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#T003#Problems in Differential Equations (J. L. Brenner)

˹ѧһ 1963 ʹ dover Ҿ㹻 2013 ѡ ͹Ҩѧ繢ͧ ⨷ͺѹ ´ҹѧ ҡ ͧӴѧ Ӣͧ¹ѹͧҷ º к͡෤ԤԸ⨷ Ẻ¡ Ẻhomogeneous ,linear 1st oe ԸաաẺ¹ ͧйҹ ʹء͹ Դդöҧ仵ԴСѺ˹ѧ礧˧͹ 555




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#O014#ҧԹõ (Table of Integrals)

ҧ⨷ѭҤԵʵ繨еͧҧԹõ ͤдǡ

ҧԹõ ҧԾѹ ٵûԾѹ ٵáԹõ


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㨿ԡ͹§¹ѡ ෤Ԥ ǹѭ⨷ԡԧ͹͡觨ԧ ѭ⨷ԡͧ㨿ԡ͹ 㨿ԡҨö⨷ԡҧ ͧ෤ԤСý֡ҧǹҹ

Strategy Design
Execution (Tactics)
Answer Checking
Further Reading

The problems and examinations in this physics course exercise not only your knowledge of physics but also your skill in solving problems. Professional physicists earn their salaries not particularly for their knowledge of physics but for their ability to solve workplace problems. This document presents tips for honing your problem solving skills. These tips and techniques will prove useful to you in your physics courses, in your other college courses, in your career, and in your everyday life.
To set the stage, I want to discuss an example of problem solving from everyday life, namely building a jigsaw puzzle. There are a number of different approaches to building a jigsaw puzzle: My approach is to first turn all the pieces face up, then put together the edge pieces to make a frame, then sort the remaining pieces into piles corresponding to small "sub-puzzles" (blue pieces over here, red pieces over there). I build the sub-puzzles, then piece the sub-puzzles together to build the whole thing. Other people have different approaches to building jigsaw puzzles, but nobody, nobody, builds a puzzle by picking up the first piece and putting it in exactly the correct position, then picking up the second piece and putting it in exactly the correct position, and so forth. Solving a jigsaw puzzle involves an approach--a strategy--and a lot of "creative fumbling" as well.

Your physics textbook contains many solved "sample problems". The solutions presented there are analogous to the completed jigsaw puzzle, with every piece in its proper position. No one solves a physics problem by simply writing down the correct equations and the correct reasoning with the correct connections the first time through, just as no one builds a jigsaw puzzle by putting every piece in its correct position the first time through. The "solved problems" in your book are extraordinarily valuable and they deserve your careful study, but they represent the end product of a problem solving session and they rarely show the process involved in reaching that end product. This document aims to expose you to the process.

Solving a physics problem usually breaks down into three stages:
1.Design a strategy.
2.Execute that strategy.
3.Check the resulting answer.
This document treats each of these three elements in turn, and concludes with a summary.
Strategy Design
Look before you leap. Whenever you face a problem, there is an immediate temptation to rush in, roll up your sleeves, and begin tinkering with it. Resist that temptation. If you start your detailed work--the execution stage--immediately, you will likely write down a lot of correct statements that do not lead to an answer. Instead, think about the problem on an overview level. What sort of conceptual tools will you need to solve the problem? What path will you take to the solution, and in what direction should you start off? Concretely, it often helps to classify your problem by its method of solution.
If you are looking for a child lost in the woods, your first step is to sit down, think about what the child probably did and where he probably is, and devise a strategy that will allow you to effectively rescue him. If, instead, you just rush about the woods in random directions, you're likely to become lost yourself.

Where are you now, and where do you want to go? Before you can design a path that takes you from the statement of the problem to its answer, you must be clear about what the situation is and what the goals are. It often helps to check off each given datum of the problem, and to underline the objective. But for getting an overall sense of the problem, nothing beats summarizing the whole situation with a diagram. The diagram will organize your work and suggest ways to proceed. One of my course graders told me that "When students draw a diagram and label it carefully, they are forced to think about what's going on, and they usually do well. If they just try a globule of math, they mess up."

Keep the goal in sight. Don't get caught in blind alleys that lead nowhere, or even in broad boulevards that lead somewhere but not to where you want to go. It sometimes helps to map a strategy backwards, by saying: "I want to find the answer Z. If I knew Y I could find Z. If I knew X I could find Y . . . " and so forth until you get back to something you are given in the problem statement.

