This document contains over 1300 math formulas organized into chapters covering topics from number sets and algebra to calculus, differential equations, and probability theory. It is intended as a complete desktop reference for students and engineers, with formulas, tables, and figures to support math learning from high school through undergraduate engineering. Formulas are structured and linked for easy navigation as an online reference guide.
This document provides a summary of the life and works of the Indian writer Mani Shankar Mukherjee, known by his pen name Mani Shankar Mukherjee. Some key points:
1. He was born in 1956 in West Bengal and worked as a journalist and professor.
2. Starting in the 1980s, he wrote extensively about culture, philosophy and social issues in India through short stories and novels.
3. His works, which number over 40,000 pages and 55,000 characters, have been translated into 63 languages.
4. They provide insight into Indian society while entertaining readers with their humor and realistic characters.
Lataif e ashrafi malfoozat e syed makhdoom ashraf 30Aale Rasool Ahmad
لطائف اشرفی حصہ اول
یا
ملفوظات سید مخدوم اشرف جہانگیرسمنانی کچھوچھوی
مترجم
حضرت شمس بریلوی
نظر ثانی
ڈاکٹر خضر نوشاہی
مدیر و ناشر
نذر اشرف شیخ محمد ہاشم رضا اشرفی
خلیفہ مجاز مخدوم المشائخ غوث الوقت سید مختار اشرف اشرف جیلانی سرکارکلاں کچھوچھو شریف
یہ کتاب تارک السلطنت غوث العالم محبوب یزدانی سلطان اوحدالدین قدوۃ الکبریٰ مخدوم سیداشرف جہانیاں جہانگیر سمنانی رضی اللہ تعالیٰ عنہ کے ملفوظات اور ارشادات و کمالات و فضائل پر مبنی ہے جسے انکے مرید خاص ابوالفضائل شیخ الاسلام والمسلمین حضرت نظام یمنی المعروف نظام حاجی غریب یمنی قدس سرہ النورانی نے مرتب کیا ۔ وہ آپکے خدمت میں تیس سال رہے۔
اپلود آن ارچیو
دیوانۂ مجمع البحرین حاجی الحرمین الشریفین اعلیٰ حضرت قدسی منزلت مخدوم الاولیاء مرشدالعالم محبوب ربانی ہم شبیہ غوث الاعظم حضرت سید شاہ ابواحمد المدعومحمد علی حسین اشرف اشرؔفی میاں الحسنی الحسینی قدس سرہ النوران
الحاج شیخ آلِ رسول احمد الصدیقی الاشرفی القادری کٹیہاری
This document provides an overview of the objectives and methods of a literature review project on traditional medicine being conducted by Dr. K.V. George of St. John's Medical College, Bangalore. The project aims to systematically review the existing body of knowledge on traditional medicine from ancient texts and evaluate evidence for treatments and practices. It involves analyzing texts to understand disease concepts, treatment approaches, and the role of physicians over time in order to glean insights for current medical education and practice. The review will synthesize information from various sources and perspectives to develop a comprehensive picture of the evolution of traditional medicine concepts.
This document summarizes training modules offered by Apec, a French organization that provides career development services and labor market research. It outlines 7 new training modules focused on topics like recruiting the next generation, integrating social media in recruiting strategy, and using social networks to build an effective sourcing strategy. The training is designed to help companies analyze recruiting trends, identify levers to optimize internal recruiting processes, develop skills, and exchange experiences with peers. Modules can be taken individually or customized for companies. Apec consultants draw on experience and innovative pedagogical methods to share solutions and facilitate exchanges.
You can find many more Government documents about aliens and UFOs here: http://alien-ufo-research.com/documents
In 1947 there was a rash of UFO sightings across the U.S. The FBI documented them. Here is one of the many files.
This document provides details about an innovative teaching manual for teaching 8th standard Social Science students about the Jallianwala Bagh massacre event in India. The document includes the name of the student teacher, school, subject, unit being taught, duration of the lesson, learning objectives, content analysis describing key terms, people and facts about the event, concept being taught, prerequisites, teaching resources, classroom interaction procedures and expected student responses. The lesson plan aims to help students understand the significance of the Jallianwala Bagh massacre event in Indian history through activities analyzing related documents and responding to discussion questions.
1. The document is a summary of the short story "The Old Men and the Sea" by K.V. Simon. It discusses the story's themes of aging, loneliness, and humanity.
2. The story portrays the struggles of elderly fishermen who are too old to fish but continue doing so out of necessity. It explores how their community views and treats them.
3. The teacher engages the students in a discussion about the themes, characters, and social commentary in the story. Students analyze aspects of the short story and reflect on its portrayal of human conditions.
1. This document is a chapter from an astrology text in Malayalam on the effects of houses. It discusses the characteristics and qualities associated with the first house, moon and signs for houses 2 through 14, and the 15th house.
2. It then provides details on the effects on personality and qualities based on the placement of planets in the different houses. For example, if the planet is in the first house, one will be self-focused, if in the second house, financially well-off, and if in the third house, communicative.
3. It also summarizes the effects of planets for the second, third and fourth signs, indicating things like occupation, family relationships and temperament
This document provides a summary of the life and works of the Indian writer Mani Shankar Mukherjee, known by his pen name Mani Shankar Mukherjee. Some key points:
1. He was born in 1956 in West Bengal and worked as a journalist and professor.
2. Starting in the 1980s, he wrote extensively about culture, philosophy and social issues in India through short stories and novels.
3. His works, which number over 40,000 pages and 55,000 characters, have been translated into 63 languages.
4. They provide insight into Indian society while entertaining readers with their humor and realistic characters.
Lataif e ashrafi malfoozat e syed makhdoom ashraf 30Aale Rasool Ahmad
لطائف اشرفی حصہ اول
یا
ملفوظات سید مخدوم اشرف جہانگیرسمنانی کچھوچھوی
مترجم
حضرت شمس بریلوی
نظر ثانی
ڈاکٹر خضر نوشاہی
مدیر و ناشر
نذر اشرف شیخ محمد ہاشم رضا اشرفی
خلیفہ مجاز مخدوم المشائخ غوث الوقت سید مختار اشرف اشرف جیلانی سرکارکلاں کچھوچھو شریف
یہ کتاب تارک السلطنت غوث العالم محبوب یزدانی سلطان اوحدالدین قدوۃ الکبریٰ مخدوم سیداشرف جہانیاں جہانگیر سمنانی رضی اللہ تعالیٰ عنہ کے ملفوظات اور ارشادات و کمالات و فضائل پر مبنی ہے جسے انکے مرید خاص ابوالفضائل شیخ الاسلام والمسلمین حضرت نظام یمنی المعروف نظام حاجی غریب یمنی قدس سرہ النورانی نے مرتب کیا ۔ وہ آپکے خدمت میں تیس سال رہے۔
اپلود آن ارچیو
دیوانۂ مجمع البحرین حاجی الحرمین الشریفین اعلیٰ حضرت قدسی منزلت مخدوم الاولیاء مرشدالعالم محبوب ربانی ہم شبیہ غوث الاعظم حضرت سید شاہ ابواحمد المدعومحمد علی حسین اشرف اشرؔفی میاں الحسنی الحسینی قدس سرہ النوران
الحاج شیخ آلِ رسول احمد الصدیقی الاشرفی القادری کٹیہاری
This document provides an overview of the objectives and methods of a literature review project on traditional medicine being conducted by Dr. K.V. George of St. John's Medical College, Bangalore. The project aims to systematically review the existing body of knowledge on traditional medicine from ancient texts and evaluate evidence for treatments and practices. It involves analyzing texts to understand disease concepts, treatment approaches, and the role of physicians over time in order to glean insights for current medical education and practice. The review will synthesize information from various sources and perspectives to develop a comprehensive picture of the evolution of traditional medicine concepts.
