# Search results for query: *

1. ### Calculus ## G(y, z)=z\frac{\partial}{\partial z}F(y, z)-F(y, z) ##?

Correct, it's pointless because it equals ## 0 ## anyways. I just feel relieved a lot more since I don't have to include the ## sec(x) ## function on part b), and it makes the proof a lot shorter than before.
2. ### Calculus ## G(y, z)=z\frac{\partial}{\partial z}F(y, z)-F(y, z) ##?

Also, isn't it ## ny^{n-1}y'=nC\frac{sin(nx+K)}{cos^2(nx+K)} ##?
3. ### Calculus ## G(y, z)=z\frac{\partial}{\partial z}F(y, z)-F(y, z) ##?

This is much easier!
4. ### Calculus ## G(y, z)=z\frac{\partial}{\partial z}F(y, z)-F(y, z) ##?

a) Observe that ## \frac{\partial}{\partial z}F(y, z)=y^{n-1}\cdot \frac{2z}{2\sqrt{y^2+z^2}}=\frac{zy^{n-1}}{\sqrt{y^2+z^2}} ##. This means ## G(y, z)=\frac{z^2\cdot y^{n-1}}{\sqrt{y^2+z^2}}-y^{n-1}\cdot \sqrt{y^2+z^2}=\frac{z^2\cdot...
5. ### Is "College Algebra" really just high school "Algebra II"?

Yes, because some students haven't taken Algebra 2 yet but they have already entered college, and this is why colleges label the name of Algebra 2 as 'College Algebra' to sound more difficult, but in reality, it's the same as Algebra 2 in high school. I don't know too much about other countries...
6. ### Difference between -3² and (-3)² ?

## -3^{2}=-9 ## but ## (-3)^2=(-3)(-3)=9 ##.
7. ### Can ChatGPT do geometry?

I don't think ChatGPT can do geometry, although it can do very simple arithmetics.
8. ### Proofs about the second-order linear differential equation?

And also ## w ## instead of the Wronskian ## W ##.
9. ### Proofs about the second-order linear differential equation?

What are characteristic curves?
10. ### Proofs about the second-order linear differential equation?

Okay, here's my revised proof on part (i). Consider the second-order linear differential equation ## \frac{d^2u}{dx^2}+\frac{fu}{2}=0, f=f(x) ##. Then ## u_{i}''+\frac{fu}{2}=0\implies u_{i}''=-\frac{f}{2}u ##, so ## -\frac{2}{f}u_{i}''=u_{i} ##. Given that ## w=\frac{u_{2}}{u_{1}} ##, we have...
11. ### Proofs about the second-order linear differential equation?

I'm working on this proof, almost done.
12. ### Proofs about the second-order linear differential equation?

I see where you got the ## -\frac{2}{f}u_{i}''=u_{i} ## from. But I don't understand why/how does ## w=\frac{u_{2}}{u_{1}}=\frac{u_{2}''}{u_{1}''} ##. And by starting both differentiations of ## \omega ##, do you mean to take up to second derivatives of ## \omega ##?
13. ### Proofs about the second-order linear differential equation?

I forgot the fact that ## f ## is a function, not a constant.
14. ### Proofs about the second-order linear differential equation?

I noticed the issue. Since both of the functions ## u_{1}(x), u_{2}(x) ## were incorrect, then how should I solve this differential equation and find these correct functions?
15. ### Proofs about the second-order linear differential equation?

Okay, I see that now. Then what's the error here? Is the characteristic equation wrong?
16. ### Proofs about the second-order linear differential equation?

Are you talking about the general solution? To which part of the problem are you referring to?
17. ### Proofs about the second-order linear differential equation?

Proof: (i) Consider the second-order linear differential equation ## \frac{d^2u}{dx^2}+\frac{fu}{2}=0, f=f(x) ##. Then ## u''+\frac{f}{2}u=0\implies r^2+\frac{f}{2}=0 ##, so ## r=\pm \sqrt{\frac{f}{2}}i ##. This implies ## u_{1}=c_{1}cos(\sqrt{\frac{f}{2}}x) ## and ##...
18. ### Can anyone please verify/confirm these derivatives?

I tried to avoid it too, but the book's problems were all written like that.
19. ### Can anyone please verify/confirm these derivatives?

Yes, I confirm.
20. ### Can anyone please verify/confirm these derivatives?