Some students find it useful to make a list of the information given and the goal to be uncovered (e.g. "given the constant acceleration, the initial velocity, and the time, find the displacement"). Others find it sufficient to write down only the goal (e.g. "to find: displacement").

Ineffective strategy. Do not page through your book looking for a magic formula that will give you the answer. Physics teachers do not assign problems in order to torture innocent young minds . . . they assign problems in order to force you into active, intimate involvement with the concepts and tools of physics. Rarely is such involvement provided by plugging numbers into a single equation, hence rarely will you be assigned a problem that yields to this attack. In those rare instances when you do face a problem that can be solved by plugging numbers into a formula, the most effective way to find that formula is by thinking about the physical principles involved, not by flipping through the pages in your book.

Make the problem more specific. You're asked to find the number of ways that M balls can be placed into N buckets. Suppose you can't even begin to map out a strategy. Then try the problem of 3 balls in 5 buckets. Solving the more specific problem will give you clues on how to solve the more general problem. And once you use those clues to solve the more general problem, you can check your solution by trying it out for the already-solved special case M=3 and N=5.

Large problems. At times you will be faced with big problems for which no method of solution is immediately apparent. In this case, break your problem into several smaller subproblems, each of which is simple enough that you know how to solve it. At this strategy-design stage it is not important that you actually solve the subproblems, but rather that you know you can solve them. You might begin by mapping out a strategy that leads nowhere, but then you haven't wasted time by implementing this strategy. Once you have mapped out a strategy that leads from the given information to the answer, you can then go back and execute the calculations. This strategy has been known from the time of the ancients under the name of "divide and conquer".

Execution (Tactics)
Eventually, of course, you do have to roll up your sleeves and tinker with the problem. As you do so, keep your strategy in mind, and keep the following tips in mind as well:
Work with symbols. Depending on the problem statement, the final answer might be a formula or a number. In either case, however, it's usually easier to work the problem with symbols and plug in numbers, if requested, only at the very end. There are three reasons for this: First, it's easier to perform algebraic manipulations on a symbol like "m" than on a value like "2.59 kg". Second, it often happens that intermediate quantities cancel out in the final result. Most important, expressing the result as an equation enables you to examine and understand it (see the section on "Answer Checking") in a way that a number alone does not permit.

(Working with symbols instead of numbers can lead to confusion as to which symbols represent given information and which represent unknown desired answers. You can resolve this difficulty by remembering--as recommended above--to "keep the goal in sight".)

Define symbols with mnemonic names. If a problem involves a helium atom colliding with a gold atom, then define mh as the mass of the helium atom and mg as the mass of the gold atom. If you instead pick the symbols m1 and m2, you stand a good chance of mixing up the symbols and their meanings as you solve the problem. And if you don't define the symbols at all, but just begin throwing around m's and M's, you'll confuse both yourself and whoever is grading your answer.

Keep packets of related variables together. In acceleration problems, the quantity (1/2)at2 comes up over and over again. This collection of variables has a simple physical interpretation, transparent dimensions, and a convenient memorable form. In short, it is easy to work with as a packet. Take advantage of this ease. Don't artificially divide this packet into pieces, or write it in an unfamiliar form like t2a/2. Packets like this come up in all aspects of physics--some are even given names (e.g. "the Bohr radius" in atomic physics). Look for these packets, think about what they are telling you, and respect their integrity.

Neatness and organization. I am not your mother, and I will not tell you how to organize either your dorm room or your problem solutions. But I can tell you that it is easier to work from neat, well-organized pages than from scribbles. I can also warn you about certain handwriting pitfalls: Distinguish carefully between t and +, between l and 1, and between Z and 2. (I write a t with a hook at the bottom, an l in script lettering, and a Z with a cross bar. You can form your own conventions.) These suggestions on neatness, organization, and handwriting do not arise from prudishness--they are practical suggestions that help avoid algebraic errors, and they are for your benefit, not mine. (On the other hand, it doesn't hurt to be neat and organized for the benefit of your grader. One course grader of mine pointed out: "If I can't read it, I can't give you credit.")