This document summarizes training modules offered by Apec, a French organization that provides career development services and labor market research. It outlines 7 new training modules focused on topics like recruiting the next generation, integrating social media in recruiting strategy, and using social networks to build an effective sourcing strategy. The training is designed to help companies analyze recruiting trends, identify levers to optimize internal recruiting processes, develop skills, and exchange experiences with peers. Modules can be taken individually or customized for companies. Apec consultants draw on experience and innovative pedagogical methods to share solutions and facilitate exchanges.
You can find many more Government documents about aliens and UFOs here: http://alien-ufo-research.com/documents
In 1947 there was a rash of UFO sightings across the U.S. The FBI documented them. Here is one of the many files.
This document provides details about an innovative teaching manual for teaching 8th standard Social Science students about the Jallianwala Bagh massacre event in India. The document includes the name of the student teacher, school, subject, unit being taught, duration of the lesson, learning objectives, content analysis describing key terms, people and facts about the event, concept being taught, prerequisites, teaching resources, classroom interaction procedures and expected student responses. The lesson plan aims to help students understand the significance of the Jallianwala Bagh massacre event in Indian history through activities analyzing related documents and responding to discussion questions.
1. The document is a summary of the short story "The Old Men and the Sea" by K.V. Simon. It discusses the story's themes of aging, loneliness, and humanity.
2. The story portrays the struggles of elderly fishermen who are too old to fish but continue doing so out of necessity. It explores how their community views and treats them.
3. The teacher engages the students in a discussion about the themes, characters, and social commentary in the story. Students analyze aspects of the short story and reflect on its portrayal of human conditions.
1. This document is a chapter from an astrology text in Malayalam on the effects of houses. It discusses the characteristics and qualities associated with the first house, moon and signs for houses 2 through 14, and the 15th house.
2. It then provides details on the effects on personality and qualities based on the placement of planets in the different houses. For example, if the planet is in the first house, one will be self-focused, if in the second house, financially well-off, and if in the third house, communicative.
3. It also summarizes the effects of planets for the second, third and fourth signs, indicating things like occupation, family relationships and temperament
This document discusses probability and how to represent the likelihood of events occurring using percentages, fractions, decimals, and probability scales. It provides examples of assigning probability values to events like the sun rising or the temperature reaching a certain level. The key aspects covered are:
1) Probability involves the chance that an event will occur, ranging from impossible (0) to certain (1)
2) The probability of an event occurring plus the probability of it not occurring must equal 1
3) Probability can be expressed in multiple ways like fractions, decimals, percentages, and on a scale
This document discusses polygons and tessellations. It defines polygons, their properties including interior angles and the relationship between number of sides and total interior angle measures. Regular polygons are introduced as those with all congruent sides and interior angles. Methods for drawing regular polygons using angles and a circumscribed circle are provided. Tessellations are defined as arrangements that cover a region without gaps or overlaps. Regular hexagons, squares, and equilateral triangles can tessellate on their own. Problem solving involves calculating the measure of each interior angle of a regular hexagon.
The document provides an introduction to the GRE quantitative reasoning section. It discusses the structure of the GRE, including section types and time limits. It then covers the main quantitative topics - arithmetic, algebra, geometry, and data analysis. Specific concepts are outlined for each topic area. Finally, it discusses quantitative question types and provides tips for scoring highly, such as learning formulas, managing time well, and practicing extensively.
The document discusses strategies and techniques for solving quantitative problems. It emphasizes that problem solving requires creativity, organization, and experience. Some key points made include: keeping track of units is an important problem solving tool; no single strategy always works so flexibility is important; understanding the context and restating the problem can help clarify the solution; and the most effective problem solvers view challenges as opportunities to improve their skills through practice.
The document is a GRE score report for Srijeet Bhattacharjee which includes his test scores from June 27, 2014. It shows his Verbal Reasoning score was 164 (94th percentile), Quantitative Reasoning score was 164 (87th percentile), and Analytical Writing score was 5.0 (93rd percentile). The report also lists the score recipients as the London School of Economics and Political Science and the University of Warwick and provides information about score sending policies and interpretations.
The document provides an overview of probability fundamentals. It discusses key concepts like probability, random variables, probability distributions, classical probability, conditional probability, Bayes' theorem, and applications of probability in areas like reliability, quality control, and queuing systems. Examples are given to illustrate probability calculations and theorems.
This document discusses fundamentals of probability, including theoretical and empirical probability. It provides examples of calculating probabilities of events using formulas that take the number of outcomes of an event over the total number of possible outcomes. Examples include probabilities of rolling certain numbers on a die, being dealt certain cards, blood types based on parent genetics, and marital status based on US census data. Assignments are provided for classwork and homework problems.
This document discusses polygons and tessellations. It defines regular and non-regular polygons and explains that a tessellation is a repetitive pattern of polygons that fit together without overlaps or gaps. Examples are provided of classifying polygons by number of sides and determining which shapes can form tessellations. Students are asked to complete examples of identifying regular and irregular polygons, determining angle measures, and finding real-world examples of tessellations.
To work as a quantitative analyst on the buy side, skills in mathematics, programming, statistics, and financial modeling are essential. Quantitative analysts build analytical models to value financial instruments and manage risk, requiring expertise in areas like derivatives pricing, portfolio construction, and statistical analysis. Strong problem-solving abilities and communication skills are also important to effectively analyze complex problems and interact with other professionals.
The document discusses key concepts in probability theory including:
1. Probability theory provides tools to quantify uncertainties and assign probabilities using classical, relative frequency, and subjective approaches.
2. Key probability terms are defined such as experiment, event, sample space, independent and dependent events, mutually exclusive events, and union, intersection, and complement of events.
3. Basic probability rules are covered including the multiplication rule for independent events, addition rule for mutually exclusive events, and how to calculate the probability of events.
Two events are independent if the occurrence of one does not affect the probability of the other. The probability of two independent events occurring is calculated by multiplying their individual probabilities. Three events are dependent if the occurrence of one affects the probability of another. The probability of dependent events requires calculating the probability of the first event multiplied by the conditional probability of the second event given the first has occurred. Conditional probability expresses the probability of one event occurring given that another event has already occurred.
This document contains examples and solutions for quantitative reasoning questions that may appear on a high school placement test. The examples cover topics like patterns in numeric and algebraic series, percentages, averages, and comparing quantities. For each question, the solution provides a step-by-step explanation of the mathematical operations used to arrive at the correct answer. Visual representations like charts and equations are employed to illustrate the underlying logic. The goal is to help students learn strategies for setting up and solving different types of math problems systematically.
The document discusses key concepts in probability, including tree diagrams, sample spaces, theoretical and experimental probability, and ratios and rates. It provides examples of how to use tree diagrams to calculate probabilities of outcomes from multiple choices. It also gives examples of calculating greatest common factors, least common multiples, and solving ratio and rate problems. Interactive questions and explanations are included throughout to illustrate the concepts.
Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; OddsJason Aubrey
This document discusses probability concepts including union, intersection, and complement of events. It begins by defining the union and intersection of two events A and B. It then provides an example calculating the probability of rolling an odd number and a prime number with a die. Next, it discusses that the probability of a union of events A and B, P(A ∪ B), equals P(A) + P(B) only if A and B are mutually exclusive. It provides the general formula for probability of a union and works through an example. Finally, it begins working through calculating the probability of drawing a jack or club from a deck of cards.
