OpenLLM-France/wikimedia huggingface dataset:
# Korean/Words/설

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 설
### Noun

* 설날. 음력설과 양력설을 통틀어 이르는 말.
    (lunar or solar) new year's day

### Relatives

* 살 (sal, "of age")

### Comparatives

* 춘절 (春節, chunjeol, "Chinese New Year")
* voorjaar #Dutch ("spring," literally, "fore of year")
* vor #German ("ahead of" cf. English "fore" Dutch "voor")
* vor #Icelandic ("spring season")
* jaro #Czech ("spring season")
* year #English ("year")
* Yule #English ("Yule")
* sol #Latin ("sun") & sol #Descendants
* ↑ Busy Koreans could do without so many artificial Sino-Korean synonyms such as: 세수 (歲首) 세시 (歲始) 세초 (歲初) 연두 (年頭) 연수 (年首) 연시 (年始) 원단 (元旦) 원신 (元新) 원일 (元日) 원정 (元正) 원조 (元朝) 정조 (正朝) 정초 (正初), etc.
* ↑ To let it be "Korean New Year" may to play it down.
* ↑ Naver: 춘절 =봄철(계절이 봄인 때). Wikt: 춘절 the spring season, the springtime Chinese New Year, Spring Festival
* ↑ "from Proto-Indo-European *yōro-, *yeh₁ro- (“year, spring”)"
* ↑ "from Old English ġēr, ġēar (“year”), from Proto-Germanic *jērą (“year”), from Proto-Indo-European *yōro-, *yeh₁ro- (“year, spring”)"
* ↑ "from Old English ġeōl, ġeola (“midwinter”)"
# Korean/Words/세

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 세
### Adjective

* [관형사] 그 수량이 셋임을 나타내는 말.
    (adj.) three

### Relatives

* 셋 (set, "three") num.

### Comparatives

* sê #Kurdish ("three")
* سه‎ #Persian (se, "three")
* се #Tajik (se, "three")
* Setar

### See also

* 네 (ne, "four") adj.
* 넷 (net, "four") num.
* ↑ Etymology The Persian word سه‌تار setâr is a combination of سه se—meaning "three"—and تار târ—meaning "string", therefore the word gives the meaning of "three-stringed" or "tri-stringed".
# Korean/Words/심다

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 심다
### Verb

* 초목의 뿌리나 씨앗 따위를 흙 속에 묻다.
    to plant, sow, cf. French semer'

### Relatives

* 씨앗 (ssiat, "seed") cf. Latin semen
* 뿌리다 (ppuri-da, "to spray, sow")
* ↑ https://ko.dict.naver.com/#/search?query=심다
* ↑ https://en.wiktionary.org/wiki/심다
* ↑ These are what Latin semino means.
# Korean/Words/아래아
## 아래아

Though uniquely outlawed of the original vowels, the so-called arae-a need be written to study Korean words better. The easiest way may be writing it as "ㅡ" while italicizing the syllable. The grave accent (`) shall stand for the arae-a in romanizing.

### Examples

* 해무리 (haemuri, "solar halo")
** 햇무리 (haetmuri)
** 희무리 (h`i-muri)
** 희ㅅ모로 (h`i-t-moro)
** 흿모로 (h`it-moro)
* 한데 (hande)
    흔듸 (h`n-d`i)
* one place
* open air
* together
# Korean/Words/세다

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 세다
### Adjective

* 힘이 많다. 물, 불, 바람 따위의 기세가 크거나 빠르다. 행동하거나 밀고 나가는 기세 따위가 강하다.
    to be powerful, strong

### Compounds

* 거세다 (geo-, "to be rough, wild")
* 굳세다 {gud-, "to be tough, willful")

### Relatives

* 세 #Etymology 1 Conjugation (se)
* 세 (勢, se, "power, force")
* 水勢 #Chinese #Japanese (Korean: 수세, suse) lit. "water power"
* 쇠 (soe, "iron, metal")