Note that ## \frac{\partial F}{\partial x}=\frac{2x}{2\sqrt{x^2+y'^2}}=\frac{x}{\sqrt{x^2+y'^2}}, \frac{\partial F}{\partial y}=0, \frac{\partial F}{\partial y'}=\frac{2y'}{2\sqrt{x^2+y'^2}}=\frac{y'}{\sqrt{x^2+y'^2}} ##. Now we have ## \frac{dF}{dx}=\frac{\partial F}{\partial x}+\frac{\partial...
21. ### Are these the correct expressions for ## dF/dy' ##?

Thank you so much for verifying!
22. ### Are these the correct expressions for ## dF/dy' ##?

I don't know either. So what should the book normally express these primes then, instead? Also, if ## exp(y') ## mean ## e^{y'} ##. Then the expression for ## dF/dy' ## is ## dF/dy'=e^{y'} ##?
23. ### Are these the correct expressions for ## dF/dy' ##?

a) ## dF/dy'=\frac{1}{4}(1+y'^2)^{\frac{-3}{4}}\cdot 2y' ## b) ## dF/dy'=cos (y') ## I just took the derivatives above and found out these expressions, but may anyone please check/verify to see if these expressions for ## dF/dy' ## are correct? Also, I do not understand part c). What does 'exp'...
24. ### Is this the correct way to find the Euler equation (strong form)?

## \eta \int_a^b e^{u} dx= \eta (e^{u(b)}-e^{u(a)}) ##
25. ### Is this the correct way to find the Euler equation (strong form)?

But where does the solution ## E(F)=e^{u} ## come from?
26. ### Is this the correct way to find the Euler equation (strong form)?

How to differentiate ## (u')^2+2u'\varepsilon \eta'+\varepsilon ^2(\eta')^2 + e^u\cdot e^{\varepsilon \eta} ##? Never mind, I got it now. It's ## 2u' \eta' +2\varepsilon (\eta')^2+\eta e^{u+ \varepsilon \eta} ##. And at ## \varepsilon=0 ##, it's ## 2u' \eta' + \eta e^{u} ##.
27. ### Is this the correct way to find the Euler equation (strong form)?

What does "compact support" mean/indicate in this problem?

I will wait.
29. ### Is this the correct way to find the Euler equation (strong form)?

May you check/verify the work and solution to see if it's correct/accurate?
30. ### Is this the correct way to find the Euler equation (strong form)?

Yes, I was working on it, but still doesn't seem to work. I will see what's wrong.
31. ### Is this the correct way to find the Euler equation (strong form)?

By the Euler's equation of the functional, we have ## J(\mathrm u)=\int ((\mathrm{u})^{2}+e^{\mathrm{u}}) \, dx ##. Then ## J(\mathrm{u}+\epsilon\eta)=\int ((\mathrm{u}'+\epsilon\eta')^{2}+e^{\mathrm{u}+\epsilon\eta}) \, dx=\int...
32. ### Prove that there are infinitely many primes of the form ## 6k+1 ##?

But how should I verify that ## N\equiv 1\pmod {6} ##? And I think I made some mistakes in my previous proof attempts, because ## p\mid N ## and ## p\mid (2p_{1}\dotsb p_{n})^{2} ## implies that ## p\mid (N-(2p_{1}\dotsb p_{n})^{2}) ##, so ## p\mid 3 ##.
33. ### Prove that there are infinitely many primes of the form ## 6k+1 ##?

I apologize for the confusion. What I meant is the symbol for expressing Legendre symbol, not divisibility.
34. ### Prove that there are infinitely many primes of the form ## 6k+1 ##?

Because ## 3\nmid 2p_{1}p_{2}\dotsb p_{n} ##. And ## p\mid N, p\mid 3 ## implies ## p\mid (N-3) ##, so ## p\mid (2p_{1}p_{2}\dotsb p_{n}) ##. Also, how should I show that ## N ## has an odd prime divisor of the form ## 6k+1 ##? At first, I thought this is so because ## N ## itself is odd. But it...
35. ### Prove that there are infinitely many primes of the form ## 6k+1 ##?

Okay, so I revised this proof: Suppose for the sake of contradiction that the only primes of the form ## 6k+1 ## are ## p_{1}, p_{2}, ..., p_{n} ##. Consider the integer ## N=4p_{1}^{2}p_{2}^{2}\dotsb p_{n}^{2}+3=(2p_{1}p_{2}\dotsb p_{n})^{2}+3 ##. Since ## N ## is odd, it follows that ## N ##...
36. ### Prove that there are infinitely many primes of the form ## 6k+1 ##?