The document provides information about quantitative aptitude study materials for the CAT exam from OneStopMBA.com. It includes 3 refresher books covering 27 math topics tested in the CAT with solved examples, chapter tests, and topic tests. The books cover number systems, percentages, permutation, combination, geometry, and functions. The materials aim to help students learn concepts, practice solving questions, and master test-taking skills for the quantitative section of the CAT exam.
Math 1300: Section 8-3 Conditional Probability, Intersection, and IndependenceJason Aubrey
The document defines conditional probability as the probability of an event occurring given that another event has already occurred. It provides an example of calculating conditional probability using a probability table and the formula P(A|B) = P(A intersect B) / P(B). The document also explains how conditional probability restricts the sample space to outcomes in the given event.
Teaching thinking and problem solving skillsAloy Salas
The document discusses teaching problem solving and thinking skills in the classroom. It outlines a five-stage model for problem solving: understanding the problem, describing barriers, identifying solutions, trying solutions, and evaluating results. It also discusses teaching algorithms, heuristics, and study skills to help students learn problem solving strategies. The goal is to engage students in authentic problem solving activities and have them explain their approaches.
This document provides information about a quantitative reasoning course. The course aims to help students gain a comprehensive understanding of mathematics and the ability to think critically and logically. It will cover topics like logical and quantitative thinking, arguments and reasoning, and the relationship between logic, science and mathematics. The course goals are to understand mathematics as a body of knowledge and a way of thinking, and to reason quantitatively on issues relevant to students and society. On completing the course, students should be able to analyze and evaluate arguments, understand mathematical concepts, and apply problem-solving skills to quantitative problems.
The document discusses fundamental counting principles for determining the number of possible outcomes of independent events. It covers permutations, or arrangements that consider order, and combinations, which do not consider order. Examples include determining the number of possible outfits, license plates, and committee selections using formulas for permutations and combinations.
This document provides an overview of probability concepts including:
- Classical probability which uses equally likely outcomes and sample spaces to calculate probabilities
- Empirical probability which is based on observed frequencies
- Addition rules for calculating probabilities of independent and dependent events
- Conditional probability which considers the probability of one event given another
- Multiplication rules for independent and dependent events
- Examples of calculating probabilities for single events, combinations of events, and conditional scenarios.
Instagram e il marketing: raccontare un brand attraverso le immagini. Documento presentato all'interno della seconda edizione del Social Birra negli spazi dell'Università La Sapienza di Roma.
This document provides an overview of non-deliverable forwards (NDFs), which are used to hedge exposures to currencies that have restrictions on foreign exchange or high political risk. NDFs are settled offshore in hard currency, avoiding local restrictions and credit risk. They allow companies and investors to hedge transaction exposures and financial investments. The document discusses when NDFs are used, how they are priced, an example of hedging a Korean equity portfolio with NDFs, and NDF documentation requirements.
This document discusses probability and how to represent the likelihood of events occurring using percentages, fractions, decimals, and probability scales. It provides examples of assigning probability values to events like the sun rising or the temperature reaching a certain level. The key aspects covered are:
1) Probability involves the chance that an event will occur, ranging from impossible (0) to certain (1)
2) The probability of an event occurring plus the probability of it not occurring must equal 1
3) Probability can be expressed in multiple ways like fractions, decimals, percentages, and on a scale
This document discusses polygons and tessellations. It defines polygons, their properties including interior angles and the relationship between number of sides and total interior angle measures. Regular polygons are introduced as those with all congruent sides and interior angles. Methods for drawing regular polygons using angles and a circumscribed circle are provided. Tessellations are defined as arrangements that cover a region without gaps or overlaps. Regular hexagons, squares, and equilateral triangles can tessellate on their own. Problem solving involves calculating the measure of each interior angle of a regular hexagon.
The document provides an introduction to the GRE quantitative reasoning section. It discusses the structure of the GRE, including section types and time limits. It then covers the main quantitative topics - arithmetic, algebra, geometry, and data analysis. Specific concepts are outlined for each topic area. Finally, it discusses quantitative question types and provides tips for scoring highly, such as learning formulas, managing time well, and practicing extensively.
The document discusses strategies and techniques for solving quantitative problems. It emphasizes that problem solving requires creativity, organization, and experience. Some key points made include: keeping track of units is an important problem solving tool; no single strategy always works so flexibility is important; understanding the context and restating the problem can help clarify the solution; and the most effective problem solvers view challenges as opportunities to improve their skills through practice.
The document is a GRE score report for Srijeet Bhattacharjee which includes his test scores from June 27, 2014. It shows his Verbal Reasoning score was 164 (94th percentile), Quantitative Reasoning score was 164 (87th percentile), and Analytical Writing score was 5.0 (93rd percentile). The report also lists the score recipients as the London School of Economics and Political Science and the University of Warwick and provides information about score sending policies and interpretations.
The document provides an overview of probability fundamentals. It discusses key concepts like probability, random variables, probability distributions, classical probability, conditional probability, Bayes' theorem, and applications of probability in areas like reliability, quality control, and queuing systems. Examples are given to illustrate probability calculations and theorems.
This document discusses fundamentals of probability, including theoretical and empirical probability. It provides examples of calculating probabilities of events using formulas that take the number of outcomes of an event over the total number of possible outcomes. Examples include probabilities of rolling certain numbers on a die, being dealt certain cards, blood types based on parent genetics, and marital status based on US census data. Assignments are provided for classwork and homework problems.
This document discusses polygons and tessellations. It defines regular and non-regular polygons and explains that a tessellation is a repetitive pattern of polygons that fit together without overlaps or gaps. Examples are provided of classifying polygons by number of sides and determining which shapes can form tessellations. Students are asked to complete examples of identifying regular and irregular polygons, determining angle measures, and finding real-world examples of tessellations.
To work as a quantitative analyst on the buy side, skills in mathematics, programming, statistics, and financial modeling are essential. Quantitative analysts build analytical models to value financial instruments and manage risk, requiring expertise in areas like derivatives pricing, portfolio construction, and statistical analysis. Strong problem-solving abilities and communication skills are also important to effectively analyze complex problems and interact with other professionals.
The document discusses key concepts in probability theory including:
1. Probability theory provides tools to quantify uncertainties and assign probabilities using classical, relative frequency, and subjective approaches.
2. Key probability terms are defined such as experiment, event, sample space, independent and dependent events, mutually exclusive events, and union, intersection, and complement of events.
3. Basic probability rules are covered including the multiplication rule for independent events, addition rule for mutually exclusive events, and how to calculate the probability of events.
Two events are independent if the occurrence of one does not affect the probability of the other. The probability of two independent events occurring is calculated by multiplying their individual probabilities. Three events are dependent if the occurrence of one affects the probability of another. The probability of dependent events requires calculating the probability of the first event multiplied by the conditional probability of the second event given the first has occurred. Conditional probability expresses the probability of one event occurring given that another event has already occurred.
This document contains examples and solutions for quantitative reasoning questions that may appear on a high school placement test. The examples cover topics like patterns in numeric and algebraic series, percentages, averages, and comparing quantities. For each question, the solution provides a step-by-step explanation of the mathematical operations used to arrive at the correct answer. Visual representations like charts and equations are employed to illustrate the underlying logic. The goal is to help students learn strategies for setting up and solving different types of math problems systematically.