### Comparatives

| Dutch: zee English: sea German: See, etc. Middle English: see Old English: sǣ Old Saxon: sēo | Danish: sø, etc. Icelandic: sjór, etc. Norwegian: sjø, etc. Swedish: sjö, etc. |

### See also

* 바다 (bada, "sea")
* ↑ sea (n.) [...] of unknown origin, outside connections "wholly doubtful" [Buck].
* ↑ La Rocco Tower with La Corbière lighthouse in the background.
* ↑ https://ko.dict.naver.com/#/search?query=세다
* ↑ https://en.wiktionary.org/wiki/세다
* ↑ 사물의 기세 따위가 몹시 거칠고 세차다. 성격 따위가 거칠고 억세다.
* ↑ 힘차고 튼튼하다. 뜻한 바를 굽히지 않고 밀고 나아가는 힘이 있다.
* ↑ This is not only the stem of the adjective 세다 at hand, but also its indicative & interrogative, non-past, informal non-polite, sentence-final form, as it were!
* ↑ IPA: /siː/
* ↑ IPA: /sɛː/, /seː/
# Korean/Words/술

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 술
### Noun

* 알코올 성분이 들어 있어 마시면 취하는 음료. 적당히 마시면 물질대사를 높이는 효과가 있다. 맥주, 청주, 막걸리 따위의 발효주와 소주, 고량주, 위스키 따위의 증류주가 있으며, 과실이나 약제를 알코올과 혼합하여 만드는 혼성주도 있다.
    liquor, alcoholic drink, strong drink

### Compounds

* 술병 (-byeong, "liquor bottle")

### Relatives

* 막걸리 (maggeolli, "Korean rice wine, makgeolli or makkoli")
* 쉬다 (swi-da, "to turn sour, spoil")

### Comparatives

* Descendants of *sūraz "sour, acidic"
* Old English: sūr
* Middle English: sur, sour
* English: sour
* Scots: sour, soor
* Old Dutch: sūr
* Middle Dutch: suur
* Dutch: zuur
* Limburgish: zoer
* Old High German: sūr
* German: sauer
* Luxembourgish: sauer
* Old Norse: súrr
* Danish: sur
* Faroese: súrur
* Icelandic: súr
* Norwegian: sur
* Swedish: sur
* ↑ https://ko.dict.naver.com/#/search?query=술
* ↑ https://en.wiktionary.org/wiki/술
* ↑ w: Liquor #Etymology The origin of "liquor" and its close relative "liquid" was the Latin verb liquere, meaning "to be fluid". According to the Oxford English Dictionary, an early use of the word in the English language, meaning simply "a liquid", can be dated to 1225. The first use the OED mentions of its meaning "a liquid for drinking" occurred in the 14th century. Its use as a term for "an intoxicating alcoholic drink" appeared in the 16th century.
# Korean/Words/쑤시다

* ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ -ㄱ-ㄲ-ㄴ-ㄷ-ㄸ-ㄹ-ㅁ-ㅂ-ㅃ-ㅅ-ㅆ-ㅇ-ㅈ-ㅉ-ㅊ-ㅋ-ㅌ-ㅍ-ㅎ- 말+/말씀+/물+/불+ a2z/Basics/News/Notes/Table/Legend/Sandbox

## 쑤시다
### Verb

* 가늘고 긴 꼬챙이 같은 것으로 다른 물체의 틈이나 구멍에 넣어 구멍을 뚫거나 속에 있는 물건을 밖으로 나오게 하다.
    to stoke, poke, peirce, thrust

### Relatives

* 부지깽이 (bujikkaeng-i, "firestick")