Proof: Suppose that the only prime numbers of the form ## 6k+1 ## are ## p_{1}, p_{2}, ..., p_{n} ##, and let ## N=4p_{1}^{2}p_{2}^{2}\dotsb p_{n}^{2}+3 ##. Since ## N ## is odd, ## N ## is divisible by some prime ## p ##, so ## 4p_{1}^{2}\dotsb p_{n}^{2}\equiv -3\pmod {p} ##. That is, ##...
37. ### I Is this the correct way to quantify these integers?

Thank you for the clarification, @andrewkirk @topsquark .
38. ### I Is this the correct way to quantify these integers?

Does the above quantifier represent/symbolize that all of the integers ## a, b, c, d ## cannot be ## 0 ##? Is this correct?
39. ### How to construct a table of all the real-valued Dirichlet characters?

So now we have that ## \chi(k)^{2}=\chi(k)\cdot \chi(k)=\chi(k\cdot k)=\chi(k^{2})=\chi(1)=1 ## for all ## k\{8, 13, 20\} ## and ## \chi(k)^{6}=\chi(k^{6})=\chi(1)=1 ## for all ## k\{2, 4, 5, 10, 11, 16, 17, 19\} ##. And this implies that ## \chi(n)=8, 13, 20 ## can either be ## -1, 1 ##. But...
40. ### How to construct a table of all the real-valued Dirichlet characters?

## 1, 8, 13, 20 ## ## 1^{2}\equiv 1\pmod {21}, 8^{2}\equiv 1\pmod {21}, 13^{2}\equiv 1\pmod {21}, 20^{2}\equiv 1\pmod {21} ##
41. ### How to construct a table of all the real-valued Dirichlet characters?

Since ## \varphi(21)=\varphi(3)\varphi(7)=2\cdot 6=12 ##, there are ## 12 ## elements such that ## G=\{1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20\} ##. So ## G ## can be generated by order ## 2 ## or ## 6 ##. And we have ## \chi(k)^{2}=\chi(k)\cdot \chi(k)=\chi(k\cdot k)=\chi(k^{2})=\chi(1)=1 ##...
42. ### How to construct a table of all the real-valued Dirichlet characters?

No. I do not know the structure of finite abelian groups. I do not know the characters of cyclic groups are. You said that my group of ## G ## is isomorphic to ## C_{6}\times C_{2} ##, which are the two cyclic groups of order ## 6 ## and ## 2 ##. But how did you get these?
43. ### How to construct a table of all the real-valued Dirichlet characters?

I already did. But I still don't understand. How should I find normal subgroups?
44. ### How to construct a table of all the real-valued Dirichlet characters?

This is my question, too. I do not know what they mean, I just posted them under the relevant equation(s) just because my book has these definitions. Since these definitions are preventing people to make sense of my question, then please ignore them. How should I find those values then, starting...
45. ### How to construct a table of all the real-valued Dirichlet characters?

\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline n & 1 & 2 & 4 & 5 & 8 & 10 & 11 & 13 & 16 & 17 & 19 & 20 \\ \hline \chi_{1}(n) & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \hline \chi_{2}(n) & 1 & -1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & -1 & 1 \\ \hline \chi_{3}(n) & 1 & 1 & 1 & -1 & 1...
46. ### A How should I write an account of prime numbers?

I wish there are more details in this question, but no. The question states: "Write an account of prime numbers in arithmetic progressions. Your account should be in the form of an essay of 500-1000 words."
47. ### A How should I write an account of prime numbers?

I am not completely sure of whether or not this will work for my purpose.
48. ### A How should I write an account of prime numbers?

How should I write an account of prime numbers in arithmetic progressions? Assuming this account should be in the form of an essay of at least ## 500 ## words. Should I apply the formula ## a_{n}=3+4n ## for ## 0\leq n\leq 2 ##? Can anyone please provide any idea(s)?
49. ### How to find the Gateaux differential of this functional?

After breaking down into smaller pieces, I got the following: \begin{align*} &(y'(x)+\tau\psi'(x))^{2}=y'^{2}(x)+2y'(x)\tau\psi'(x)+\tau^{2}\psi'^{2}(x)\\ &\omega^{2}(y(x)+\tau\psi(x))^{2}=\omega^{2}y^{2}(x)+2\tau\psi(x)\omega^{2}+\omega^{2}\tau^{2}\psi^{2}(x)\\...
50. ### How to find the Gateaux differential of this functional?

Do you mean ## S(y+\tau\psi)=\int_{a}^{b}\frac{d}{d\tau}(y'^{2}+\omega^{2}y^{2}+2yx^{4})dx ##? If so, then how to evaluate this?