The document discusses key concepts in probability, including tree diagrams, sample spaces, theoretical and experimental probability, and ratios and rates. It provides examples of how to use tree diagrams to calculate probabilities of outcomes from multiple choices. It also gives examples of calculating greatest common factors, least common multiples, and solving ratio and rate problems. Interactive questions and explanations are included throughout to illustrate the concepts.
Math 1300: Section 8 -2 Union, Intersection, and Complement of Events; OddsJason Aubrey
This document discusses probability concepts including union, intersection, and complement of events. It begins by defining the union and intersection of two events A and B. It then provides an example calculating the probability of rolling an odd number and a prime number with a die. Next, it discusses that the probability of a union of events A and B, P(A ∪ B), equals P(A) + P(B) only if A and B are mutually exclusive. It provides the general formula for probability of a union and works through an example. Finally, it begins working through calculating the probability of drawing a jack or club from a deck of cards.
The document provides information about quantitative aptitude study materials for the CAT exam from OneStopMBA.com. It includes 3 refresher books covering 27 math topics tested in the CAT with solved examples, chapter tests, and topic tests. The books cover number systems, percentages, permutation, combination, geometry, and functions. The materials aim to help students learn concepts, practice solving questions, and master test-taking skills for the quantitative section of the CAT exam.
Math 1300: Section 8-3 Conditional Probability, Intersection, and IndependenceJason Aubrey
The document defines conditional probability as the probability of an event occurring given that another event has already occurred. It provides an example of calculating conditional probability using a probability table and the formula P(A|B) = P(A intersect B) / P(B). The document also explains how conditional probability restricts the sample space to outcomes in the given event.
Teaching thinking and problem solving skillsAloy Salas
The document discusses teaching problem solving and thinking skills in the classroom. It outlines a five-stage model for problem solving: understanding the problem, describing barriers, identifying solutions, trying solutions, and evaluating results. It also discusses teaching algorithms, heuristics, and study skills to help students learn problem solving strategies. The goal is to engage students in authentic problem solving activities and have them explain their approaches.
This document provides information about a quantitative reasoning course. The course aims to help students gain a comprehensive understanding of mathematics and the ability to think critically and logically. It will cover topics like logical and quantitative thinking, arguments and reasoning, and the relationship between logic, science and mathematics. The course goals are to understand mathematics as a body of knowledge and a way of thinking, and to reason quantitatively on issues relevant to students and society. On completing the course, students should be able to analyze and evaluate arguments, understand mathematical concepts, and apply problem-solving skills to quantitative problems.
The document discusses fundamental counting principles for determining the number of possible outcomes of independent events. It covers permutations, or arrangements that consider order, and combinations, which do not consider order. Examples include determining the number of possible outfits, license plates, and committee selections using formulas for permutations and combinations.
This document provides an overview of probability concepts including:
- Classical probability which uses equally likely outcomes and sample spaces to calculate probabilities
- Empirical probability which is based on observed frequencies
- Addition rules for calculating probabilities of independent and dependent events
- Conditional probability which considers the probability of one event given another
- Multiplication rules for independent and dependent events
- Examples of calculating probabilities for single events, combinations of events, and conditional scenarios.
Instagram e il marketing: raccontare un brand attraverso le immagini. Documento presentato all'interno della seconda edizione del Social Birra negli spazi dell'Università La Sapienza di Roma.
This document provides an overview of non-deliverable forwards (NDFs), which are used to hedge exposures to currencies that have restrictions on foreign exchange or high political risk. NDFs are settled offshore in hard currency, avoiding local restrictions and credit risk. They allow companies and investors to hedge transaction exposures and financial investments. The document discusses when NDFs are used, how they are priced, an example of hedging a Korean equity portfolio with NDFs, and NDF documentation requirements.
This document provides information about the NEBOSH International Diploma in Occupational Health and Safety qualification, including an introduction to the benefits and requirements of the diploma. It outlines the qualification structure, units, assessment methods, policies, and sample examination question papers. The diploma is designed to provide knowledge of international management of health and safety principles and practice.
Ketterer Specialfahrzeuge AG introduces the new Ketterer Continental motorhome. At just over 10 meters long, it combines compact size with luxurious and spacious living spaces. Key features include a lounge-like living room, kitchen/dining area, bathroom with shower and toilet, bedroom, and optional pop-out living space extension. Even before its official release, the Ketterer Continental won an award for innovation in new mobility.
This is a sem-annually questionnaire put forward to north European companies in China, The purpose is to follow how these companies are viewing the future for their companies in China. The answers are then analysed and an index is the reult.
This document provides information about MediMultilingual, including its location, services, projects for 2019-2020, and collaboration with other organizations.
The document summarizes the activities of the 49th Annual Conference of the Indian Association of Pediatric Surgeons held in Bengaluru from January 5-8, 2023. Over 500 pediatric surgeons from across India participated in the conference. Various scientific sessions were held covering topics like advances in pediatric surgery, minimal access surgery, pediatric urology and oncology. Several awards and prizes were also distributed during the closing ceremony to recognize outstanding contributions in the field of pediatric surgery. The conference helped in exchanging new ideas and keeping participants abreast with the latest global developments and innovations in pediatric surgery.
This is the compilation of articles by late B.Chandrasekhar,a spirited human rights activist, sharp intellectual, impressive speaker and Special Public Prosecutor in the Tsundur massacre case.
The document discusses two temples - Qutub Minar (qP~üF) in Delhi built in 1172 CE and Quwwat-ul-Islam Mosque (qP~üZ) in Delhi built in 6063 AH. Qutub Minar is 72 meters tall with 379 steps while Quwwat-ul-Islam Mosque was the first mosque built in India. The archaeological survey details the findings at both sites, listing the materials and architectural features discovered.
2015 2016 manmatha samvatsara kaalanirnaya panchangam l s siddhanthySiddhanthygaru Ls
Sr manmatha nama samvatsara kala nirnaya panchangam 2015-2016
DAIVAGNA. L.S.SIDDHANTHY
శ్రీ కంచి కామకోటి పీఠ పరిపాలిత శ్రీ మన్మథ నామ సంవత్సర కాలనిర్ణయ పంచాంగం
"దైవజ్ఞ సుబ్రహ్మణ్య సిద్ధాన్తి"
సృష్ట్యాది గత సౌరాబ్దాః ౧౯౫,౫౮,౮౫,౧౧౬
వర్తమాన మహాయుగ గతాబ్దాః ౩౮,౯౭,౨౧౫
వర్తమాన కలియుగ గతాబ్దాః ౫,౧౧౬
శాలివాహన శక గతాబ్దాః ౧౯౩౭
శ్రీమత్ శంకరాచార్య అవతార గతాబ్దాః ౨౦౮౭
ప్రభవాది ౨౯
పంచాంగము శ్రౌత స్మార్తాది సకల సత్కర్మానుష్టానములకు ఆధారభూతమైనది.
జటిల గణిత సాధ్యమైన పంచాంగణనము బహుప్రాచీన కాలము నుండి వారి వారి సాంప్రదాయములను అనుసరించి చేయబడుచున్నవి.
కాగా కొన్ని పంచాంగములయందు గ్రహణాది ప్రత్యక్ష గోచారములు కూడా తప్పిపోవు ప్రమాదములు మనము చూచుచున్నాము.