Avoid needless conversions. If the problem gives you one length in meters and another in inches, then it's probably best to convert all lengths to meters. But if all the lengths are in inches, then there's no need to convert everything to meters--your answer should be in inches. In fact, you might not actually need to convert. For example, perhaps two lengths are given in inches and the final answer turns out to depend only on the ratio of those two lengths. In that case, the ratio is the same whether the lengths going into the ratio are inches or meters. It's easy to make arithmetic errors while doing conversions. If you don't convert, then you don't make those errors!

Keep it simple. I will not assign baroque problems that require tortuous explanations and pages of algebra. If you find yourself working in such a way, then you're on the wrong path. The cure is to stop, go back to the beginning, and start over with a new strategy. (Generations of students have kept track of this rule by remembering to KISS: Keep It Simple and Straightforward.)

Answer Checking
Checking your answer does not mean comparing it to the answer in the back of the book. It means finding the characteristics of your answer and comparing them to the characteristics that you expect. Some of your problems--particularly the ones assigned early in the course--will actually lead you through the checking stage in order to familiarize you with the process. Other problems will leave it to you to perform this check. In either case, checking your answer is not just good problem solving practice that helps you gain points on problem assignments and on exams. The checking stage builds familiarity with the content of physics and the character of problem solutions, and hence develops your intuition to make solving other problems--and learning more physics--easier. (See Daniel F. Styer, "Guest comment: Getting there is half the fun", American Journal of Physics 64 (1998) 105-106.)
Dimensional analysis. Suppose you find a formula for distance (in, say, meters) in terms of some information about velocity (meters/second), acceleration (meters/second2), and time (seconds). If your formula is correct then all of the dimensions on the right hand side must cancel so as to end up with "meters".

Numerical reasonableness. If your problem asks you to find the mass of a squirrel, do you find a mass of 1,970 kilograms? Even worse, do you find a mass of -1,970 kilograms?

[Reasonable speeds. "My calculations give me a speed of 23 m/s. Is this reasonable?" It's hard for most people to get a feel for the reasonableness of speeds expressed in meters per second. Until this qualitative feel develops, Americans should check for reasonableness by converting speeds in meters per second to speeds in miles per hour: simply double the number (20 m/s is about 40 mi/hr). Non-Americans should convert to kilometers per hour: simply quadruple the number (20 m/s is about 80 km/hr).]

Algebraically possible. Would evaluating your formula ever lead you to divide by zero or take the square root of negative number?

Functionally reasonable. Does your answer depend on the given quantities in a reasonable way? For example, you might be asked how far a projectile travels after it is launched at a given speed with a given angle. Common sense says that if the initial speed is increased (keeping the angle constant) then the distance traveled will increase. Does your formula agree with common sense?

Limiting values and special cases. In the projectile travel distance problem mentioned above, the range is obviously zero for a vertical launch. Does your formula give this result? If you solve a problem regarding two objects, does it give the proper result when the two objects have equal masses? When one of them has zero mass (i.e. does not exist)?

Symmetry. Problems often have geometrical symmetry from which you can determine the direction of a vector but not its magnitude. More often they have a "permutation" symmetry: If your problem has two objects, you can call the cube "object number 1" and the sphere "object number 2" but your final answer had better not depend upon how you numbered your objects. (That is, it should give the same answer if every "1" is changed to a "2" and vice versa.)

Specify units. "The distance is 5.72" is not an answer. Is that 5.72 miles, 5.72 meters, or 5.72 inches? Similarly, if the answer is a vector, both magnitude and direction must be specified. (The direction may be drawn into a diagram rather than stated explicitly.)

Significant figures. Any number that comes from an experiment comes with some uncertainty. Most of the numbers in this course come with three significant figures. If a ball rolls 3.24 meters in 2.41 seconds, then report its speed as 1.34 m/s, not 1.34439834 m/s. Most introductory physics courses do not require a formal or technical error analysis, but you should avoid inaccurate statements like the second quotient above.

Large problems. If you break up your large problem into several subproblems, as recommended above, then check your results at the end of each subproblem. If your answer to the second subproblem passes its checks, but your answer to the third subproblem fails its checks, then your execution error almost certainly falls within the third subproblem. Knowing its general location, you can quickly go back and correct the error, so its effects will not propagate on to the remaining subproblems. This can be a real time-saver.

The problems in your physics course can be fun and exciting. Approach them in the spirit of exploration and they will not disappoint you! 1. Strategy design a. Classify the problem by its method of solution.
b. Summarize the situation with a diagram.
c. Keep the goal in sight (perhaps by writing it down).

