Lecture 2: Review of Prerequisites. Table of contents


 Kathryn Morgan
 1 years ago
 Views:
Transcription
1 Math 348 Fall 217 Lecture 2: Review of Prerequisites Disclaimer. As we have a textbook, this lecture note is for guidance and supplement only. It should not be relied on when preparing for exams. In this lecture we review the prerequisites: Multivariable calculus, linear algebra, and basics of dierential equations. We will not be able to cover everything that will be needed through the semester. We will only cover the basics, and leave some more sophisticated topics such as inverse/implicit function theorem for later lectures to explain. Warning. In this lecture note we omit many technical assumptions such as dierentiability/integrability of functions when stating formulas and theorems, because in 348 they are almost always satised. However please do not quote the statements here for your other courses. Table of contents Lecture 2: Review Multivariable calculus The Euclidean space R n Operations on R n Functions from R m to R n Taking derivatives of functions from R m to R n Linear algebra Operations on matrices Linear dependence and independence Dierential equations The simplest and second simplest dierential equations Existence and uniqueness of a general rst order ODE
2 Dierential Geometry of Curves & Surfaces 1. Multivariable calculus Multivariable calculus studies functions from R m to R n with either m > 1 or n >1. In 348 we focus on functions from R to R 2 /R 3 (curves) and from R 2 to R 3 (surfaces). In the following we review concepts, theorems, and techniques that are important to us The Euclidean space R n R n is the mathematical representation of a ndimensional at space through setting up n orthogonal axes. Along each axes, we dene a unit vector: e 1 ; :::; e n. Then each point in the space is identied by an ntuple (x 1 ; :::; x n ). It is important to realize that R n can be interpreted in two ways, in the context of describing a moving particle (along a curve, on a surface, etc.). R n as possible locations. At each moment, the location of the particle is identied with a point in R n : At time t, the particle is at (x 1 ; :::; x n ). R n as possible velocities. To completely describe the motion of this particle, we also need to know its velocity. The velocity is also described by an ntuple (v 1 ; :::; v n ). Such ntuples are called vectors. Thus at each point x in the location space R n, we overlay a velocity space whose mathematical representation is also R n. This overlay will later be called the tangent space of the location space at x. Such distinction seems a bit silly now, but will be very useful later when we curve the location space. The key idea of dierential geometry is that when the location space is curved, the velocity space stays at. To visualize, think of a 2dimensional surface as the curved locations space, then the velocity space at a location is the tangent plane of the surface there Operations on R n When R n is interpreted as the overlaying velocity space, the most important concepts are Norm. When v = (v 1 ; :::; v n ) describes a velocity vector, its speed is given by its norm: p kvk := v v n : (1) Inner product. The inner product of two vectors u = (u 1 ; :::; u n ) and v = (v 1 ; :::; v n ) is dened as u v := u 1 v u n v n : (2) The importance of inner product comes from the following property: Proposition 1. Let be the angle between two vectors u = (u 1 ; :::; u n ) and v = (v 1 ; :::; v n ), then cos = u v kuk kvk : (3) 2
3 Math 348 Fall 217 Exercise 1. Obviously is not uniquely dened. What does this not matter? In particular, uv =()u?v, that is u is perpendicular(orthogonal) to v. Cross product/vector product. Let u=(u 1 ;u 2 ;u 3 );v =(v 1 ;v 2 ;v 3 ) be two vectors in R 3. Their cross product (also called vector product) is dened as u v := (u 2 v 3 u 3 v 2 ; u 3 v 1 u 1 v 3 ; u 1 v 2 u 2 v 1 ): (4) Exercise 2. Prove the following. i. u v = v u; ii. Let a; b 2 R be arbitrary. Then (a u + b w) v = a u v + b w v: (5) iii. u v = if and only if u and v are parallel. iv. (u v) u = ; (u v) v =. v. (u v) w = (v w) u = (w u) v. Also note that where is the angle between u; v. ku vk = kuk kvk sin (6) Remark 2. It turns out that reasonable crossproducts can only be dened in R 3 and R 7. See W. S. Massey, Cross products of vectors in higher dimensional Euclidean spaces, The American Mathematical Monthly, 9(1): , When R n is interpreted as the underlying location space, the most important concepts are Distance. Let x = (x 1 ; :::; x n ) and y = (y 1 ; :::; y n ). For now, the only distance we know is the distance along straight lines, given by the Pythagorean theorem: p d(x; y) = (x 1 y 1 ) (x n y n ) 2 : (7) Remark 3. We see that d(x; y) = kx yk. However this will cease to be true when the location space is curved. One important theme in 348 is to generalize (7) for curved spaces. Area. The foundation of the denition of area is the following agreement: i. The unit square has area 1. ii. The area of a union of disjoint regions is the sum of the areas of these regions. 1. The idea of the proof is that if a cross product can be dened on R n such that i) v w is bilinear in v and w, ii) v w is perpendicular to both v; w, iii) jv wj 2 + (v w) 2 = kvk 2 kwk 2, then R n+1 can be made into a normed division ring with identity. By Hurwitz's Theorem n + 1 = 1; 2; 4; 8 and the conclusion follows. 3