### Comparatives

| 불쑤시개(bul-ssusigae, "firestick") | Ditch stoken "to poke, stoke" |
| 불쏘시개(bul-ssosigae, "kindler, firewood") | Ditch stoken "to light, to start (fire)" |
* ↑ https://ko.dict.naver.com/#/search?query=쑤시다
* ↑ https://en.wiktionary.org/wiki/쑤시다
* ↑ Possibly from the short 부 (bu) of 불(bul, "fire") + the sound shift of 지팡이(jipang-i, "stick")
* ↑ This is cognate with the above.
# MATLAB/Vectors

Quick way to show vector algebra in Matlab with matrices:
EDU>> a = [ 1 2; 3 4 ]
a =
EDU>> b = [1;0]
b =
EDU>> a\b
ans =
EDU>> quit
# MATLAB/Sound/Just and equal tuning

The two codes play the same riff, both using just and equal tuning. The only difference between the two versions is whether the just or equal tuned riff is first. If the variable testnote equals true, then major sixth above the first note is played at the end, for comparison.
On our speaker, it feels as if the sharper note is also louder. If that is true, then this is not a true test of whether the human ear prefers the just major sixth.
The codes can be copied and pasted into MATLAB either directly, or by going into Edit mode. The advantage of the latter is that if you select only the section, you can "select all" and then delete the first and last lines (<pre> ... <\pre>) after pasting.
# Occupational Epidemiology/Research tools/Ongoing projects/Mental Health in Seafaring/Draft Protocol
## Title page

* Article title
* Authors
* Afiliaciones
* Contact, name, mail, afiliacion

## Abstract

100-150 palabras

##### Expected results
## Introduction Background/rationale

* Explain the scientific background and rationale for the investigation being done

## Objectives

* State the research questions, including any prespecified hypotheses

## Methods
##### Study design

* Present key elements of study design early in the paper

##### Setting

* Describe the setting, locations, and relevant dates, including periods of recruitment, exposure, follow-up, and data collection

##### Participants

* (a) Give the eligibility criteria, and the sources and methods of selection of participants

##### Variables

* Clearly define all outcomes, exposures, predictors, potential confounders, and effect modifiers. Give diagnostic criteria, if applicable

##### Data sources/ measurement

* For each variable of interest, give sources of data and details of methods of assessment (measurement). Describe comparability of assessment methods if there is more than one group

##### Bias

* Describe any efforts to address potential sources of bias

##### Study size

* Explain how the study size was arrived at
* Quantitative variables
* Explain how quantitative variables were handled in the analyses. If applicable, describe which groupings were chosen and why

##### Statistical methods

* a) Describe all statistical methods, including those used to control for confounding
* (b) Describe any methods used to examine subgroups and interactions
* (c) Explain how missing data were addressed
* (d) If applicable, describe analytical methods taking account of sampling strategy
* (e) Describe any sensitivity analyses

## Ethical requirements

The ethical rules for database research in Denmark and for the University of Southern Denmark are complied with. Confidentiality in handling personal information is done according to the rules set out by the Danish Data Protection Agency. There are no personal sensitive information included so approval from the Ethics Committee or written informed consent is not necessary. Data and results will be used for the public health master thesis in the University of Southern Denmark.

## Informed consent

The participants will be explained the purpose and details of the study through consent from the beginning of the questionnaire before the start. All of information from the participants will be confidential and only used for scientific purpose.

##### Chronogram

Where and when will the tasks be completed
# Occupational Epidemiology/Research tools/Ongoing projects/Mental Health in Seafaring in Thailand/Draft Protocol
## Title page

* Article title
* Authors
* Afiliaciones
* Contact, name, mail, afiliacion

## Abstract

100-150 palabras

##### Expected results
## Introduction Background/rationale

* Explain the scientific background and rationale for the investigation being done

## Objectives

* State the research questions, including any prespecified hypotheses

## Methods
##### Study design

* Present key elements of study design early in the paper

##### Setting

* Describe the setting, locations, and relevant dates, including periods of recruitment, exposure, follow-up, and data collection