2. Execution tactics a. Work with symbols.
b. Keep packets of related variables together.
c. Be neat and organized.
d. Keep it simple.

3. Answer checking a. Dimensionally consistent?
b. Numerically reasonable (including sign)?
c. Algebraically possible? (Example: no imaginary or infinite answers.)
d. Functionally reasonable? (Example: greater range with greater initial speed.)
e. Check special cases and symmetry.
f. Report numbers with units specified and with reasonable significant figures.

Further Reading
The classic exploration of mathematical problem solving technique is George Polya, How To Solve It (Princeton University Press, Princeton, New Jersey, 1957).
More mundane and somewhat pedantic, but nevertheless valuable, is Donald Scarl, How To Solve Problems: For Success in Freshman Physics, Engineering, and Beyond, third edition (Dosoris Press, Glen Cove, New York, 1993).
Study of the following books will help develop your general (as opposed to strictly mathematical) problem-solving skills: James L. Adams, Conceptual Blockbusting: A Guide to Better Ideas (Norton, New York, 1980),
Berton Roueche, The Medical Detectives (Times Books, New York, 1980) and The Medical Detectives, volume II (Dutton, New York, 1984),
Martin Gardner, Aha! Insight (Freeman, New York, 1978),
Donald J. Sobol, Two-Minute Mysteries,
Arthur Conan Doyle, Sherlock Holmes stories,
Agatha Christie, Hercule Poirot stories, particularly Murder on the Orient Express.
Entry into recent literature on physics problem solving skills is provided by Frederick Reif, "Understanding and teaching important scientific thought processes", American Journal of Physics 63 (1995) 17-35 (especially section V),
Rolf Plotzner, The Integrative Use of Qualitative and Quantitative Knowledge in Physics Problem Solving (Peter Lang, Frankfurt am Main, 1994).

Why do I assign problems in physics courses? How can working these problems help you?

Philosophy. Listening to lectures, reading books, running computer simulations, performing experiments, participating in discussions . . . all these are fine tools for learning physics. But you will not really become familiar with the subject until you get it under your skin by working problems. The problems in a physics course do not simply test your comprehension of the material that you learned in the text. Instead they are an important component of the learning experience, designed to extend and solidify your grasp of the concepts and content of physics. Solving problems is a more active, and hence more effective, learning technique than reading text or listening to lectures.

In answering your homework (and exam) problems, you must show your work. That is, you must present your evidence and your reasoning as well as your final conclusion. (This rule holds for all intellectual discourse. For example, suppose you are asked in an English Literature course to consider Melville's influence on American literature. You march off to the library and -- after considerable reading and analysis -- conclude that "Moby Dick changed the entire landscape of the American novel." If you typed up this single sentence and submitted it as your paper, your professor would be unimpressed.) Exactly how much detail should you give in presenting your reasoning? A good rule of thumb is to present enough detail that you could reconstruct your thought processes two or three months later, when you're using your solutions to study for the final exam.

Difficulty. No one should expect to score 100% on homework. Some of the problems are deliberately very challenging. Everyone can use improvement, and the problems are a relatively painless way for me to challenge you and show you how to improve. I assign problems that will expose you to many fascinating phenomena and useful devices. The exams I set for you are much easier because they serve an entirely different purpose. Homework is a way for me to show you some of the many vistas of physics that you don't know, whereas exams are a way for you to show me the many aspects of physics that you do know. Before each exam I will distribute a practice exam so that you will have some idea of what to expect in terms of length and difficulty.

Warm up exercises. A problem arises from my assigning such interesting homework problems. They tend to be harder than plodding, mechanical problems, and in particular, they tend to need many steps in their solution. Hence I will, for many assigned problems, suggest "warm up exercises" that are more mechanical, simpler, and involve fewer steps than the assigned problems. Working the warm-ups is not required, and if you do work them you should not hand them in. But if you find that a particular assigned problem is too difficult, then try the associated warm-ups. Working them should give you practice that will be directly relevant in helping you solve the associated assigned problem.

Model solutions. I distribute model solutions to the homeworks on the day that I collect them from you. My model solutions are only that: models. You might have solved the problem in a completely different way that is actually superior to the way I chose. I urge you to scan the model solutions on the day I give them out . . . the problems will be fresh in your mind and you'll learn from my solutions more readily. If you find that you solved the problem (correctly!) in a way different from mine, then please do let me know about it. One of the great joys I find in teaching is to learn from my students, and it happens more frequently than you might think.

