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1. ### MIT OCW, 8.02 Electromagnetism: Potential for an Electric Dipole

Here is a depiction of the problem a) The potential at any point P due to a charge q is given by ##\frac{kq}{r}=\frac{kq}{\lvert \vec{r}_s-\vec{r}_P \rvert}##, where ##r## is the distance from the charge to point P, which is the length of the vector difference between ##\vec{r}_s##, the...
2. ### Chemistry Creating KF Molecule from Neutral Atoms: A Process of Ionization and Binding

Let's think now about the energy to create a KF molecule from neutral atoms. First we need to ionize both K and F: K loses an electron and F gains an electron. Then we have to bring the ions together. The first ionization energy of K is 418 kJ/mol and for F is 1681 kJ/mol. The electron...
3. ### Checking the Solution to -√2: Is It Right?

The answer at the end of the book says ##-\sqrt{2}##. Is this correct or is my solution correct? Here is a depiction of the path where we are integrating
4. ### MIT OCW, 8.02, Electromagnetism: Charged Cylindrical Shell

Indeed, thanks. So the integral itself is correct? It's just a question of solving it.
5. ### MIT OCW, 8.02, Electromagnetism: Charged Cylindrical Shell

Here is figure 2.16.6 Here is the picture I drew to set up the problem My first question is if the reasoning and integrals are correct. I used Maple to compute the three integrals. The first two result in 0, which makes sense by symmetry. Maple can't seem to solve the last integral.
6. ### Solving for Simple Harmonic Motion: A Picture Problem

Here is a picture of the problem It is not clear to me how to really prove that the equation for ##\theta(t)## is simple harmonic motion, and what the period of this motion is.
7. ### I Spivak, Ch. 20: Understanding a step in the proof of lemma

Looks like the answer is a silly oversight. Since the induction is over ##n##, this includes the part about the function ##R## being ##(n+1)##-times differentiable. After the inductive hypothesis, we want to prove the result for some ##k+1##. To do this we assume that the function ##R## is...
8. ### I Spivak, Ch. 20: Understanding a step in the proof of lemma

In Chapter 20 of Spivak's Calculus is the lemma shown below (used afterward to prove Taylor's Theorem). My question is about a step in the proof of this lemma. Here is the proof as it appears in the book My question is: how do we know that ##(R')^{n+1}## is defined in ##(2)##? Let me try to...
9. ### LaTeX Issue with Latex Equations: I have a screen recording of the problem

Is it the case that someone has looked into it and there is no easy solution, or is it just an issue on the backburner? If the latter, is it possible for a regular user to look into it and possibly create a pull request?
10. ### LaTeX Issue with Latex Equations: I have a screen recording of the problem

My posts all include many equations written in Latex. It seems to me like there are a few bugs related to usage of Latex specifically when just starting a new thread. I've made a short 2 minute screen recording to show the issue I face every single time I want to start a new thread, and the way...
11. ### Spivak, Ch 5 Limits, Problems 10c: Proving limit relationship

Perhaps not obvious but also not not obvious. I am only using concepts from the current and previous chapter I am on in Spivak, so no bijections yet. Also not inverse functions, which is what I think might be needed to consider the case of replacing ##x^3## with ##\sin{x}##. However, using...
12. ### Spivak, Ch 5 Limits, Problems 10c: Proving limit relationship

c) Why is the assertion ##\lim\limits_{x \to 0} f(x) = \lim\limits_{x \to 0} f(x^3)## obvious? First of all I don't think it is obvious but here is an explanation of why the limits are the same. ##\lim\limits_{x\to0} f(x^3)=l_2## means we are looking at points with ##x## close to zero and...
13. ### Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits

It's the proof of this, which was a step in the main proof. It's required for the latter.
14. ### Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits

The proposition should actually be with a ##\leq##, ie ##a-\delta \leq f(x)##. Is this what you mean?
15. ### Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits

Proving that ##a-\delta<f(x)## seems like a side-proof in the main proof, not the main thing to be proved. Here's an attempt at a proof: Proposition: ##b<x \implies f(b) \leq f(x)## Proof: Assume ##0<b<x<1##. If ##x## contains no digit 7 in its decimal expansion then ##f(x)=x>b \geq f(b)...
16. ### Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits

It's Spivak actually, and he's not expecting an epsilon-delta proof in this case; he's simply asking where the limit exists, not a proof. This epsilon-delta madness is the product of my curiosity.
17. ### Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits

Let me try and fill in the parts that you left implicit for the case of an ##a## with a decimal expansion with no digit ##7## in it, ending in zeros, e.g. ##a=0.6##. If ##x>a## then ##x\geq f(x)>a=f(a)##. This means that ##f(x)## is always between ##x## and ##a##, so ##f(x)-a## is between 0...
18. ### Spivak, Ch. 5 Limits, Problem 3 viii: Prove a limit of a function

If ##0<a<1##, then ##\forall \epsilon>0## choose ##\delta = \sqrt{a}\epsilon##. Then ##|x-a|<min(a,\delta) \implies |\sqrt{x}-\sqrt{a}|<\epsilon## My god, so simple.
19. ### Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits

The proof I presented was for the specific case of a number ##a## ending in ##6\bar{9}## (originally I specified this as ending in ##7\bar{0}##). These are points at the left end of one of the infinite intervals composing the graph of ##f##. ##a=0.800005## isn't part of the available values in...
20. ### Spivak, Ch. 5 Limits, Problem 3 viii: Prove a limit of a function