4 Dierential Geometry of Curves & Surfaces iii. Moving a region around rigidly (that is not changing the distance of any pair of two points in the region) does not change its area. From these one could develop the whole theory of integration in twovariable calculus. Example 4. Let x = (x 1 ; x 2 ; x 3 ); y = (y 1 ; y 2 ; y 3 ) in R 3. Then the area of the triangle (; ; ) (x 1 ; x 2 ; x 3 ) (y 1 ; y 2 ; y 3 ) (; ; ) is given by 1 kx yk. 2 Volume. The foundation of the denition of volume is the following agreement: i. The unit cube has volume 1. ii. The volume of a union of disjoint regions is the sum of the volumes of these regions. iii. Moving a region around rigidly (that is not changing the distance of any pair of two points in the region) does not change its volume. With these agreement one could develop the whole theory of integration in threevariable calculus. Example 5. Let x = (x 1 ; x 2 ; x 3 ); y = (y 1 ; y 2 ; y 3 ); z = (z 1 ; z 2 ; z 3 ) in R 3. Then the volume of the parallelopiped spanned by these three vectors is given by V = j(x y) zj: (8) Exercise 3. Check your old textbook and explain what the sign of (x y) z means Functions from R m to R n Let R m be a region in R m. A function f: 7! R n is a correspondence between points in and points in R n satisfying the following: Each x 2 corresponds to exactly one y 2 R n. One classies functions into two categories: linear and nonlinear. In 348, we will see that linear functions are often considered in the context of velocity spaces. The most important functions are linear functions (linear transformations) and bilinear forms. Linear functions. A function f: 7! R n is linear if and only if for every a; b 2 R and every x; y 2, one has f(a x + b y) = a f(x) + b f(y): (9) Exercise 4. Prove that if f is linear then f() =. Exercise 5. Let f: R 7! R be linear. Prove that there is a constant c 2 R such that f(x) = c x. Notation. We often denote a linear function by a capital letter such as T First letter of transformation. 4
5 Math 348 Fall 217 Proposition 6. Let T : R m 7! R n be linear. Then there are m n numbers a 11 ; a 12 ; :::; a 1m ; a 21 ; :::; a n1 ; :::; a nm 2 R, such that for every v 2, the following holds for w = T (v): w 1 = a 11 v a 1m v m (1) w n = a n1 v a nm v m : (11) In other words, we have T (v) = A v where A is the matrix Exercise 6. Prove Proposition a 11 a 1m a n1 a nm Bilinear forms. A bilinear form is a special type of nonlinear function: B:R 2n 7!R, satisfying for every a; b 2 R and every u; v; w 2 R n, 1 A. B(a u + b w; v) = a B(u; v) + b B(w; v); (12) B(u; a w + b v) = a B(u; w) + b B(u; v): (13) Exercise 7. Prove that the inner product: f(u; v) := u v is a bilinear form. Exercise 8. Prove or disprove: The area A(u; v) := ku vk is a bilinear form. Proposition 7. Let B: R 2n 7! R be a bilinear form. Then there are n 2 numbers a 11 ; :::; a 1n ; a 21 ; :::; a n1 ; :::; a nn such that Exercise 9. Let A := inner product. n B(u; v) = X i=1 a11 a n1 a nn 1.4. Taking derivatives of functions from R m to R n Differentiability. Xn u i a ij v j : (14) j=1 1 A. Prove that B(u; v) = u A v = (A u) v where denotes Definition 8. A function f: R m 7! R n is dierentiable at x 2 if and only if the following holds: There is a linear function T : R m 7! R n such that kf(x + v) f(x) T (v)k lim = : (15) v! kvk The linear function T is called the dierential of f at x and will be denoted Df(x). Remark 9. Note that f is dened on which belongs to the location space R m, and f takes value in another location space R n ; while T is dened on R m which is the velocity overlay space on top of, centered at x, and takes value in another velocity space centered at f(x). 5
6 Dierential Geometry of Curves & Surfaces By Proposition 6 there is a matrix A such that T (v) = A v for all v 2 R m. The numbers a ij in the matrix A is given by the partial derivatives of f at x: Definition 1. Let f: R m 7! R. Its jth partial derivative at x = (x 1 ; :::; x m ) 2 is dened (x) := df(x 1; :::; x j 1 ; t; x j+1 ; :::; x m ) j t=xj : (16) Exercise 1. Prove that a j (x). Note. This matrix (representation of the dierential of f) is called the Jacobian matrix of f at x. Exercise 11. Let f: R m 7! R n be dierentiable everywhere. Prove that if Df(x) = for all x 2 R m, then f is a constant. Chain Rule. Let f(x): R m 7! R n and g(y): R n 7! R k. Then the composite function g f: R m 7! R k is dened as (g f)(x) = g(f(x)): (17) The partial derivatives of g f can be calculated f) i = X j i k(x) : j Exercise 12. Find a calculus textbook and work on a few chain rule problems. Higher order derivatives. We consider the simplest case f: R m 7! R. At every x 2, it has a dierential which is a linear function Df(x): R m 7! R. As this linear function can be represented by a 1 m matrix, it can also be identied with an mvector. Thus at each point x 2, we have a vector g(x) 2 R m. Dierentiating this function g we obtain the second order derivative of f at x, denoted D 2 f(x). Exercise 13. Prove or disprove: Let x 2 be arbitrary. The second order derivative of f can be seen as a bilinear form on R 2m = R m R m, where each R m is the velocity space at x. Exercise 14. Prove that the matrix representing the bilinear form is given by a ij : j As we can see here, for functions between R m and R n with m or n > 1, higher order derivatives get more and more complicated. However there is one exception. Exercise 15. What happens if we take successive derivatives of a function f: R 7! R n? Note. The matrix representation of D 2 f(x) is called the Hessian matrix of f at x. Exercise 16. What are the Jacobian and Hessian matrices for f: R 7! R? Taylor expansion. 6