Sample problem. For concreteness, many of the tips below refer to the following problem (Halliday, Resnick, and Walker, Fundamentals of Physics, sixth edition 22-9):

Two free point charges +q and +4q are a distance L apart. A third charge is placed so that the entire system is in equilibrium. (a) Find the location, magnitude, and sign of the third charge. (b) Show that the equilibrium of the system is unstable.

Explain. When you write up solutions to problems, be sure to explain your reasoning. Don't just give me the final numerical answer or the end formula . . . I already know what it is! Instead I'm interested in seeing how you overcome the roadblocks that get in your way as you progress through the problem. An appendix in the text by Halliday, Resnick, and Walker lists "Answers to odd-numbered questions, exercises, and problems". Be aware that these are merely skeleton answers, and I am interested in a full solutions, like the model solutions that I hand out to you or that you can find in the "sample problems" of the text. The benefits that accrue from active problem solving come only if you supply the reasoning yourself. The "answers" at the end of the book will help you learn physics if you work through the problem yourself and then use the skeleton answer to check your reasoning. If you instead look up the answer before attempting the problem, the "answers" section will actually be an impediment to your learning.

The explanation does not need to be terribly long or detailed, but it must exist if you are to earn full credit. For example, in the sample problem the third charge must be located on the axis: Otherwise the total force on the first two charges would have a vertical component and hence could not equal zero. You can get away with a statement as simple as "The equilibrium position is on the axis", but you can't just omit it.

Map out the logic. This point is similar to the one above. If you are using Coulomb's law to find the force between two particles of charge +q and +4q separated by a distance L, then you might say "F = k q1q2/r2 = k 4q2/L2", or "From Coulomb's law, F = k 4q2/L2", but don't just say "k 4q2/L2" without any indication of what you are calculating or how you are calculating it. (Note, however, that it is perfectly all right to have a quantity like the above pointing to a force arrow in a diagram. In that case the diagram, rather than words or equations, provides the context.)

Diagrams and definitions. In almost all cases, the first step in solving a problem is to draw a diagram showing the geometry of the situation. The diagram will organize your work and point out ways to proceed.

The diagram is often a good place to define variables that you will need as well. Using the sample problem as an example again, remember that you need to find the location of the third charge. Saying x = L/3 is not an answer unless you have first defined x to be the distance from the charge +q rather than the distance from the charge +4q. The easiest way to do this is through a diagram.

Do full problem for full credit. The sample problem has two parts. Part (a) is clearly the main part of the problem, but it's not the only part of the problem! For full credit, you must do part (b) as well. If a problem asks you to derive an equation and discuss the result in the limit that the charge vanishes, then you have to supply the discussion to get full credit. Sentences count just as much as equations and numbers!

Do partial problem for partial credit. If you can't solve a problem completely, then hand in a start. If you have a plan for solving the problem but can't execute it, then hand in the plan. If you can't find the magnitude of the electric field but can find its direction, then tell me the direction. If a problem has two variables x and y, and you can solve it only in the case that x = 2y, then hand in the solution for the special case that you have solved. You will even get some points for saying nothing more than "This problem can be solved using Coulomb's Law, but I can't figure out the details." In the world of physics research, it often happens that the questions change as you work on the answers, and you can use the same philosophy in this course. (If you follow this advice and solve a problem related to but different from the one that I assigned, then please point this out explicitly in your solution.)

Citation. If you use a specialized result from your text, then cite it. No need to cite momentum conservation, but there is a need to cite an equation giving the final velocity of the target particle in a one-dimensional elastic collision with the target initially stationary . . . now there's a specialized result!

If you can't solve the problem then at least do something: sketch the situation and define a few relevant variables. State a relevant principle. If you come up with a silly result (e.g. a negative kinetic energy), then tell me that it's silly and you've earned a point (or two). If you run out of time, then write a sentence about how you would solve the problem if you did have time. Writing down your thoughts can clarify them and lead you to your goal. If nothing else, they might earn you points.

Mechanics. The problem sets are graded by a student working under my close supervision. I have the final say on your homework grade, so if you feel that the grader has been unfair or arbitrary or wrong, then see me and I might change your grade. (In practice, however, I rarely do so because I give the grader very detailed instructions.) Sometimes I will ask the grader to look at only about half the problems that I assign to you. This way, he or she can go through the sets quickly and get your solutions back to you soon enough that you can learn effectively from them.