ఇట్టి దోషములు తప్పిదములు రాకుండా ఉండాలని కంచి పరమాచార్యుల వారు సుమారు ౧౧౦ సంవత్సరముల నుండి జ్యోతిష పంచాంగ పండిత సదస్సులను భారతదేశములోని ప్రముఖులైన జ్యోతిష, పంచాంగ, ఖగోళ, తర్క, మీమాంస, వ్యాకరణ, సంస్కృత, స్కంధత్రయ, మతత్రయ పండితులను, ధర్మశాస్త్ర పండితులను పిలచి శ్రీమఠమున నిర్వహించడము అవిచ్ఛినన్నముగా జరుగుతున్నది.
అట్టి సభయందు తెలంగాణ రాష్ట్రం, నల్లగోడ జిల్లా, ఆలేరు వాస్తవ్యులు బ్రహ్మశ్రీ దైవజ్ఞ సుబ్రహ్మణ్య సిద్ధాంతి గారు ప్రతి సంవత్సరము గణిస్తూ వెలువరిస్తున్న శ్రీ కాలనిర్ణయ పంచాంగములోని పండుగల, మౌఢ్య, పుష్కర, సం
The high-level summary is:
1. The document provides a list of names in various languages.
2. The names appear to be of people or characters from different cultures and time periods.
3. The list spans 3 pages and includes over 100 names.
The document provides a summary of a publication. It states that the information in the publication was prepared carefully using reliable sources, but errors could remain. Any opinions or recommendations reflect the author's judgment and may change without notice. No responsibility is accepted for losses resulting from using this information.
2014 15 subhmanya siddhanthy panchangam jaya nama samvatsaramSiddhanthygaru Ls
This document discusses the importance of education and knowledge. It notes that education helps remove ignorance and superstitions from society. It also states that education provides people with skills to earn a living and contribute to society. The document urges all members of society to value education and make efforts to become educated. It highlights that an educated society leads to greater progress and development.
The document appears to be notes or excerpts from a diary or journal written in Arabic. It discusses daily activities and reflections. Specific events mentioned include going to work, visiting family, and thinking about the past. The writing has a contemplative tone and focuses on personal experiences and thoughts.
Noah introduces himself as a database professor. He discusses passing information between pages and what is stored in a database. He posts information to demonstrate how data can be passed and retrieved from a database.
1. HõOk (fish farming) provides essential nutrition and income. Various fish like Rohu, Catla, Mrigal, Silver carp and Grass carp are commonly farmed.
2. For fish farming, a pond area is prepared by removing weeds and levelling the bottom. Fingerlings are stocked at the rate of 25,000 to 30,000 per hectare. Supplementary feeding is provided twice a day.
3. For cage culture in lakes, 20-30 cages are installed and stocked with fingerlings. After 6-8 months of rearing, the fish are harvested and sold in local markets.
This document provides information about a teacher recruitment service called myEDmatch. It describes how myEDmatch allows schools to access a national database of teacher candidates, market job openings to teachers across the country, and identify teachers that fit the mission and culture of the school through a matching algorithm. It also explains how schools and teachers can build online profiles, search for candidates that meet their criteria, and communicate with favorites to streamline the hiring process. Testimonials from recruiters praise the high quality interface and responsive staff.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
Communicating effectively and consistently with students can help them feel at ease during their learning experience and provide the instructor with a communication trail to track the course's progress. This workshop will take you through constructing an engaging course container to facilitate effective communication.
Gender and Mental Health - Counselling and Family Therapy Applications and In...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
36. CHAPTER 3. GEOMETRY
26
165. ÑÖÜO
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37. CHAPTER 3. GEOMETRY
27
171.
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3.2 Isosceles Triangle
=
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38. CHAPTER 3. GEOMETRY
28
174. ÄO~i += =
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39. CHAPTER 3. GEOMETRY
29
177.
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178.
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179. ~Pi = =
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180.
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41. CHAPTER 3. GEOMETRY
31
185. i~ï=çÑ=`çëáåÉë=
α−+= ÅçëÄÅOÅÄ~ OOO
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187.
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189.
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42. CHAPTER 3. GEOMETRY
32
191. β=γ= ëáåÅëáåÄÜ~ I=
α=γ= ëáåÅëáå~ÜÄ I=
α=β= ëáåÄëáå~ÜÅ K=
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192.
Q
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193. ~ã
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43. CHAPTER 3. GEOMETRY
33
195.
O
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p ÅÄ~
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O
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páÇÉ=çÑ=~=ëèì~êÉW=~=
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44. CHAPTER 3. GEOMETRY
34
196. O~Ç = ==
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197.
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198.
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199. ~Qi = =
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45. CHAPTER 3. GEOMETRY
35
202.
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205. °=β+α NUM =
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206. ( )OOO
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47. CHAPTER 3. GEOMETRY
37
210. °=β+α NUM =
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211. OO
O
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N ~QÇÇ =+ =
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212.
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=α= =
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213.
O
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=== =
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214. ~Qi = =
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215. α== ëáå~~Üp O
I==
ONÇÇ
O
N
p = K=
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49. CHAPTER 3. GEOMETRY
39
=
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=
218.
O
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+
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219. O
Å~ÄÇ += =
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220. ( )OO
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Q
N
ÅÜ −−= =
=
221.
( )( )Ä~ÅOÄ~ÅO
Å~ÄÅ
o
O
−++−
+
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222. èÜÜ
O
Ä~
p =⋅
+
= =
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50. CHAPTER 3. GEOMETRY
40
3.11 Isosceles Trapezoid with
Inscribed Circle
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iÉÖW=Å=
jáÇäáåÉW=è=
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o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=o=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=ê=
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Figure 22.
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223. ÅOÄ~ =+ =
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224. Å
O
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+
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225. OOO
ÅÜÇ += =
=
51. CHAPTER 3. GEOMETRY
41
226.
O
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O
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ê == =
=
227.
~
Ä
S
Ä
~
U
Ä~
ÅÜ
ÜO
Å
~Ä
Å
N
O
Å
êQ
ÅÇ
ÜO
ÅÇ
o OO
O
++
+
=+=+=== =
=
228. ( ) ÅQÄ~Oi =+= =
=
229.
( )
O
iê
ÅÜèÜ
O
~ÄÄ~
Ü
O
Ä~
p ===
+
=⋅
+
= ==
=
=
=
3.12 Trapezoid with Inscribed Circle
=
_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=
i~íÉê~ä=ëáÇÉëW=ÅI=Ç=
jáÇäáåÉW=è=
^äíáíìÇÉW=Ü=
aá~Öçå~äëW= ON ÇIÇ =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
52. CHAPTER 3. GEOMETRY
42
=
=
Figure 23.
=
230. ÇÅÄ~ +=+ =
=
231.
O
ÇÅ
O
Ä~
è
+
=
+
= =
=
232. ( ) ( )ÇÅOÄ~Oi +=+= =
=
233. èÜÜ
O
ÇÅ
Ü
O
Ä~
p =⋅
+
=⋅
+
= I==
ϕ= ëáåÇÇ
O
N
p ON K=
=
=
=
3.13 Kite
=
páÇÉë=çÑ=~=âáíÉW=~I=Ä=
aá~Öçå~äëW= ON ÇIÇ =
^åÖäÉëW= γβα II =
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
53. CHAPTER 3. GEOMETRY
43
=
=
Figure 24.
=
234. °=γ+β+α PSMO =
=
235. ( )Ä~Oi += =
=
236.
O
ÇÇ
p ON
= =
=
=
=
3.14 Cyclic Quadrilateral
páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=
aá~Öçå~äëW= ON ÇIÇ =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =
fåíÉêå~ä=~åÖäÉëW= δγβα III =
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=
54. CHAPTER 3. GEOMETRY
44
=
=
Figure 25.