7 Math 348 Fall 217 Let f: R m 7! R. Then there holds f(x + v) = f(x) + Df(x)(v) D2 f(x)(v; v) + R (2) where lim v! krk kvk 2 =. (2) is the Taylor expansion of f at x to second order. Exercise 17. Prove the following. Examples. m Df(x)(v) = X v i ; D 2 f(x)(v; v) = i i=1 Xm 2 j v i v j : (21) Example 11. Let f(x; y):= e xy. We calculate its Taylor expansion at (; ) to second order. We have Therefore the expansion reads f(; ) = e = y exy (; ) @y = x exy (; ) 2 = 2 y2 e xy ) = 2 f f = exy f (; ) = 2 = 2 x2 e xy f (; ) = f(u; v) = f(; ) + Df(; )(u; v) D2 f(; )(u; v) + R = 1 + u v + R: (22) Remark 12. (Interpreting f and Df) Functions are often called maps. For classical dierential geometry this is pertinent. Consider, e.g., Google Map with GPS. This can be seen as a function f from R 2 to the earth surface. Thus f(x ) is the real world location of a car that is at location x on the map. More importantly, if the car symbol is moving with velocity v on the map, the velocity of the car itself in the real world is given by [Df(x )](v) (remember that Df(x ) is a function!). It is woth mentioning that, although the car is located in a curved surface (earth surface), its possible velocities at a particular location (time) form a plane, the tangent plane of earth surface at location f(x ). 2. Linear algebra The fundamental idea of calculus is to study functions through their rst and second (or higher) derivatives. We have seen that such derivatives can be viewed as linear transformations between R m and R n. Therefore to understand them we need linear algebra. 7
8 Dierential Geometry of Curves & Surfaces 2.1. Operations on matrices Determinant. Determinant is dened for square matrices. Determinant is uniquely dened. Let f(v 1 ; :::; v n ): R n R n 7! R be such that i. f is linear in each of the v i 's. That is, e.g., f is a linear function of v 1 when v 2 ; :::; v n are xed. ii. f(e 1 ; :::; e n ) = 1. Where e i is the vector with one 1 at the ith entry and 's elsewhere. iii. f changes sign when two columns are switched. e.g. f(v 1 ; v 2 ; v 3 ; :::; v n ) = f(v 2 ; v 1 ; v 3 ; :::; v n ). Then f(v 1 ; :::; v n ) = det V where V is the matrix with v 1 as its rst column, v 2 as its second column, and so on. Exercise 18. Prove the above claim for n = 2; Let A := a 11 a 12 a 21 a 22. Then det A := a 11 a 22 a 12 a 21 : (23) Exercise 19. Let u; v; x; y 2 R 3. Prove that u x v x (u v) (x y) = det u y v y 3 3. Let A a 1 11 a 12 a 13 a 21 a 22 a 23 A. Then a 31 a 32 a 33 : (24) det A := a 11 a 22 a 33 + a 12 a 23 a 31 + a 21 a 32 a 13 a 13 a 22 a 31 a 12 a 21 a 33 a 23 a 32 a 11 : Exercise 2. Let u; v; w 2 R 3. Let A u 1 v 1 w 1 u 2 v 2 w 2 u 3 v 3 w 3 (25) 1 A. Prove that det A = (u v) w. A square matrix A is invertible, that is there is a matrix B such that A B = B A = I A if and only if det A =/. Eigenvalues and eigenvectors. In most cases the matrix A under study is the mathematical representation of a linear function T. We could try to understand an n n matrix through its n 2 entries. However a more ecient way is through its eigenvalues/eigenvectors. The eigenvalues of an n n matrix A are the solutions to the equation det( I A) = : (26) 8
9 Math 348 Fall 217 The eigenvectors corresponding to an eigenvalue are those vectors satisfying ( I A) v = () A v = v: (27) In many situations, one could choose n linearly independent eigenvectors fv 1 ; :::; v n g to form a coordinate system, that is every vector u 2 R n can be uniquely represented as u = a 1 v a n v n where a 1 ; :::; a n 2 R. Then the linear function T can be very easily understood: T (u) = 1 a 1 v n a n v n : (28) The most useful situation for us is when A is symmetric: a ij = a ji for all i; j = 1; 2; :::; n. In this case we can choose n eigenvectors v 1 ; :::; v n satisfying i. v i?v j, for all i =/ j; ii. kv i k = 1 for all i = 1; 2; :::; n Linear dependence and independence Linear dependence/independence. Let v 1 ; :::; v k 2 R n. We say they are linearly dependent if there are a 1 ; :::; a k 2 R, not all equal to, such that a 1 v a k v k = : (29) If no such a 1 ; :::; a k exist, we say the vectors are linearly independent. Exercise 21. Prove that, if v 1 ; :::; v k are linearly independent, then the following holds. If a 1 ; :::; a k 2 R satisfy a 1 v a k v k =, then a 1 = a 2 = = a k =. Exercise 22. Prove that if v 1 ; :::; v k are linearly dependent, then at least one of them is redundant in the sense that it equals a linear combination of the other vectors. n vectors v 1 = (v 11 ; :::; v 1n ); :::; v n = (v n1 ; :::; v nn ) 2 R n are linearly independent if and only if det A =/ where A v 1 11 v 1n A: (3) v n1 v nn n vectors v 1 ; :::; v n 2 R n are linearly independent if and only if they form a base of R n, that is every u 2 R n can be written uniquely as a linear combination of v 1 ; :::; v n. 3. Dierential equations A dierential equation is an equation involving the derivative (and/or higher order derivatives) of functions The simplest and second simplest dierential equations In the following x(t): R 7! R is a single variable function. The simplets ODE. 9