I didn't succeed with this suggestion. Here is an alternative way I found: Let (i) ##0<a<1## (ii) ##\delta<min(a,1-a) \leq 0.5##. $$|x-a|<\delta \implies a-\delta<x<a+\delta$$ $$0<a\leq 0.5 \implies \delta < a \implies 0<x<2a<1$$ $$0.5 \leq a < 1 \implies \delta < 1-a \implies 0<2a-1<x<1$$...
21. ### Spivak, Ch. 5 Limits, Problem 3 viii: Prove a limit of a function

It seems to be tricky to show that ##\lim\limits_{x \to a} \sqrt{x}=\sqrt{a}## for a general ##a##. More specifically, I think I can prove it for ##a \geq 1##, but having trouble with ##0 < a < 1##. The proof I gave for ##a=1## is valid for ##a \geq 1## with slight modifications: Let...
22. ### Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits

Not sure if you saw this, but this was my initial sketch and attempt above. It is convoluted. I'm actually going to come back to this problem at a later time, I've spent too long on it today! Proposition: ##\lim\limits_{x\to a}f(x)## exists for any number ##a## that is not on the right (closed)...
23. ### Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits

The first part yes, and this is why the intervals are closed on the right. About the second part (##f(0.abc...hij8)=0.abc...hij8##) I am not sure. This number ends in a string of zeros, so it is equal to another number that ends in ##7\bar{9}##. On the line ##y=x##, at ##x=0.7\bar{9}##, there...
24. ### Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits

Not quite. See my updated version above. The limit exists except where ##x## is of the form ##0.abc...hij7\bar{9}##, ie it ends in ##7\bar{9}##
25. ### Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits

Indeed, I've spent a few hours on this problem and have not yet understood exactly what is meant by the passage I quoted. I understand that ##0.7\bar{9}## and ##0.8\bar{0}## are equal. It's more murky what it means that "we will always use the one ending in 9's". Does this mean that the notation...
26. ### Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits

Missed this message earlier :) Here is the exact wording:
27. ### Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits

I mean actually formally prove using an epsilon-delta proof. From simply looking at the graph and the function definition, I would say the limit exists for every ##x## except at places such as ##0.8##. The reason is that the limit from below exists and is equal to 0.7, but the limit from above...
28. ### Spivak, Ch. 5 Limits, Problem 3 viii: Prove a limit of a function

Just to be extra precise about the absolute value issue. Isn't it always implicit that there is the ##0<## portion? Absolute value is always non-negative, by definition. To make sure ##x \geq 0## it is only necessary to write $$|x-1|<\delta\leq 1$$
29. ### Spivak, Ch. 5 Limits, Problem 3 viii: Prove a limit of a function

I've been thinking that maybe I should read a book or take a course on mathematical logic, to really learn to write mathematics correctly (is that the best way to learn the actual "grammar" of math, if I can call it that?). I actually understand what you mean with the points you brought up...
30. ### Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits

Isn't it rather ##\forall x: 0.7 \leq x \leq 0.7\bar{9}, f(x)=0.7##? (Note that I made the interval closed not open). The solution manual graph that I posted actually shows ##\forall x: 0.7 < x \leq 0.7\bar{9}, f(x)=0.7##. This is actually one question I have: why is the interval open on the...
31. ### Spivak, Ch. 5 Limits, Problem 3 viii: Prove a limit of a function

In general, if we write ##|x-1|<\delta## this means that ##1-\delta<x<1+\delta##. If ##\delta>1## then ##x## can be negative. If we write ##0<\delta<min(1,\epsilon)## this implies that ##x\geq 0## (as I showed in the previous post). Indeed I forgot to put in the ##0<## part. If we write...
32. ### Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits

$$f(0.73)=0.7<0.73$$
33. ### Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits

I believe the x-axis is vertical here. The graph is composed of i) an infinite number of intervals that start on the ##y=x## line and finish at some ##x## with a decimal expansion ending in ##7\bar{9}##. E.g., from ##0.67## to ##0.67\bar{9}## which is considered ##0.68##. Other examples of...
34. ### Spivak, Ch. 5 Limits, Problem 3 viii: Prove a limit of a function

##\forall \epsilon>0##, we are looking for ##\delta>0:|x-1|<\delta \implies|\sqrt{x}-1|<\epsilon## The domain of ##f## is ##[0,+\infty)## so ##x\geq 0##. Consider ##0<\delta \leq 1 ## $$|x-1|<\delta \implies 1-\delta<x<1+\delta \implies x \geq 0$$...
35. ### Going through Spivak's Calculus, every chapter, every problem

Going through Spivak's Calculus, every chapter, every problem

49. ### Electric field created by two charged circular arcs?

I can't believe I missed the integral limits! My god. I am aware of the symmetry of the problem. At this stage I am doing the full calculations to get practice doing the calculus. But I will try to just use symmetry to also get practice with identifying such shortcuts from now on as well.
50. ### Electric field created by two charged circular arcs?

The strategy will be to figure out what ##dq##, ##\hat{r}_{dq,p}##, and ##r_{dq,p}## are, plug them into the expression for ##d\vec{E}_{p_r}##, then integrate over ##d\vec{E}_{p_r}## to obtain ##\vec{E}_{p_r}##, the electric field at ##P## due to the arc on the right. Then I will repeat the...