
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### Bridging the gap of understanding function terminology in math for a programmer.

I'm a computer programmer by profession with no formal CS education. When I read in mathematics the terminology used around a function, I get confused. For example, I was reading up on some calc and ...
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### Is there a(n elementary) function whose derivative we cannot integrate?

Say, for example, I take a reasonably-complicated function $f(x)=\tanh[\ln(x^x)]$, and differentiate it to get $$f'(x)=\frac{4x^{2x} [1+\ln(x)]}{(x^{2x}+1)^2}.$$ Now, to integrate this, I imagine, ...
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### What is the difference between a calculus and an algebra? [duplicate]

You can have a lambda calculus, the calculus of the real numbers or a logical calculus but on the other hand you could also have an algebra of sets, a Lie algebra, or a linear algebra. Is there any ...
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### Advice for self-studying Inequalities and Calculus

I'm interested in self-studying the following books over the next year or so: Spivak's Calculus (I'm already in Ch. 5 and it is very slow going) The Cauchy-Schwarz Master Class by J. Michael Steele ...
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### Why is a straight line the shortest distance between two points?

The first application I was shown of the calculus of variations was proving that the shortest distance between two points is a straight line. Define a functional measuring the length of a curve ...
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### Is Courant's Introduction to Calculus and Analysis still up-to-date?

I just found this marvelous book and I think that it's the best book in this category, but I'm worried that it is not up-to-date. I've heard that Hardy's A Course of Pure Mathematics has some switched ...
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### Is there a reason that sine substitution is preferred to cosine substitution?

When integrating by trigonometric substitution that has $\sqrt{a^2-x^2}$ in the integrand, the general recommended approach is to make the substitution $x = a \sin\theta$, and this substitution is the ...
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### I want to learn math from ground-up, basic to advanced, beginner to expert

I want to learn math. I've learned math long time ago, but i hardly remember anything. I really want to relearn but have no idea where to begin. I want to learn math by reading through good books, ...
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### Structured Self-Learning Program for Calculus I & II

I'm interested in a organised program which comprehensively covers the topics of Calculus I and Calculus II. I've recently finished taking my secondary school's university-level Calculus I course, ...
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### How should this definition of a family with corresponding index set be interpreted?

Does the following definition of a family imply that there exist a surjection from $I$ onto the family ? Or should it be interpreted as a injective partial function from $I$ to the family ? ...
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### Beautiful Indefinite Integrals. [closed]

These are some of the integrals with beautiful solutions I came across- $$\int \frac{x^2}{(x\sin x+\cos x)^2} dx$$ $$\int\frac {1}{\sin^3x+\cos^3x} dx$$ $$\int \frac{1}{x^4+1}dx$$ I'd love if you ...
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### How to practice applied mathematics calculation skill

As a natural science student in university, you may encounter so many problems that might require a deep understanding in integrating skills and series calculation. But as many of the college students ...
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### Does Discrete Math require Calculus?

So I'm transferring to UC Berkeley and they require Discrete Math or 1 other math class for me to declare my Psychology major. So it is highly recommended to take Calculus for maturity before ...
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### How can we relate calculus, trigonometry etc in real life

I have always wondered what does trigonometry, calculus, logarithms solve real world problems? Where do they apply in real life? Is there any simple book where I can understand it?
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### Most important things to be proficient in before Calculus 1?

What are the main things one should be proficient in before taking Calculus 1? Please be specific.
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### l'Hopital's questionable premise?

Historians widely report that l'Hopital's 1696 book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes contains a questionable premise expressed by an equation of type $x+dx=x$ ...
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### Errors of Euler interpretation?

To complement the recent post on Euler's errors, I would pose the following question: what common errors of Euler interpetation appear in the literature? What errors are attributed to Euler's work in ...
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### Cycloidal coincidence?

For the cycloid $$x=a(t-\sin t)\ ,\quad y=a(1-\cos t)$$ we have, as is easily seen, $$\frac{dx}{dt}=y\ .$$ Does this have any geometrical or physical significance? Or is it just a meaningless ...
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I'm reading Thomas Calculus now but I don't think it includes Mellin transform or Riemann-Stieltjes integration... Can you recommend an advanced calculus book which includes all of this stuff?
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### Jumping back into Calculus III

At the age of 30 I am going back to school for Electrical Engineering. Because of the way higher education works, all of my previous college coursework is being transferred, which does not allow you ...
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### What is the most elementary but still correct according to the most rigorous standard proof of the isoperimetric inequality?