10 Dierential Geometry of Curves & Surfaces Consider dx = f(t): (31) Its solution, by denition, is the indenite integral Z x(t) = f(s) ds (32) which by the fundamental theorem of calculus further equals Z t f(s) ds + C (33) where C is an arbitrary constant. If we further specify x(t ) = x, then this constant is xed. Exercise 23. What is the solution to Exercise 24. Solve dx = f(t); x(t ) = x? (34) dx = f(x); x(t ) = x : (35) The next simplest ODE. Consider + c x(t) = f(t); x() = x : (36) where c 2 R. Multiplying both sides by e ct and apply Leibniz rule of dierentiation of product of functions, we reach d (ect x(t)) = e ct f(t); (37) which is the simplest ODE now. The solution of (37) is Z t e ct x(t) = x + e cs f(s) ds (38) which now gives Exercise 25. Solve In general, the equation Z t x(t) = e ct x + e c(s t) f(s) ds: (39) + c x(t) = f(t); x(t ) = x : (4) + c(t) x(t) = f(t) (41) could be solved by multiplying both sides by e C(t) where C(t) satises C (t) = c(t). Exercise 26. Solve + c(t) x(t) = f(t); x() = x : (42) 1
11 Math 348 Fall Existence and uniqueness of a general rst order ODE General situation we face in 348. In 348 we often need to study systems of dierential equations: dx 1 (t) dx n (t) = f 1 (x 1 (t); :::; x n (t)); x 1 () = x 1 (43) = f n (x 1 (t); :::; x n (t)): x n () = x n (44) Taking advantage of multivariable calculus, we can rewrite the above to a more compact form: = f(x(t)); x() = x (45) where x: R 7! R n, f: R n 7! R n, and x = (x 1 ; :::; x n ). The one ODE system that we could really solve. For general f there is little hope explicitly solving (45). However there is a special, very useful, situation that we could solve. Consider the case where f i (x 1 ; :::; x n ) = a i1 x a in x n ; i = 1; 2; :::; n (46) where a ij 2 R for all i; j = 1; 2; :::; n. In other words, we have = A x(t); x() = x : (47) Now to further simplify the situation, we assume A has n linearly independent eigenvectors v 1 ; :::; v n. Writing x(t) = z 1 (t) v z n (t) v n and x = z 1 v z n v n we obtain dz i (t) = i z i (t); z i () = z i : (48) Exercise 27. Justify (48). Exercise 28. Finish the solution. Existence and uniqueness theorem. Although there is little hope explicitly solving (45), we have the following qualitative understanding. Theorem 13. Let f: R n 7! R n be dierentiable with continuous derivatives. Then there is T > and a unique function x(t) satisfying x() = x, and for all < t < T, = f(x): (49) 11
12 Dierential Geometry of Curves & Surfaces Exercise 29. Let A(t) a 1 11(t) a 1n (t) a n1(t) a nn(t) for every i; j = 1; 2; :::; n. Prove that the solution to exists and is unique. A be such that a ij (t) are continuous functions of t = A(t) x(t) (5) Remark 14. We will see later that existence of curves with certain desirable properties would be equivalent to the existence of solution to an ODE of the form (45). Then Theorem 13 would guarantee us that this curve exists. 12
DSGA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DSGA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More informationLecture 6: Contraction mapping, inverse and implicit function theorems
Lecture 6: Contraction mapping, inverse and implicit function theorems 1 The contraction mapping theorem De nition 11 Let X be a metric space, with metric d If f : X! X and if there is a number 2 (0; 1)
More informationDefinition 5.1. A vector field v on a manifold M is map M T M such that for all x M, v(x) T x M.
5 Vector fields Last updated: March 12, 2012. 5.1 Definition and general properties We first need to define what a vector field is. Definition 5.1. A vector field v on a manifold M is map M T M such that
More informationLectures 15: Parallel Transport. Table of contents
Lectures 15: Parallel Transport Disclaimer. As we have a textbook, this lecture note is for guidance and supplement only. It should not be relied on when preparing for exams. In this lecture we study the
More informationMath 413/513 Chapter 6 (from Friedberg, Insel, & Spence)
Math 413/513 Chapter 6 (from Friedberg, Insel, & Spence) David Glickenstein December 7, 2015 1 Inner product spaces In this chapter, we will only consider the elds R and C. De nition 1 Let V be a vector
More informationOR MSc Maths Revision Course
OR MSc Maths Revision Course Tom Byrne School of Mathematics University of Edinburgh t.m.byrne@sms.ed.ac.uk 15 September 2017 General Information Today JCMB Lecture Theatre A, 09:3012:30 Mathematics revision
More information7. Let X be a (general, abstract) metric space which is sequentially compact. Prove X must be complete.
Math 411 problems The following are some practice problems for Math 411. Many are meant to challenge rather that be solved right away. Some could be discussed in class, and some are similar to hard exam
More informationCS 246 Review of Linear Algebra 01/17/19
1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector
More informationAlgebra II. Paulius Drungilas and Jonas Jankauskas
Algebra II Paulius Drungilas and Jonas Jankauskas Contents 1. Quadratic forms 3 What is quadratic form? 3 Change of variables. 3 Equivalence of quadratic forms. 4 Canonical form. 4 Normal form. 7 Positive
More informationMath 2912: Final Exam Solutions Northwestern University, Winter 2016
Math 292: Final Exam Solutions Northwestern University, Winter 206 Determine whether each of the following statements is true or false f it is true, explain why; if it is false, give a counterexample
More informationCourse Summary Math 211
Course Summary Math 211 table of contents I. Functions of several variables. II. R n. III. Derivatives. IV. Taylor s Theorem. V. Differential Geometry. VI. Applications. 1. Best affine approximations.
