Differential equations

http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/video-lectures/embed16/

 

Listen to the part of the lecture and try to answer questions.

 

 

Transcript - Lecture 16

1] What was the mistake the professor made last time?

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2] Can students find these examples on the Web?

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3] What is the funny topic the professor is going to introduce?

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4] Is this the only course whose subject is differential equations?

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5] What is the easiest kind of differential equation?

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6] What is the antiderivative of x?

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7] What is the purpose of the technique called substitution?

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8] What is the opposite of the annihilation operator?

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9] In which branch of physics are these operators used?

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10] Is this differential equation difficult to solve?

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Now consider the following Qs. In case you are not sure, study the following text.

1.     What is calculus?

2.     What is the difference between real and complex analysis?

3.      What is the relation between the Riemann hypothesis and complex analysis?

4.      What does it mean when a system is described as „dynamic“?

5.     What does the chaos theory study?

6.     What does the functional analysis study?

Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and real-valued functions is known as real analysis, with complex analysis the equivalent field for the complex numbers. The Riemann hypothesis, one of the most fundamental open questions in mathematics, is drawn from complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.

Differential equation

From Wikipedia, the free encyclopedia

7.      Have a look at the text and try to fill in the missing words. First, try to guess, than consult the list of words.

A differential equation is a mathematical equation for an unknown function of one or several a) …………..that relates the values of the function itself and its derivatives of various orders.

Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of

partial differential equations.

Differential equations arise in many areas of science and technology: whenever a deterministic relationship involving some continuously varying quantities (b) …… by functions and their rates of change in c)…… and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the d)……. of a body is described by its position and e)…….. as the time varies. Newton's Laws allow one to relate the position, velocity, f)…….. and various forces acting on the body and state this relation as a differential equation for the unknown position of the body as a function of g)…... In some cases, this differential equation (called an equation of motion) may be h)…….. explicitly.

a) variables           derivatives            equations

b) described          modelled                drawn

c)  time                force                     space

d)  state                position               motion

e)  acceleration         velocity                 position

f)   velocity          acceleration          time

g)  time              space                gravity

h)   counted        solved              guessed

 

 

WORD STUDY: Prefixes are important means of creating new words, usually the opposits. There are some words from the text, try to supply prefixes forming new expressions.

 

dependent  ..............................                      partial .........................................

proportional ..........................                       significant ..................................

known ....................................                       real ............................................

natural ...................................                       continuous ................................

changing ................................                       finite ........................................

predictable ...........................                        important .............................                       

                                        Selected problems of English Grammar

 

                                                               ARTICLES

    Adapted from Trzeciak Jerzy, Writing Mathematical Papers in English. European Mathematical society, 1995

 

  Indefinite article     a) Instead of number one

                                    b) Meaning member of a class of objects, some, one of

                                    c) In definitions of classes of objects (many objects with the

                                        given property)

                                    d) In the plural – when you are referring to each element of a class

                                    e) In front of an adjective which is intended to mean “having this

                                        particular quality”

 

 

Definite article          a) Meaning mentioned earlier, that

                                    b) In front of a noun referring to a single, uniquely determined

                                        object (i.e. definition)

                                    c) In the construction the + property + of+ object

                                    d) In front of a cardinal number if it embraces all objects

                                         considered

                                     e) In front of an ordinal number

                                     f) In front of surnames used attributively

                                     g) In front of a noun in the plural if you are referring to a class of

                                         objects as a whole, and not to particular members of the class

 

 

Article omission     a) In front of nouns referring to activities

                                 b) In front of nouns referring to properties if you mention no

                                      particular object

                                 c) After certain expressions with of

                                 d) In front of numbered objects

                                 e) To avoid repetition

                                 f) In front of surnames in the possessive

                                 g) In some expressions describing a noun, especially after with and of

                                 h) After forms of have

                                 i) In front of the name of a mathematical discipline  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Exercise. Fill in a definite, indefinite or zero article.

 

 

1)      Theorem 7 has been extended to …….class of boundary value problems.

2)      We call C …… module of ellipticity.

3)      ……. First Poisson integral in (4) converges to g.

4)      Thus A is the smallest possible extension in which …..differentiation is possible.

5)      By ……duality we easily obtain the following theorem.

6)      …….more general theory must be sought to account for these irregularities.

7)      …….direct sums exist in the category of abelian groups.

8)      ……..section 4 gives a concise presentation of this problem.

9)      This idea comes from the game theory (homological algebra).

10)  It has ………..finite norm.

11)  The direct sum and ……direct product.

12)  It has …………cardinality c.

13)  ……remarkable feature of the solution should be stressed.

14)  In particular, …….closed sets are Borel sets.

15)  …….Borel measurable functions are often called Borel mappings.

16)  ………existence of test functions is not evident.

17)  ……..second statement follows immediately from the first.

18)  …….real measures form a subclass of ……..complex ones.

19)  ……..Gauss theorem

20)  …….two groups have been shown to have the same number of generators.

21)  The equation (3) has …… unique solution g for every f.

22)  This map extends to all of M an …….obvious fashion.

23)  The four centers lie in ……plane.

24)  The right-hand side of (4) is then …….bounded function.

25)  Let f and g be ……functions such that

26)  This is easily seen to be ……equivalence relation.

27)  For this, we introduce ……….auxiliary variable z.

28)  Then x is ……centre of an open ball U.

29)  This reduces the solution to ………division by Px.

30)  …….property (iii) is called the triangle inequality.

31)  It has …….compact support.

32)  The equation of …….motion.

33)  The order and …….symbol of a distribution

34)  This has been proved in ……..part (a) of the proof.

35)  The hypothesis of …….positivity

36)  Here we do not require ……translation invariance.

37)  …….chapter will be devoted to the study of …….expanding maps.

38)  The set of ……….points with distance 1 from K.

39)  We wish to find …….solution of (6) which is of the form

40)  ……..intersection of a decreasing family of such sets is convex.

41)  Using …….standard inner product we may identify

42)  Each of …..three products on the right of (4) satisfies

43)  ……Dirichlet problem

44)  Let us now state ……corollary of ……Lebesgue’ s theorem for

45)  After …..change of variable in the integral we get