Can you write the most elementary proof of the isoperimetric inequality (but still correct according to the most rigorous standard )? $$l^2> 4πA$$
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### Best proof of some theorems in calculus

I would like to choose (among the miriads of proofs) a well-structured, elegant, neat, clear proof of the first fundamental theorem of calculus; the second fundamental theorem of calculus; the mean ...
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### Where is the Pinching/Squeeze theorem in Spivak Calculus?

So I got Spivak Calculus 3. Edition. I'm starting it now but I want to know if there is the squeeze theorem clearly explained in the book, as I can't find it in the Appendix(pinching theorem, ...
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### Borsuk–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 is there no universally accepted mathematical definition of turbulence?

Why is there no universally accepted mathematical equation for the definition of turbulent flow?
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### Calculus book with good explanation on squeeze theorem?

I'm trying to learn squeeze theorem and it's quite difficult to get the concepts. Could anyone suggest some good books which explain this in depth, along with limits, continuity, etc? So far, I have ...
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### Which book among these would you recommend for first year calculus?

I'm struggling a bit with functions(limits, squeeze theorem, etc). I have done some research and found a list of books on calculus but I'm not sure which one would be better suited for me, so I would ...
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### Applications of inflection points

Recently, I was teaching maxima, minima and inflection points to first year engineering students. I motived extrema by giving practical examples of optimization problems, but when a colleague asked me ...
868 views

### Why limits work

I'm currently a first year student in electrical engineering and computer science. I know how to compute limits, derivatives, integrals with respect to one variable that is things from one variable ...
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### Complex polynomial decomposition - Residue Theory

I am given the following function: $R(z) = (z^2-9)/(z^2+9)^2$ I need to let $R = P/Q$ be a rational function with $deg P < deg Q$. I will let $ξ$ be a pole of $R$ and the coefficient of $1/(z-ξ)$ ...
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### If $Z$ is an admissible function, does $f(z) = f(|z|)$?

If $Z$ is an admissible function, does $f(z) = f(|z|)$? For example, if $f(z) = x^3 + 1$ and I am given $2$ points $z_1 = -1+i\sqrt3$ and $z_2 = -1-i\sqrt3$, can I just find the moduli and use that ...
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### Disproving the mean-value theorem of calculus to complex functions?

I'm defining a function $f(z) = z^3 + 1$, and I will let $2$ points $z_1 = (-1+i\sqrt3)/2$ and $z_2 = (-1-i\sqrt3)/2$ I am trying to show that there is no point $w$ on the line segment from $z_1$ to ...
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### A beautiful identity of $\sin(x)$ [duplicate]

When I was in high school, I had proved that $$\sin^2(x)-\sin^2(y)=\sin(x-y)\sin(x+y)$$ I think it is beautiful since it resembles the identity $a^2-b^2=(a+b)(a-b)$. But I can not find it in ...
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### Intuitive idea behind the probability density function

as an application of Calculus, I am currently teaching some material about continuous random variables. My main example is the height $X$ of a French male chosen randomly in the French population. ...
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### Was the Weierstrass function constructed or discovered?

Reading Halmos' I want to be a mathematician, he mentions a continuous function without a tangent. Naturally, I was curious to see how such a function could possibly exist, and I imagined it to be ...
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### Why has $\int \sin (\sin x) dx$ not been solved yet?

I have Calculus 2 background, so please try to keep your answers around that level. I inly want a brief explanation. What is it about $\sin (\sin x)$ that makes it difficult to integrate? Also, what ...
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### Calculator similar to Desmos but for 3D

Is there a calculator with functionality similar to Desmos but in 3 dimensions? I am looking to learn about families of quadric surfaces so I am looking for a 3D calculator with sliders.
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### Transitioning to Higher Level Mathematics

I am just finishing grade 12 pre-calculus at my school and have strong interest in math. The problem is, it seems some important elements of higher level math are not in my schools curriculum that are ...
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### $\epsilon, \delta$…So what?

Over the course of my studies I often encounter phrases in reference material of the type "and this avoids the need for using $\epsilon$, $\delta$ definitions" or "by this we can omit those ...
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### Difficulty in learning calculus 3 years after one has learned calculus 2?

Academically speaking, I'm a late bloomer. Long story short, I was a bad kid, teachers told me I wasn't good at math and I believed them. Years later, as an almost 30 year old man, I started going ...
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### What does “area” really mean?

My professor had an interesting statement at the beginning of first year integral calculus. What does area really mean? How do we know that the area of a circle is $\pi r^2$? Archimedes used ...
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### Knowledge of proofs and proof-writing before studying spivak's or apostol's Calculus

I want to start reading spivak or apostol calculus. But I believe a certain level of proof knowledge is essential to solve the problems. Do these books teach you how to prove results or do we learn ...