More informationChapter 7. Extremal Problems. 7.1 Extrema and Local Extrema
Chapter 7 Extremal Problems No matter in theoretical context or in applications many problems can be formulated as problems of finding the maximum or minimum of a function. Whenever this is the case, advanced
More informationDifferentiation. f(x + h) f(x) Lh = L.
Analysis in R n Math 204, Section 30 Winter Quarter 2008 Paul Sally, email: sally@math.uchicago.edu John Boller, email: boller@math.uchicago.edu website: http://www.math.uchicago.edu/ boller/m203 Differentiation
More informationSMSTC (2017/18) Geometry and Topology 2.
SMSTC (2017/18) Geometry and Topology 2 Lecture 1: Differentiable Functions and Manifolds in R n Lecturer: Diletta Martinelli (Notes by Bernd Schroers) a wwwsmstcacuk 11 General remarks In this lecture
More information4 Linear Algebra Review
Linear Algebra Review For this topic we quickly review many key aspects of linear algebra that will be necessary for the remainder of the text 1 Vectors and Matrices For the context of data analysis, the
More informationLectures 18: Gauss's Remarkable Theorem II. Table of contents
Math 348 Fall 27 Lectures 8: Gauss's Remarkable Theorem II Disclaimer. As we have a textbook, this lecture note is for guidance and supplement only. It should not be relied on when preparing for exams.
More information4.3  Linear Combinations and Independence of Vectors
 Linear Combinations and Independence of Vectors De nitions, Theorems, and Examples De nition 1 A vector v in a vector space V is called a linear combination of the vectors u 1, u,,u k in V if v can be
More informationThe following definition is fundamental.
1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic
More informationREVIEW OF DIFFERENTIAL CALCULUS
REVIEW OF DIFFERENTIAL CALCULUS DONU ARAPURA 1. Limits and continuity To simplify the statements, we will often stick to two variables, but everything holds with any number of variables. Let f(x, y) be
More informationProblem Set 9 Due: In class Tuesday, Nov. 27 Late papers will be accepted until 12:00 on Thursday (at the beginning of class).
Math 3, Fall Jerry L. Kazdan Problem Set 9 Due In class Tuesday, Nov. 7 Late papers will be accepted until on Thursday (at the beginning of class).. Suppose that is an eigenvalue of an n n matrix A and
More informationTHEODORE VORONOV DIFFERENTIABLE MANIFOLDS. Fall Last updated: November 26, (Under construction.)
4 Vector fields Last updated: November 26, 2009. (Under construction.) 4.1 Tangent vectors as derivations After we have introduced topological notions, we can come back to analysis on manifolds. Let M
More informationMath 361: Homework 1 Solutions
January 3, 4 Math 36: Homework Solutions. We say that two norms and on a vector space V are equivalent or comparable if the topology they define on V are the same, i.e., for any sequence of vectors {x
More informationOur goal is to solve a general constant coecient linear second order. this way but that will not always happen). Once we have y 1, it will always
October 5 Relevant reading: Section 2.1, 2.2, 2.3 and 2.4 Our goal is to solve a general constant coecient linear second order ODE a d2 y dt + bdy + cy = g (t) 2 dt where a, b, c are constants and a 0.
More informationb 1 b 2.. b = b m A = [a 1,a 2,...,a n ] where a 1,j a 2,j a j = a m,j Let A R m n and x 1 x 2 x = x n
Lectures 2: Linear Algebra Background Almost all linear and nonlinear problems in scientific computation require the use of linear algebra These lectures review basic concepts in a way that has proven
More informationTEST CODE: PMB SYLLABUS
TEST CODE: PMB SYLLABUS Convergence and divergence of sequence and series; Cauchy sequence and completeness; BolzanoWeierstrass theorem; continuity, uniform continuity, differentiability; directional
More informationMath113: Linear Algebra. Beifang Chen
Math3: Linear Algebra Beifang Chen Spring 26 Contents Systems of Linear Equations 3 Systems of Linear Equations 3 Linear Systems 3 2 Geometric Interpretation 3 3 Matrices of Linear Systems 4 4 Elementary
More informationVector Calculus. Lecture Notes
Vector Calculus Lecture Notes Adolfo J. Rumbos c Draft date November 23, 211 2 Contents 1 Motivation for the course 5 2 Euclidean Space 7 2.1 Definition of n Dimensional Euclidean Space........... 7 2.2
More informationSeveral variables. x 1 x 2. x n
Several variables Often we have not only one, but several variables in a problem The issues that come up are somewhat more complex than for one variable Let us first start with vector spaces and linear
More information11 a 12 a 21 a 11 a 22 a 12 a 21. (C.11) A = The determinant of a product of two matrices is given by AB = A B 1 1 = (C.13) and similarly.