=
237. °=δ+β=γ+α NUM =
=
238. míçäÉãó∞ë=qÜÉçêÉã=
ONÇÇÄÇ~Å =+ =
=
239. ÇÅÄ~i +++= =
=
240.
( )( )( )
( )( )( )( )ÇéÅéÄé~é
ÅÇ~ÄÄÅ~ÇÄÇ~Å
Q
N
o
−−−−
+++
= I==
ïÜÉêÉ=
O
i
é = K=
=
241. ϕ= ëáåÇÇ
O
N
p ON I==
( )( )( )( )ÇéÅéÄé~ép −−−−= I==
ïÜÉêÉ=
O
i
é = K=
=
=
=
55. CHAPTER 3. GEOMETRY
45
3.15 Tangential Quadrilateral
=
páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=
aá~Öçå~äëW= ON ÇIÇ =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=
=
=
=
=
Figure 26.
=
242. ÇÄÅ~ +=+ =
=
243. ( ) ( )ÇÄOÅ~OÇÅÄ~i +=+=+++= =
=
244.
( ) ( )
éO
éÄ~Ä~ÇÇ
ê
OOO
O
O
N −+−−
= I==
ïÜÉêÉ=
O
i
é = K==
=
56. CHAPTER 3. GEOMETRY
46
245. ϕ== ëáåÇÇ
O
N
éêp ON =
=
=
=
3.16 General Quadrilateral
=
páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=
aá~Öçå~äëW= ON ÇIÇ =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =
fåíÉêå~ä=~åÖäÉëW= δγβα III =
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
======= =
=
Figure 27.
=
246. °=δ+γ+β+α PSM =
=
247. ÇÅÄ~i +++= =
=
57. CHAPTER 3. GEOMETRY
47
248. ϕ= ëáåÇÇ
O
N
p ON =
=
=
=
3.17 Regular Hexagon
=
páÇÉW=~=
fåíÉêå~ä=~åÖäÉW=α =
pä~åí=ÜÉáÖÜíW=ã=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=
=
=
=
=
Figure 28.
=
249. °=α NOM =
=
250.
O
P~
ãê == =
58. CHAPTER 3. GEOMETRY
48
251. ~o = =
=
252. ~Si = =
=
253.
O
PP~
éêp
O
== I==
ïÜÉêÉ=
O
i
é = K=
=
=
=
3.18 Regular Polygon
=
páÇÉW=~=
kìãÄÉê=çÑ=ëáÇÉëW=å=
fåíÉêå~ä=~åÖäÉW=α =
pä~åí=ÜÉáÖÜíW=ã=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=
=
=
59. CHAPTER 3. GEOMETRY
49
=
=
Figure 29.
=
254. °⋅
−
=α NUM
O
Oå
=
=
255. °⋅
−
=α NUM
O
Oå
=
=
256.
å
ëáåO
~
o
π
= =
=
257.
Q
~
o
å
í~åO
~
ãê
O
O
−=
π
== =
=
258. å~i = =
=
259.
å
O
ëáå
O
åo
p
O
π
= I==
Q
~
oééêp
O
O
−== I==
60. CHAPTER 3. GEOMETRY
50
ïÜÉêÉ=
O
i
é = K==
=
=
=
3.19 Circle
=
o~ÇáìëW=o=
aá~ãÉíÉêW=Ç=
`ÜçêÇW=~=
pÉÅ~åí=ëÉÖãÉåíëW=ÉI=Ñ=
q~åÖÉåí=ëÉÖãÉåíW=Ö=
`Éåíê~ä=~åÖäÉW=α =
fåëÅêáÄÉÇ=~åÖäÉW=β =
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
260.
O
ëáåoO~
α
= =
=
=
=
Figure 30.
=
69. CHAPTER 3. GEOMETRY
59
289.
Q
~P
p
O
_ = =
=
290. O
~Pp = =
=
291.
OS
~
Üp
P
N
s
P
_ == K==
=
=
=
3.26 Regular Pyramid
=
páÇÉ=çÑ=Ä~ëÉW=~=
i~íÉê~ä=ÉÇÖÉW=Ä=
eÉáÖÜíW=Ü=
pä~åí=ÜÉáÖÜíW=ã==
kìãÄÉê=çÑ=ëáÇÉëW=å==
pÉãáéÉêáãÉíÉê=çÑ=Ä~ëÉW=é=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ëéÜÉêÉ=çÑ=Ä~ëÉW=ê=
^êÉ~=çÑ=Ä~ëÉW= _p =
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
70. CHAPTER 3. GEOMETRY
60
=
=
Figure 41.
=
292.
Q
~
Äã
O
O
−= =
=
293.
å
ëáåO
~
å
ëáåÄQ
Ü
OOO
π
−
π
= =
=
294. éã~ÄQå~
Q
N
å~ã
O
N
p OO
i =−== =
=
295. éêp_ = =
=
296. i_ ppp += =
=
297. éêÜ
P
N
Üp
P
N
s _ == ==
=
=
=
71. CHAPTER 3. GEOMETRY
61
3.27 Frustum of a Regular Pyramid
=
_~ëÉ=~åÇ=íçé=ëáÇÉ=äÉåÖíÜëW=
åPON
åPON
ÄIIÄIÄIÄ
~II~I~I~
K
K
=
eÉáÖÜíW=Ü=
pä~åí=ÜÉáÖÜíW=ã==
^êÉ~=çÑ=Ä~ëÉëW= Np I= Op =
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =
mÉêáãÉíÉê=çÑ=Ä~ëÉëW= Nm I= Om =
pÅ~äÉ=Ñ~ÅíçêW=â=
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
Figure 42.
=
298. â
~
Ä
~
Ä
~
Ä
~
Ä
~
Ä
å
å
P
P
O
O
N
N
====== K =
=
72. CHAPTER 3. GEOMETRY
62
299. O
N
O
â
p
p
= =
=
300.
( )
O
mmã
p ON
i
+
= =
=
301. ONi pppp ++= =
=
302. ( )OONN pppp
P
Ü
s ++= =
=
303. [ ]ON
O
N
ââN
P
Üp
~
Ä
~
Ä
N
P
Üp
s ++=
++= =
=
=
=
3.28 Rectangular Right Wedge
=
páÇÉë=çÑ=Ä~ëÉW=~I=Ä=
qçé=ÉÇÖÉW=Å=
eÉáÖÜíW=Ü=
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =
^êÉ~=çÑ=Ä~ëÉW= _p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
73. CHAPTER 3. GEOMETRY
63
=
=
Figure 43.
=
304. ( ) ( )OOOO
i Å~ÜÄÄÜQÅ~
O
N
p −++++= =
=
305. ~Äp_ = =
=
306. i_ ppp += =
=
307. ( )Å~O
S
ÄÜ
s += =
=
=
=
3.29 Platonic Solids
=
bÇÖÉW=~=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
78. CHAPTER 3. GEOMETRY
68
3.31 Right Circular Cylinder with
an Oblique Plane Face
=
o~Çáìë=çÑ=Ä~ëÉW=o=
qÜÉ=ÖêÉ~íÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= NÜ =
qÜÉ=ëÜçêíÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= OÜ =
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =
^êÉ~=çÑ=éä~åÉ=ÉåÇ=Ñ~ÅÉëW= _p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
Figure 48.
=
324. ( )ONi ÜÜop +π= =
=
325.