Your solutions do not need to be obsessively neat, but they do need to be legible . . . particularly your name! One former grader for this course said "If I can't read it, I can't give you credit." I know of no one (certainly not myself) who can solve the problems in this course on the first shot: you'll need to work the problems first in rough draft and then copy out a version to submit for grading. The copying out is not just for neatness. It helps you consolidate your thoughts and brings the logic of your solution into focus. Please staple your problems together, for otherwise the pages are likely to become separated from each other and you will get credit for only the first page of your answers.

The problems in a physics course are not dry appendages designed to keep you indoors on sunny days. They are exciting, dynamic, and central to the course structure. Enjoy them!


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#O013#Theoretical Physicist

Theoretical physics is a branch of physics which employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena.

The advancement of science depends in general on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigor while giving little weight to experiments and observations.For example, while developing special relativity, Albert Einstein was concerned with the Lorentz transformation which left Maxwell's equations invariant, but was apparently uninterested in the MichelsonMorley experiment on Earth's drift through a luminiferous ether. On the other hand, Einstein was awarded the Nobel Prize for explaining the photoelectric effect, previously an experimental result lacking a theoretical formulation.

A physicist is a scientist who does research in physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made (particle physics) to the behavior of the material Universe as a whole (cosmology).

The term "physicist" was coined by William Whewell in his 1840 book The Philosophy of the Inductive Sciences.

Most material a student encounters in the undergraduate physics curriculum is based on discoveries and insights of a century or more in the past. Alhazen's intromission theory of light was formulated in the 11th century; Newton's laws of motion and Newton's law of universal gravitation were formulated in the 17th century; Maxwell's equations, 19th century; and quantum mechanics, early 20th century. The undergraduate physics curriculum generally includes the following range of courses: chemistry, classical physics, kinematics, astronomy and astrophysics, physics laboratory, electricity and magnetism, thermodynamics, optics, modern physics, quantum physics, nuclear physics, particle physics, and solid state physics. Undergraduate physics students must also take extensive mathematics courses (calculus, differential equations, linear algebra, complex analysis, etc.), and computer science and programming. Undergraduate physics students often perform research with faculty members.

Many positions, especially in research, require a doctoral degree. At the Master's level and higher, students tend to specialize in a particular field. Fields of specialization include experimental and theoretical astrophysics, atomic physics, molecular physics, biophysics, chemical physics, medical physics, condensed matter physics, cosmology, geophysics, gravitational physics, material science, nuclear physics, optics, particle physics, and plasma physics. Post-doctoral experience may be required for certain positions.

The highest honor awarded to physicists is the Nobel Prize in Physics, awarded since 1901 by the Royal Swedish Academy of Sciences.

The three major employers of career physicists are academic institutions, government laboratories, and private industries, with the largest employer being the last. Many trained physicists, however, apply their skills to other activities, in particular to engineering, computing, and finance, often quite successfully. Some physicists take up additional careers where their knowledge of physics can be combined with further training in other disciplines, such as patent law in industry or private practice. In the United States, a majority of those in the private sections having a physics degree actually work outside the fields of physics, astronomy and engineering altogether.

Nobel laureate Sir Joseph Rotblat has suggested that physicists going into employment in scientific research should honour a Hippocratic Oath for Scientists.

Well, I think Life has no meaning, Life or human existence has no real meaning or purpose because human existence occurred out of a random chance in nature, and anything that exists by chance has no intended purpose

but I want to be a greatest physicist anyway.


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#O012#ʾھ Թҵ

ʾھҹ ҹ Թҵ
ʾھõ õ Թҵ
ʾھ Թҵ
ˡڢ ʾھءڢ Թҵ.

鹵ѳзء駻ǧ ѧ.

õդѺ Mr.Feynman
ʾھ Թҵ

͸ҵ ԡФԵʵ ÷⨷ԡФԵʵ ⨷ҡöʹء ¡͹ͺͧ´ ҡҡ֧ѹͺ ͵˹ѡ
ʾھ Թҵ ҡͺ ҡ⨷ ҷ駡ҧѹСҧ׹ӡáҹн֡⨷

¹ҵö١ѧѺҾФ˹ѡ֧§лªͧѹ Ҩ¹ǤԴҡѡȨҡ͹䢡ͧ¤ ֧ش˹觤سе˹ѡ



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