C PROPERTIES OF MATRICES 697 to whether the permutation i 1 i 2 i N is even or odd, respectively Note that I =1 Thus, for a 2 2 matrix, the determinant takes the form A = a 11 a 12 = a a 21 a 11 a 22 a
More information. Consider the linear system dx= =! = " a b # x y! : (a) For what values of a and b do solutions oscillate (i.e., do both x(t) and y(t) pass through z
Preliminary Exam { 1999 Morning Part Instructions: No calculators or crib sheets are allowed. Do as many problems as you can. Justify your answers as much as you can but very briey. 1. For positive real
More informationLecture 13  Wednesday April 29th
Lecture 13  Wednesday April 29th jacques@ucsdedu Key words: Systems of equations, Implicit differentiation Know how to do implicit differentiation, how to use implicit and inverse function theorems 131
More informationLS.1 Review of Linear Algebra
LS. LINEAR SYSTEMS LS.1 Review of Linear Algebra In these notes, we will investigate a way of handling a linear system of ODE s directly, instead of using elimination to reduce it to a single higherorder
More informationMulti Variable Calculus
Multi Variable Calculus Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 3, 03 Functions from R n to R m So far we have looked at functions that map one number to another
More informationChapter 3. Differentiable Mappings. 1. Differentiable Mappings
Chapter 3 Differentiable Mappings 1 Differentiable Mappings Let V and W be two linear spaces over IR A mapping L from V to W is called a linear mapping if L(u + v) = Lu + Lv for all u, v V and L(λv) =
More informationa11 a A = : a 21 a 22
Matrices The study of linear systems is facilitated by introducing matrices. Matrix theory provides a convenient language and notation to express many of the ideas concisely, and complicated formulas are
More information1. Bounded linear maps. A linear map T : E F of real Banach
DIFFERENTIABLE MAPS 1. Bounded linear maps. A linear map T : E F of real Banach spaces E, F is bounded if M > 0 so that for all v E: T v M v. If v r T v C for some positive constants r, C, then T is bounded:
More informationB553 Lecture 3: Multivariate Calculus and Linear Algebra Review
B553 Lecture 3: Multivariate Calculus and Linear Algebra Review Kris Hauser December 30, 2011 We now move from the univariate setting to the multivariate setting, where we will spend the rest of the class.
More informationAnalysis3 lecture schemes
Analysis3 lecture schemes (with Homeworks) 1 Csörgő István November, 2015 1 A jegyzet az ELTE Informatikai Kar 2015. évi Jegyzetpályázatának támogatásával készült Contents 1. Lesson 1 4 1.1. The Space
More informationMath 212Lecture 8. The chain rule with one independent variable
Math 212Lecture 8 137: The multivariable chain rule The chain rule with one independent variable w = f(x, y) If the particle is moving along a curve x = x(t), y = y(t), then the values that the particle
More informationMultivariable Calculus
2 Multivariable Calculus 2.1 Limits and Continuity Problem 2.1.1 (Fa94) Let the function f : R n R n satisfy the following two conditions: (i) f (K ) is compact whenever K is a compact subset of R n. (ii)
More informationRecitation 9: Probability Matrices and Real Symmetric Matrices. 3 Probability Matrices: Definitions and Examples
Math b TA: Padraic Bartlett Recitation 9: Probability Matrices and Real Symmetric Matrices Week 9 Caltech 20 Random Question Show that + + + + +... = ϕ, the golden ratio, which is = + 5. 2 2 Homework comments
More informationThe Derivative. Appendix B. B.1 The Derivative of f. Mappings from IR to IR
Appendix B The Derivative B.1 The Derivative of f In this chapter, we give a short summary of the derivative. Specifically, we want to compare/contrast how the derivative appears for functions whose domain
More informationLecture two. January 17, 2019
Lecture two January 17, 2019 We will learn how to solve rstorder linear equations in this lecture. Example 1. 1) Find all solutions satisfy the equation u x (x, y) = 0. 2) Find the solution if we know
More informationEXERCISE SET 5.1. = (kx + kx + k, ky + ky + k ) = (kx + kx + 1, ky + ky + 1) = ((k + )x + 1, (k + )y + 1)
EXERCISE SET 5. 6. The pair (, 2) is in the set but the pair ( )(, 2) = (, 2) is not because the first component is negative; hence Axiom 6 fails. Axiom 5 also fails. 8. Axioms, 2, 3, 6, 9, and are easily
More informationMath 216 Final Exam 14 December, 2017
Math 216 Final Exam 14 December, 2017 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that
More informationAPPENDIX A. Background Mathematics. A.1 Linear Algebra. Vector algebra. Let x denote the ndimensional column vector with components x 1 x 2.
APPENDIX A Background Mathematics A. Linear Algebra A.. Vector algebra Let x denote the ndimensional column vector with components 0 x x 2 B C @. A x n Definition 6 (scalar product). The scalar product
More informationTwo hours THE UNIVERSITY OF MANCHESTER. 19 May 2017 XX:00 XX:00
Two hours THE UNIVERSITY OF MANHESTER INTRODUTION TO GEOMETRY 19 May 217 XX: XX: Answer ALL FIVE questions in Section A (5 marks in total). Answer TWO of the THREE questions in Section B (3 marks in total).
More informationIntroduction to Linear Algebra. Tyrone L. Vincent
Introduction to Linear Algebra Tyrone L. Vincent Engineering Division, Colorado School of Mines, Golden, CO Email address: tvincent@mines.edu URL: http://egweb.mines.edu/~tvincent Contents Chapter. Revew
More informationg(t) = f(x 1 (t),..., x n (t)).