O
ONOO
_
O
ÜÜ
ooop
−
+π+π= =
=
79. CHAPTER 3. GEOMETRY
69
326.
−
++++π=+=
O
ONO
ON_i
O
ÜÜ
ooÜÜoppp =
=
327. ( )ON
O
ÜÜ
O
o
s +
π
= =
=
=
=
3.32 Right Circular Cone
o~Çáìë=çÑ=Ä~ëÉW=o=
aá~ãÉíÉê=çÑ=Ä~ëÉW=Ç=
eÉáÖÜíW=e=
pä~åí=ÜÉáÖÜíW=ã=
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =
^êÉ~=çÑ=Ä~ëÉW= _p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
Figure 49.
80. CHAPTER 3. GEOMETRY
70
328. OO
oãe −= =
=
329.
O
ãÇ
oãpi
π
=π= =
=
330. O
_ op π= =
=
331. ( )
+π=+π=+=
O
Ç
ãÇ
O
N
oãoppp _i =
=
332. eo
P
N
ep
P
N
s O
_ π== =
=
=
=
3.33 Frustum of a Right Circular Cone
=
o~Çáìë=çÑ=Ä~ëÉëW=oI=ê=
eÉáÖÜíW=e=
pä~åí=ÜÉáÖÜíW=ã=
pÅ~äÉ=Ñ~ÅíçêW=â=
^êÉ~=çÑ=Ä~ëÉëW= Np I= Op =
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
81. CHAPTER 3. GEOMETRY
71
=
=
Figure 50.
=
333. ( )OO
êoãe −−= =
=
334. â
ê
o
= =
=
335. O
O
O
N
O
â
ê
o
p
p
== =
=
336. ( )êoãpi +π= =
=
337. ( )[ ]êoãêopppp OO
iON +++π=++= =
=
338. ( )OONN pppp
P
Ü
s ++= =
=
339. [ ]ON
O
N
ââN
P
Üp
ê
o
ê
o
N
P
Üp
s ++=
++= =
=
=
=
82. CHAPTER 3. GEOMETRY
72
3.34 Sphere
=
o~ÇáìëW=o=
aá~ãÉíÉêW=Ç=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
Figure 51.
=
340. O
oQp π= =
=
341. po
P
N
Ç
S
N
eo
P
Q
s PP
=π=π= =
=
=
=
3.35 Spherical Cap
o~Çáìë=çÑ=ëéÜÉêÉW=o=
o~Çáìë=çÑ=Ä~ëÉW=ê=
eÉáÖÜíW=Ü=
^êÉ~=çÑ=éä~åÉ=Ñ~ÅÉW= _p =
^êÉ~=çÑ=ëéÜÉêáÅ~ä=Å~éW= `p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
85. CHAPTER 3. GEOMETRY
75
===== =
=
Figure 54.
=
349. oÜOpp π= =
=
350. ( )O
O
O
NONp êêoÜOpppp ++π=++= =
=
351. ( )OO
O
O
N ÜêPêPÜ
S
N
s ++π= =
=
=
=
3.38 Spherical Wedge
=
o~ÇáìëW=o=
aáÜÉÇê~ä=~åÖäÉ=áå=ÇÉÖêÉÉëW=ñ=
aáÜÉÇê~ä=~åÖäÉ=áå=ê~Çá~åëW=α=
^êÉ~=çÑ=ëéÜÉêáÅ~ä=äìåÉW= ip =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
86. CHAPTER 3. GEOMETRY
76
=
=
Figure 55.
=
352. ñoO
VM
o
p O
O
i =α
π
= =
=
353. ñoOo
VM
o
op OO
O
O
+π=α
π
+π= =
=
354. ño
P
O
OTM
o
s P
P
=α
π
= =
=
=
=
3.39 Ellipsoid
=
pÉãá-~ñÉëW=~I=ÄI=Å=
sçäìãÉW=s=
87. CHAPTER 3. GEOMETRY
77
======= =
=
Figure 56.
=
355. ~ÄÅ
P
Q
s π= =
=
=
=
Prolate Spheroid
=
pÉãá-~ñÉëW=~I=ÄI=Ä=E Ä~ > F=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
356.
+π=
É
É~êÅëáå~
ÄÄOp I==
ïÜÉêÉ=
~
Ä~
É
OO
−
= K=
=
357. ~Ä
P
Q
s O
π= =
=
97. CHAPTER 4. TRIGONOMETRY
87
4.4 Trigonometric Functions of Common
Angles
381. =
°α = ê~Çα = αëáå = αÅçë = αí~å = αÅçí αëÉÅ = αÅçëÉÅ =
M= M= M= N= M= ∞= N= ∞=
PM=
S
π
=
O
N
=
O
P
=
P
N
= P =
P
O
= O=
QR=
Q
π
=
O
O
=
O
O
= N= N= O = O =
SM=
P
π
=
O
P
=
O
N
= P =
P
N
= O=
P
O
=
VM=
O
π
= N= M= ∞ = M= ∞ = N=
NOM=
P
Oπ
=
O
P
=
O
N
− = P− =
P
N
− O− =
P
O
=
NUM= π= M= N− = M= ∞ = N− = ∞ =
OTM=
O
Pπ
= N− = M= ∞= M= ∞= N− =
PSM= πO = M= N= M= ∞ = N= ∞ =
=
=
=
=
=
=
=
=
=
=
=
=
=
98. CHAPTER 4. TRIGONOMETRY
88
382. =
°α = ê~Çα = αëáå = αÅçë = αí~å = αÅçí =
NR=
NO
π
=
Q
OS −
=
Q
OS +
= PO− = PO+ =
NU=
NM
π
=
Q
NR −
=
Q
RONM +
R
ROR−
= ROR+ =
PS=
R
π
=
Q
RONM −
Q
NR +
=
NR
RONM
+
−
RONM
NR
−
+
=
RQ=
NM
Pπ
=
Q
NR +
=
Q
RONM −
RONM
NR
−
+
NR
RONM
+
−
=
TO=
R
Oπ
=
Q
RONM +
Q
NR −
= ROR+ = R
ROR−
=
TR=
NO
Rπ
=
Q
OS +
=
Q
OS −
= PO+ = PO− =
=
=
=
4.5 Most Important Formulas
=
383. NÅçëëáå OO
=α+α =
=
384. Ní~åëÉÅ OO
=α−α =
=
385. NÅçíÅëÅ OO
=α−α =
=
386.
α
α
=α
Åçë
ëáå
í~å =
105. CHAPTER 4. TRIGONOMETRY
95
431.
O
í~åN
O
í~åN
Åçë
O
O
α
+
α
−
=α =
=
432.
O
í~åN
O
í~åO
í~å
O α
−
α
=α =
=
433.
O
í~åO
O
í~åN
Åçí
O
α
α
−
=α =
=
=
=
4.14 Transforming of Trigonometric
Expressions to Product
=
434.
O
Åçë
O
ëáåOëáåëáå
β−αβ+α
=β+α =
=
435.
O
ëáå
O
ÅçëOëáåëáå
β−αβ+α
=β−α =
=
436.
O
Åçë
O
ÅçëOÅçëÅçë
β−αβ+α
=β+α =
=
437.
O
ëáå
O
ëáåOÅçëÅçë
β−αβ+α
−=β−α =
=
125. CHAPTER 5. MATRICES AND DETERMINANTS
115
aÉíÉêãáå~åíëW=aI= ña I= óa I= òa ==
j~íêáÅÉëW=^I=_I=u=
=
=
544.