##### Participants

* (a) Give the eligibility criteria, and the sources and methods of selection of participants

##### Variables

* Clearly define all outcomes, exposures, predictors, potential confounders, and effect modifiers. Give diagnostic criteria, if applicable

##### Data sources/ measurement

* For each variable of interest, give sources of data and details of methods of assessment (measurement). Describe comparability of assessment methods if there is more than one group

##### Bias

* Describe any efforts to address potential sources of bias

##### Study size

* Explain how the study size was arrived at
* Quantitative variables
* Explain how quantitative variables were handled in the analyses. If applicable, describe which groupings were chosen and why

##### Statistical methods

* a) Describe all statistical methods, including those used to control for confounding
* (b) Describe any methods used to examine subgroups and interactions
* (c) Explain how missing data were addressed
* (d) If applicable, describe analytical methods taking account of sampling strategy
* (e) Describe any sensitivity analyses

## Informed consent

The participants will be explained the purpose and details of the study through confirming the consent from the beginning of the questionnaire before the start. All of information from the participants will be confidential and only used for scientific purpose.

## Ethical requirements

The ethical rules for database research in Denmark and for the University of Southern Denmark are complied with. Confidentiality in handling personal information is done according to the rules set out by the Danish Data Protection Agency. There are no personal sensitive information included so approval from the Ethics Committee or written informed consent is not necessary. Data and results will be used for the public health master thesis in the University of Southern Denmark.

## Informed consent

The participants will be explained the purpose and details of the study through consent from the beginning of the questionnaire before the start. All of information from the participants will be confidential and only used for scientific purpose.

##### Chronogram

Where and when will the tasks be completed
# Occupational Epidemiology/Research tools/Ongoing projects/Oral Health for fishermen
#### Project title
## Oral health, habits and practice of dental hygiene among seafarers
##### Investigator:

Tuan Nguyen Anh Pham

##### Facilitators

OJ/FB

#### Tasks to be done

* Create an account in Wikiversity .
* Select and complete relevant methodology courses: Theoretical Program - open here
* Draft Protocol
* Revision of the literature
* Collection of data
* Analysis of data
* Draft Article guideline IMRAD
* Dissemination of results

##### References

How to write medical articles
# Occupational Epidemiology/Research tools/Ongoing projects/Fatal injuries in fishing in Bangladesh/Protocol for fatal accidents in fishing study
## Title page

* Article title
* Authors
* Afiliaciones
* Contact, name, mail, afiliacion

## Abstract

100-150 palabras

##### Expected results
## Introduction Background/rationale

* Explain the scientific background and rationale for the investigation being done

## Objectives

* State the research questions, including any prespecified hypotheses

## Methods
##### Study design

* Present key elements of study design early in the paper

##### Setting

* Describe the setting, locations, and relevant dates, including periods of recruitment, exposure, follow-up, and data collection

##### Participants

* (a) Give the eligibility criteria, and the sources and methods of selection of participants

##### Variables

* Clearly define all outcomes, exposures, predictors, potential confounders, and effect modifiers. Give diagnostic criteria, if applicable

##### Data sources/ measurement

* For each variable of interest, give sources of data and details of methods of assessment (measurement). Describe comparability of assessment methods if there is more than one group

##### Bias

* Describe any efforts to address potential sources of bias

##### Study size

* Explain how the study size was arrived at
* Quantitative variables
* Explain how quantitative variables were handled in the analyses. If applicable, describe which groupings were chosen and why

##### Statistical methods

* a) Describe all statistical methods, including those used to control for confounding
* (b) Describe any methods used to examine subgroups and interactions
* (c) Explain how missing data were addressed
* (d) If applicable, describe analytical methods taking account of sampling strategy
* (e) Describe any sensitivity analyses

## Ethical requirements

The ethical rules for database research in Denmark and for the University of Southern Denmark are complied with. Confidentiality in handling personal information is done according to the rules set out by the Danish Data Protection Agency. There are no personal sensitive information included so approval from the Ethics Committee or written informed consent is not necessary. Data and results will be used for the public health master thesis in the University of Southern Denmark.