Reading: [Simon] p. 313333, 833836. 0.1 The Chain Rule Partial derivatives describe how a function changes in directions parallel to the coordinate axes. Now we shall demonstrate how the partial derivatives
More informationLinear Algebra Review (Course Notes for Math 308H  Spring 2016)
Linear Algebra Review (Course Notes for Math 308H  Spring 2016) Dr. Michael S. Pilant February 12, 2016 1 Background: We begin with one of the most fundamental notions in R 2, distance. Letting (x 1,
More informationDr. Allen Back. Sep. 10, 2014
Dr. Allen Back Sep. 10, 2014 The chain rule in multivariable calculus is in some ways very simple. But it can lead to extremely intricate sorts of relationships (try thermodynamics in physical chemistry...
More informationReview and problem list for Applied Math I
Review and problem list for Applied Math I (This is a first version of a serious review sheet; it may contain errors and it certainly omits a number of topic which were covered in the course. Let me know
More informationTHE INVERSE FUNCTION THEOREM
THE INVERSE FUNCTION THEOREM W. PATRICK HOOPER The implicit function theorem is the following result: Theorem 1. Let f be a C 1 function from a neighborhood of a point a R n into R n. Suppose A = Df(a)
More informationNOTES ON MULTIVARIABLE CALCULUS: DIFFERENTIAL CALCULUS
NOTES ON MULTIVARIABLE CALCULUS: DIFFERENTIAL CALCULUS SAMEER CHAVAN Abstract. This is the first part of Notes on Multivariable Calculus based on the classical texts [6] and [5]. We present here the geometric
More information2.2. Show that U 0 is a vector space. For each α 0 in F, show by example that U α does not satisfy closure.
Hints for Exercises 1.3. This diagram says that f α = β g. I will prove f injective g injective. You should show g injective f injective. Assume f is injective. Now suppose g(x) = g(y) for some x, y A.
More informationVector Geometry. Chapter 5
Chapter 5 Vector Geometry In this chapter we will look more closely at certain geometric aspects of vectors in R n. We will first develop an intuitive understanding of some basic concepts by looking at
More information4.1 Distance and Length
Chapter Vector Geometry In this chapter we will look more closely at certain geometric aspects of vectors in R n. We will first develop an intuitive understanding of some basic concepts by looking at vectors
More informationLecture 23: 6.1 Inner Products
Lecture 23: 6.1 Inner Products WeiTa Chu 2008/12/17 Definition An inner product on a real vector space V is a function that associates a real number u, vwith each pair of vectors u and v in V in such
More informationMarch 4, 2019 LECTURE 11: DIFFERENTIALS AND TAYLOR SERIES.
March 4, 2019 LECTURE 11: DIFFERENTIALS AND TAYLOR SERIES 110211 HONORS MULTIVARIABLE CALCULUS PROFESSOR RICHARD BROWN Synopsis Herein, we give a brief interpretation of the differential of a function
More informationMath Ordinary Differential Equations
Math 411  Ordinary Differential Equations Review Notes  1 1  Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x
More informationNotes on the matrix exponential
Notes on the matrix exponential Erik Wahlén erik.wahlen@math.lu.se February 14, 212 1 Introduction The purpose of these notes is to describe how one can compute the matrix exponential e A when A is not
More informationMath Linear Algebra II. 1. Inner Products and Norms
Math 342  Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,
More informationREVIEW FOR EXAM III SIMILARITY AND DIAGONALIZATION
REVIEW FOR EXAM III The exam covers sections 4.4, the portions of 4. on systems of differential equations and on Markov chains, and..4. SIMILARITY AND DIAGONALIZATION. Two matrices A and B are similar
More informationEconomics 204 Fall 2013 Problem Set 5 Suggested Solutions
Economics 204 Fall 2013 Problem Set 5 Suggested Solutions 1. Let A and B be n n matrices such that A 2 = A and B 2 = B. Suppose that A and B have the same rank. Prove that A and B are similar. Solution.
More informationLinear Algebra March 16, 2019
Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented
More informationIntroduction  Motivation. Many phenomena (physical, chemical, biological, etc.) are model by differential equations. f f(x + h) f(x) (x) = lim
Introduction  Motivation Many phenomena (physical, chemical, biological, etc.) are model by differential equations. Recall the definition of the derivative of f(x) f f(x + h) f(x) (x) = lim. h 0 h Its
More informationContents. 2 Partial Derivatives. 2.1 Limits and Continuity. Calculus III (part 2): Partial Derivatives (by Evan Dummit, 2017, v. 2.
Calculus III (part 2): Partial Derivatives (by Evan Dummit, 2017, v 260) Contents 2 Partial Derivatives 1 21 Limits and Continuity 1 22 Partial Derivatives 5 23 Directional Derivatives and the Gradient
More informationLecture 8: The First Fundamental Form Table of contents
Math 38 Fall 2016 Lecture 8: The First Fundamental Form Disclaimer. As we have a textbook, this lecture note is for guidance and sulement only. It should not be relied on when rearing for exams. In this
More information2. Linear algebra. matrices and vectors. linear equations. range and nullspace of matrices. function of vectors, gradient and Hessian
FE661  Statistical Methods for Financial Engineering 2. Linear algebra Jitkomut Songsiri matrices and vectors linear equations range and nullspace of matrices function of vectors, gradient and Hessian
More informationMathematical Analysis Outline. William G. Faris
Mathematical Analysis Outline William G. Faris January 8, 2007 2 Chapter 1 Metric spaces and continuous maps 1.1 Metric spaces A metric space is a set X together with a real distance function (x, x ) d(x,
More informationLecture 1: Systems of linear equations and their solutions
Lecture 1: Systems of linear equations and their solutions Course overview Topics to be covered this semester: Systems of linear equations and Gaussian elimination: Solving linear equations and applications
More informationLecture 7. Econ August 18
Lecture 7 Econ 2001 2015 August 18 Lecture 7 Outline First, the theorem of the maximum, an amazing result about continuity in optimization problems. Then, we start linear algebra, mostly looking at familiar
More informationLinear Algebra: Linear Systems and Matrices  Quadratic Forms and Deniteness  Eigenvalues and Markov Chains
Linear Algebra: Linear Systems and Matrices  Quadratic Forms and Deniteness  Eigenvalues and Markov Chains Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 3, 3 Systems
More information25. Chain Rule. Now, f is a function of t only. Expand by multiplication:
25. Chain Rule The Chain Rule is present in all differentiation. If z = f(x, y) represents a twovariable function, then it is plausible to consider the cases when x and y may be functions of other variable(s).