=+
=+
OOO
NNN
ÇóÄñ~
ÇóÄñ~
I==
a
a
ñ ñ
= I=
a
a
ó
ó
= =E`ê~ãÉê∞ë=êìäÉFI==
ïÜÉêÉ==
NOON
OO
NN
Ä~Ä~
Ä~
Ä~
a −== I==
NOON
OO
NN
ñ ÄÇÄÇ
ÄÇ
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126. CHAPTER 5. MATRICES AND DETERMINANTS
116
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127. CHAPTER 5. MATRICES AND DETERMINANTS
117
ïÜÉêÉ==
=
ååOåNå
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K
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128. 118
Chapter 6
Vectors
=
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sÉÅíçêëW=ì
r
I= î
r
I= ï
r
I= ê
r
I=
→
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sÉÅíçê=äÉåÖíÜW= ì
r
I= î
r
I=£=
råáí=îÉÅíçêëW= á
r
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r
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r
W= NNN wIvIu =
`ççêÇáå~íÉë=çÑ=îÉÅíçê= î
r
W= OOO wIvIu =
pÅ~ä~êëW=λ Iµ=
aáêÉÅíáçå=ÅçëáåÉëW= αÅçë I= βÅçë I= γÅçë =
^åÖäÉ=ÄÉíïÉÉå=íïç=îÉÅíçêëW=θ =
=
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6.1 Vector Coordinates
=
550. råáí=sÉÅíçêë=
( )MIMINá =
r
I=
( )MINIMà =
r
I=
( )NIMIMâ =
r
I=
Nâàá ===
rrr
K=
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551. ( ) ( ) ( )âòòàóóáññ^_ê MNMNMN
rrrr
−+−+−==
→
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129. CHAPTER 6. VECTORS
119
======= =
=
Figure 73.
=
552. ( ) ( ) ( )O
MN
O
MN
O
MN òòóóññ^_ê −+−+−==
→
r
=
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553. fÑ= ê^_
r
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→
I=íÜÉå= ê_^
r
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→
K=
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Figure 74.
=
554. α= Åçëêu
r
I=
β= Åçëêv
r
I=
γ= Åçëêw
r
K=
145. CHAPTER 7. ANALYTIC GEOMETRY
135
615. `áêÅìãÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=íÜÉ=páÇÉ=mÉêéÉåÇáÅìä~ê======================
_áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ=
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Nóñ
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ñ
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N
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ó
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+
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======== =
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Figure 93.
=
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146. CHAPTER 7. ANALYTIC GEOMETRY
136
616. lêíÜçÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^äíáíìÇÉëF=çÑ=~=qêá~åÖäÉ=
Nóñ
Nóñ
Nóñ
Nóññó
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ñ
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O
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+
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Nñóóñ
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ó
PP
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O
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Figure 94.
=
617. ^êÉ~=çÑ=~=qêá~åÖäÉ=
( ) ( )
NPNP
NONO
PP
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NN
óóññ
óóññ
O
N
Nóñ
Nóñ
Nóñ
O
N
p
−−
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±=±= =
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147. CHAPTER 7. ANALYTIC GEOMETRY
137
618. ^êÉ~=çÑ=~=nì~Çêáä~íÉê~ä=
( ) ( )( ) ( )( )[ ++−++−±= POPOONON óóññóóññ
O
N
p =
( )( ) ( )( )]NQNQQPQP óóññóóññ +−++−+ =
=
=== =
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kçíÉW=få=Ñçêãìä~ë=SNTI=SNU=ïÉ=ÅÜççëÉ=íÜÉ=ëáÖå=EHF=çê=E¥F=ëç=
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( )NOON
O
O
O
N ÅçëêêOêê^_Ç ϕ−ϕ−+== =
=
157. CHAPTER 7. ANALYTIC GEOMETRY
147
636. m~ê~ääÉä=iáåÉë=
qïç=äáåÉë= NN Äñâó += =~åÇ= OO Äñâó += =~êÉ=é~ê~ääÉä=áÑ==
ON ââ = K=
qïç= äáåÉë= M`ó_ñ^ NNN =++ = ~åÇ= M`ó_ñ^ OOO =++ = ~êÉ=
é~ê~ääÉä=áÑ=
O
N
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=
637. mÉêéÉåÇáÅìä~ê=iáåÉë=
qïç=äáåÉë= NN Äñâó += =~åÇ= OO Äñâó += =~êÉ=éÉêéÉåÇáÅìä~ê=áÑ==
N
O
â
N
â −= =çêI=Éèìáî~äÉåíäóI= Nââ ON −= K=
qïç= äáåÉë= M`ó_ñ^ NNN =++ = ~åÇ= M`ó_ñ^ OOO =++ = ~êÉ=
éÉêéÉåÇáÅìä~ê=áÑ=
M__^^ ONON =+ K=
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158. CHAPTER 7. ANALYTIC GEOMETRY
148
=
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Figure 110.
=
638. ^åÖäÉ=_ÉíïÉÉå=qïç=iáåÉë=
ON
NO
ââN
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O
O
O
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O
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N
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161. CHAPTER 7. ANALYTIC GEOMETRY
151
642. qÜêÉÉ=mçáåí=cçêã
M
Nóñóñ
Nóñóñ
Nóñóñ
Nóñóñ
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O
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643. m~ê~ãÉíêáÅ=cçêã
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íÅçëoñ
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Mcbóañ^ó^ñ OO
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>+ FK==
qÜÉ=ÅÉåíÉê=çÑ=íÜÉ=ÅáêÅäÉ=Ü~ë=ÅççêÇáå~íÉë=( )ÄI~ I=ïÜÉêÉ==
^O
a
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162. CHAPTER 7. ANALYTIC GEOMETRY
152
^O
^cQba
o
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−+
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pÉãáã~àçê=~ñáëW=~=
pÉãáãáåçê=~ñáëW=Ä=
cçÅáW= ( )MIÅcN − I= ( )MIÅcO =
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Figure 115.
163. CHAPTER 7. ANALYTIC GEOMETRY
153
646. ~Oêê ON =+ I=
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650. m~ê~ãÉíêáÅ=cçêã
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169. CHAPTER 7. ANALYTIC GEOMETRY
159
=
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bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ
O
é
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~O
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é = K=
bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ
170. CHAPTER 7. ANALYTIC GEOMETRY
160
O
é
ó −= I=
`ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë=
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é
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171. CHAPTER 7. ANALYTIC GEOMETRY
161
+
O
é
óIñc MM I=
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172. CHAPTER 7. ANALYTIC GEOMETRY
162
670. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=
( ) ( ) ( )O
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O
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λ+
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λ+
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ïÜÉêÉ=
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174. CHAPTER 7. ANALYTIC GEOMETRY
164
672. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí=
O
ññ
ñ ON
M
+
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p ++= K=
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qÜÉ=îçäìãÉ=çÑ=~=íÉíê~ÜÉÇêçå=ïáíÜ=îÉêíáÅÉë= ( )NNNN òIóIñm I=
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s
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175. CHAPTER 7. ANALYTIC GEOMETRY
165
==== =
=
Figure 126.
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Ma`ò_ó^ñ =+++ =
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181. CHAPTER 7. ANALYTIC GEOMETRY
171
===== =
=
Figure 132.
=
683. aáÜÉÇê~ä=^åÖäÉ=_ÉíïÉÉå=qïç=mä~åÉë=
fÑ=íÜÉ=éä~åÉë=~êÉ=ÖáîÉå=Äó==
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