## Informed consent

The participants will be explained the purpose and details of the study through consent from the beginning of the questionnaire before the start. All of information from the participants will be confidential and only used for scientific purpose.

##### Chronogram

Where and when will the tasks be completed
# Occupational Epidemiology/Research tools/Ongoing projects/Mental Health in Seafaring Iceland

MENTAL HEALTH IN ICELANDIC SEAFARING

#### Investigators:

Dr. Ing. Stefán Einarsson and Dr. phil. Valdimar Briem

#### Scientific advisor

Dr.med.Olaf Jensen

#### Tasks to be done

* Create an account in Wikiversity .
* Select and complete relevant methodology courses: Theoretical Program - open here
* Draft Protocol to be developed
* Copenhagen-Psychosocial-Questionnaire-COPSOQ-II
* Questions about ship and person to be added at the beginning of the COPSQII
* Revision of the literature
* Collection of data
* Analysis of data
* The STROBE statement guide for writing of the article
* Dissemination of results

#### References

* BBC news on maritime mental health advice based on the research Nov 2019
* Seafarers mental health and wellbeing SIRC research Nov 2019
# Occupational Epidemiology/Research tools/Ongoing projects/Fatal injuries in fishing in Bangladesh/Project calendar
## Chronogram Bangladesh program 2019-2020

| | Nov | Dec | Jan | Oct |
| Prepare contract with budget | x | x | | |
| Prepare teaching materials | x | x | x | |
| Protocol+questionnaire fatal accidents | x | x | x | |
| Data collection in the villages | | | | |
| Travel and field work in Bangladesh OJ | | | | x |
# Propositional logic/De Morgan's laws/2/Proof/Exercise

Prove, with help of truth tables, the (generalized) De Morgan's laws, namely that the statements
    ${\left(\alpha \wedge \neg {\left(\beta \vee \gamma \right)}\right)}\leftrightarrow {\left({\left(\alpha \wedge \neg \beta \right)}\wedge {\left(\alpha \wedge \neg \gamma \right)}\right)}$
and
    ${\left(\alpha \wedge \neg {\left(\beta \wedge \gamma \right)}\right)}\leftrightarrow {\left((\alpha \wedge \neg \beta )\vee {\left(\alpha \wedge \neg \gamma \right)}\right)}$
are tautologies.
# Proper time

Proper time refers to the passage of time as experienced by the thing being observed specifically at its location. For example, an astronaught in orbit will experience time differently than an identical twin remaining on the planet. The time experience for the astronaught at his location such as his rate of aging or his wristwatch's time may be thought of as measures of his proper time. Lets say the astronaught uses coordinates $\left(ct',x',y',z'\right)$, then his proper time may be designated by $\tau$ is given by
* $\tau =t'|_{(x',y',z')=(0,0,0)}$
If we observe him with coordinates we designate unprimed and describe his path through spacetime called his world line, with the line element, then that line element gives us the flow of his proper time according to
* $ds^{2}=dc\tau ^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }$
For example, lets say the astronaught moved about in flat spacetime and that we use an inertial frame to observe him so that the line element corresponds to the metric of special relativity which we use to describe his path
* $ds^{2}=dc\tau ^{2}=dct^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)$
factoring out our coordinate time differential yields
* $dc\tau ^{2}=\left(1-{\frac {v^{2}}{c^{2}}}\right)dct^{2}$
simplify
* $dt={\frac {d\tau }{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$
which is special relativity's time dilation formula.
# Propositional logic/De Morgan's laws/Proof/Exercise

