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 Apr17 comment Explanation for the next steps of chaplygin dipole @JuanSimões, I'm sorry, I tried to understand it and I didn't succeed. I'm trying to find another material that is related to the link you posted. thank you from the deep of my heart! you help me a lot!! Apr17 comment Explanation for the next steps of chaplygin dipole @JuanSimões, ok, I will read about that. and you answered about that: "I think v is the two first elements of (2.2) [thanks to your link]. am I right?" thank you very much!! you don't know how much you help me to understand it! Apr17 comment Explanation for the next steps of chaplygin dipole @JuanSimões, wow, the first step is by definition! thanks! (2.3) is a choice (thank you again), and I guess (2.4) is also, right? so (2.5) is a derivative of which v (once by r and once by theta)? I think v is the two first elements of (2.2) [thanks to your link]. am I right? Apr17 comment Explanation for the next steps of chaplygin dipole @JuanSimões, thank you.. to be honest I didn't understand the step between (1.2) to (2.2). and why he chose the function of Psi and this Omega in (2.3)? thank you very much! Feb15 comment Bernoulli integral (conservation of energy) ok, so my last question: does first solution contain the full solution? thank you for all your comments. Feb15 comment Bernoulli integral (conservation of energy) thank you Tom! I didn't try to prove that.. what's about what I wrote in the topic, under the title: First solution ? Feb15 comment Bernoulli integral (conservation of energy) I updated my topic :) Feb14 comment Bernoulli integral (conservation of energy) thank you very much Tom! I think I will show them the both solutions, so I have one proof (of you). about the second proof: do you have a link with a solution of rewriting Euler equations to form (2.13)? and then the second solution will be the rewriting Euler equations to form (2.13) and after that I will write (2.14)-(2.19). BTW, what's about (2.1)-(2.12)? I don't need this? about your question, The lecturer wants me to lecture about this topic.. so I will have to learn the both solutions (I will learn it by myself when I get them).. thank you very very much! Nov28 comment Given a set $A$ such that for any family of sets $F$: if $\cup F = A$ then $A \in F$. I know why but i dont know how to explain it.. the answer is: A doesn't belong to F because of the definition of F.. Nov28 comment Given a set $A$ such that for any family of sets $F$: if $\cup F = A$ then $A \in F$. can you give me an exaple of A and F? I think I dont really know what is F.. thank you! Nov28 comment Given a set $A$ such that for any family of sets $F$: if $\cup F = A$ then $A \in F$. @DonAntonio, you are right. I translated the word to english and use a group instead of a set. Nov25 comment Prove that if $|a_n-a_{n-1}| < \frac{1}{2^{n+1} }$ and $a_0=\frac12$, then $\{a_n\}$ converges to \$0