Fonction vectorielle de Leibniz
គេហៅអនុគម៍វ៉ិចទ័រលិបនីស គឺជាអនុគមន៍ On appelle fonction vectorielle de Leibniz l'application \(\vec{f}\) qui à tout point \(\mathrm{M}\) du plan (ou de l'espace ) associe le vecteur $\vec{f}(M)=\alpha \overrightarrow{M A}+\beta \overrightarrow{M B}+\gamma \overrightarrow{M C}$ où \(\alpha, \beta,\) y sont trois réels fixés et \(A, B, C\) trois points fixés du plan (ou de l'espace).
Propriétés : Pour tout couple de points $(M,N)$ du plan on a: \[ \vec{f}(N)-\vec{f}(M)=(\alpha+\beta+\gamma) \overrightarrow{N M} \]
Démonstration $$\begin{array}{ll} \vec{f}(N)&=&\alpha \overrightarrow{N A}+\beta \overrightarrow{N B}+\gamma \overrightarrow{N C}\\ &=& \alpha \overrightarrow{N M}+\alpha \overrightarrow{M A}+\beta \overrightarrow{N M}+\beta \overrightarrow{M B}+\gamma \overrightarrow{N M}+\gamma \overrightarrow{M C}\\ &=& (\alpha+\beta+\gamma) \overrightarrow{N M}+\alpha \overrightarrow{M A}+\beta \overrightarrow{M B}+\gamma \overrightarrow{M C}\\ &=& (\alpha+\beta+\gamma) \overrightarrow{N M}+\vec{f}(M) \end{array}$$
Barycentre de deux points
Définition 1: On appelle barycentre des points \(A\) et \(B\) ( ou \(A\) et \(B\) deux points du plan ou de I'espace ) affectés respectivement des coefficients \(\alpha, \beta\) ( ou \(\alpha, \beta\) sont des réels tels que \(\alpha+\beta \neq 0\) l'unique point \(G\) tel que \(\alpha \overrightarrow{G A}+\beta \overrightarrow{G B}=\overrightarrow{0}\)
- Si la somme \(\alpha+\beta+\gamma\) des coefficients est nulle alors les vecteurs \(\vec{f}(\mathrm{M})\) et \(\vec{f}\) (N) sont égaux et la fonction \(\vec{f}\) est constante ($\vec{f}(\mathrm{M})$ est un vecteur constant ne dépendant pas de $\mathrm{M}$)
- si cette somme est non nulle \(\alpha+\beta+\gamma \neq 0\) \begin{eqnarray*} \vec{f}(M)=\vec{f}(N) &\Longleftrightarrow & (\alpha+\beta+\gamma) \overrightarrow{N M} = \overrightarrow{0}\\ &\Longleftrightarrow& \overrightarrow{N M}=\overrightarrow{0}\\ &\Longleftrightarrow & N=M \end{eqnarray*} la fonction $\vec{f}$ est donc bijective dans ce cas et le vecteur nul $\overrightarrow{0}$ admet donc un unique antécédent appelé barycentre des points $A, B, C$ affectés des coefficients $\alpha, \beta, \gamma$ et dans ce cas comme $$\vec{f}(M)=\vec{f}(N)+(\alpha+\beta+\gamma) \overrightarrow{M N}$$ Pour $\mathrm{N}=\mathrm{G}$ on a: \begin{eqnarray*} \vec{f}(M) &=& \vec{f}(G)+(\alpha+\beta+\gamma) \overrightarrow{M G}\\ &=& \vec{f}(G)+(\alpha+\beta+\gamma) \overrightarrow{M G}\\ &=& \overrightarrow{0}+(\alpha+\beta+\gamma) \overrightarrow{M G}\\ &=& (\alpha+\beta+\gamma) \overrightarrow{M G} \end{eqnarray*}
Remarque: on peut généraliser la définition de cette fonction, quelque soit le nombre $n$ de points $(n \geq 2)$. Pour $\vec{f}(M)=\alpha_{1} \overrightarrow{M A_{1}}+\alpha_{2} \overrightarrow{M A_{2}}+\ldots+\alpha_{n} \overrightarrow{M A_{n}}$ les propriétés restes analogues.
Définition 2: On appelle barycentre des points A et B ( ou A et B deux points du plan ou de l'espace) affectés respectivement des coefficients \(\alpha, \beta\) ( ou \(\alpha, \beta\) sont des réels tels que \(\alpha+\beta \neq 0\) ) l'unique point \(G\) tel que pour tout point \(M\) du plan ou de l'espace on a \(:(\alpha+\beta) \overrightarrow{M G}=\alpha \overrightarrow{M A}+\beta \overrightarrow{M B}\) (2) Pour placer le point \(G,\) on peut prendre \(M=A d^{\prime}\) où \(: \quad \overrightarrow{A G}=\frac{\beta}{\alpha+\beta} \cdot \overrightarrow{A B}\)
Why I created this site: I was tired of an extremely confusing system of handwritten notes, word documents, Evernote, iPhone notes, and PDFs, so I decided to make a more sustainable solution. The first major problem with my notes is my handwriting, which isn’t great and is probably getting worse (thanks, Internet). The second is my organization skills, which aren’t great and are definitely getting worse.
In Spring 2017, I pretty much gave up at having any sort of “system” at all. That semester, I pretty much just carried around one tiny moleskine notebook, which contained everything from chord charts to random thoughts to de-classified notes from all my classes. I realized this actually wasn’t too bad of an idea, since at least I knew where everything was. If only I could Ctrl+F my notebook…
So that’s what this site is, a virtual version of my notebook, in the form of a Github repository for ease of use. There’s also a script on the backend that automatically renders the notes into PDFs using LaTeX for downloading and printing purposes. Hopefully I’ll be killing fewer trees from now on.
The site is always under construction, so check back for regular updates.
What I’ve Been Up To Lately
For some miscellaneous math writing, check out my post where I look into the surprisingly fascinating history of “the quintic equation”. You can also check my notes page for some of my notes about various math topics. I’m currently studying for an upcoming round of comprehensive exams. You can check out my notes here.
For some writing about music, check out my post analyzing a short chord progression in a Kendrick Lamar song or my review of the new Haim album. I also wrote a post breaking down 5 EPs you definitely haven’t listened to from the first half of 2017.
You can also check out some various other coding projects here.
Recent Posts
- A Note on the Variation of Parameters Method | 11/01/17
- Group Theory, Part 3: Direct and Semidirect Products | 10/26/17
- Galois Theory, Part 1: The Fundamental Theorem of Galois Theory | 10/19/17
- Field Theory, Part 2: Splitting Fields; Algebraic Closure | 10/19/17
- Field Theory, Part 1: Basic Theory and Algebraic Extensions | 10/18/17