Prove, with help of truth tables, the De Morgan's laws, namely that the statements
    $\neg {\left(\beta \vee \gamma \right)}\leftrightarrow {\left(\neg \beta \wedge \neg \gamma \right)}$
and
    $\neg {\left(\beta \wedge \gamma \right)}\leftrightarrow {\left(\neg \beta \vee \neg \gamma \right)}$
are tautologies.
# Online Teaching

resources, guides and tools for enabling online teaching.
Check whether your university has already some guides, on our University guides page.
# One over a trillion/One over a billion/Amount/Exercise

How many trillionths does it take to reach one billionth?
# PlanetPhysics/Transformation Between Cartesian Coordinates and Polar Coordinates

From the definition of a contravariant vector (contravariant tensor of rank 1)
   ${\bar {T}}^{i}=T^{j}{\frac {\partial {\bar {x}}^{i}}{\partial x^{j}}}$
we get the transformation matrix from the partial derivatives
   $A_{ij}={\frac {\partial {\bar {x}}^{i}}{\partial x^{j}}}$
In order to calculate the transformation matrix, we need the equations relating the two coordinates systems. For cartesian to polar, we have
$r={\sqrt {x^{2}+y^{2}}}$
$\theta =tan^{-1}\left({\frac {y}{x}}\right)$
and for polar to cartesian
$x=r\cos \theta$
$y=r\sin \theta$
So if we designate $(x,y)$ as the bar coordinates, then the transformation components from polar coordinates $(r,\theta )$ to cartesian coordinates $(x,y)$ is calculated as
$A_{11}={\frac {\partial {\bar {x}}^{1}}{\partial x^{1}}}={\frac {\partial x}{\partial r}}=\cos \theta$
$A_{12}={\frac {\partial {\bar {x}}^{1}}{\partial x^{2}}}={\frac {\partial x}{\partial \theta }}=-r\sin \theta$
$A_{21}={\frac {\partial {\bar {x}}^{2}}{\partial x^{1}}}={\frac {\partial y}{\partial r}}=\sin \theta$
$A_{22}={\frac {\partial {\bar {x}}^{2}}{\partial x^{2}}}={\frac {\partial y}{\partial \theta }}=r\cos \theta$
The components from cartesian coordinates to polar coordinates transform the same way, but now the polar coordinates have the bar
$B_{11}={\frac {\partial {\bar {x}}^{1}}{\partial {x}^{1}}}={\frac {\partial r}{\partial x}}={\frac {x}{\sqrt {x^{2}+y^{2}}}}$
$B_{12}={\frac {\partial {\bar {x}}^{1}}{\partial {x}^{2}}}={\frac {\partial r}{\partial y}}={\frac {y}{\sqrt {x^{2}+y^{2}}}}$
$B_{21}={\frac {\partial {\bar {x}}^{2}}{\partial {x}^{1}}}={\frac {\partial \theta }{\partial x}}=-{\frac {y}{x^{2}+y^{2}}}$
$B_{22}={\frac {\partial {\bar {x}}^{2}}{\partial {x}^{2}}}={\frac {\partial \theta }{\partial y}}={\frac {x}{x^{2}+y^{2}}}$
In summary, the {\mathbf components of contravariant vectors} in cartesian coordinates and polar coordinates transform between each other according to
$\left[{\begin{matrix}x\\y\end{matrix}}\right]=\left[{\begin{matrix}\cos \theta &-r\sin \theta \\\sin \theta &r\cos \theta \end{matrix}}\right]\left[{\begin{matrix}r\\\theta \end{matrix}}\right]$
$\left[{\begin{matrix}r\\\theta \end{matrix}}\right]=\left[{\begin{matrix}{\frac {x}{\sqrt {x^{2}+y^{2}}}}&{\frac {y}{\sqrt {x^{2}+y^{2}}}}\\-{\frac {y}{x^{2}+y^{2}}}&{\frac {x}{x^{2}+y^{2}}}\end{matrix}}\right]\left[{\begin{matrix}x\\y\end{matrix}}\right]$