More informationLecture Notes on Metric Spaces
Lecture Notes on Metric Spaces Math 117: Summer 2007 John Douglas Moore Our goal of these notes is to explain a few facts regarding metric spaces not included in the first few chapters of the text [1],
More informationMath 2912: Lecture Notes Northwestern University, Winter 2016
Math 2912: Lecture Notes Northwestern University, Winter 2016 Written by Santiago Cañez These are lecture notes for Math 2912, the second quarter of MENU: Intensive Linear Algebra and Multivariable Calculus,
More informationMATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (inclass portion) January 22, 2003
MATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (inclass portion) January 22, 2003 1. True or False (28 points, 2 each) T or F If V is a vector space
More information1 Last time: leastsquares problems
MATH Linear algebra (Fall 07) Lecture Last time: leastsquares problems Definition. If A is an m n matrix and b R m, then a leastsquares solution to the linear system Ax = b is a vector x R n such that
More informationVector Space Basics. 1 Abstract Vector Spaces. 1. (commutativity of vector addition) u + v = v + u. 2. (associativity of vector addition)
Vector Space Basics (Remark: these notes are highly formal and may be a useful reference to some students however I am also posting Ray Heitmann's notes to Canvas for students interested in a direct computational
More informationMath 140A  Fall Final Exam
Math 140A  Fall 2014  Final Exam Problem 1. Let {a n } n 1 be an increasing sequence of real numbers. (i) If {a n } has a bounded subsequence, show that {a n } is itself bounded. (ii) If {a n } has a
More informationMath (P)Review Part II:
Math (P)Review Part II: Vector Calculus Computer Graphics Assignment 0.5 (Out today!) Same story as last homework; second part on vector calculus. Slightly fewer questions Last Time: Linear Algebra Touched
More informationMA2AA1 (ODE s): The inverse and implicit function theorem
MA2AA1 (ODE s): The inverse and implicit function theorem Sebastian van Strien (Imperial College) February 3, 2013 Differential Equations MA2AA1 Sebastian van Strien (Imperial College) 0 Some of you did
More informationLinear Algebra, 4th day, Thursday 7/1/04 REU Info:
Linear Algebra, 4th day, Thursday 7/1/04 REU 004. Info http//people.cs.uchicago.edu/laci/reu04. Instructor Laszlo Babai Scribe Nick Gurski 1 Linear maps We shall study the notion of maps between vector
More informationP = A(A T A) 1 A T. A Om (m n)
Chapter 4: Orthogonality 4.. Projections Proposition. Let A be a matrix. Then N(A T A) N(A). Proof. If Ax, then of course A T Ax. Conversely, if A T Ax, then so Ax also. x (A T Ax) x T A T Ax (Ax) T Ax
More informationTangent Planes, Linear Approximations and Differentiability
Jim Lambers MAT 80 Spring Semester 00910 Lecture 5 Notes These notes correspond to Section 114 in Stewart and Section 3 in Marsden and Tromba Tangent Planes, Linear Approximations and Differentiability
More informationNumerical Methods for Inverse Kinematics
Numerical Methods for Inverse Kinematics Niels Joubert, UC Berkeley, CS184 20081125 Inverse Kinematics is used to pose models by specifying endpoints of segments rather than individual joint angles.
More informationMATH 304 Linear Algebra Lecture 23: Diagonalization. Review for Test 2.
MATH 304 Linear Algebra Lecture 23: Diagonalization. Review for Test 2. Diagonalization Let L be a linear operator on a finitedimensional vector space V. Then the following conditions are equivalent:
More informationLINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM
LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator
More informationReview problems for MA 54, Fall 2004.
Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on
More information7.5 Partial Fractions and Integration
650 CHPTER 7. DVNCED INTEGRTION TECHNIQUES 7.5 Partial Fractions and Integration In this section we are interested in techniques for computing integrals of the form P(x) dx, (7.49) Q(x) where P(x) and
More informationProblems in Linear Algebra and Representation Theory
Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific
More informationMath 443 Differential Geometry Spring Handout 3: Bilinear and Quadratic Forms This handout should be read just before Chapter 4 of the textbook.
Math 443 Differential Geometry Spring 2013 Handout 3: Bilinear and Quadratic Forms This handout should be read just before Chapter 4 of the textbook. Endomorphisms of a Vector Space This handout discusses
More informationExercises for Multivariable Differential Calculus XM521
This document lists all the exercises for XM521. The Type I (True/False) exercises will be given, and should be answered, online immediately following each lecture. The Type III exercises are to be done
More informationMath 1060 Linear Algebra Homework Exercises 1 1. Find the complete solutions (if any!) to each of the following systems of simultaneous equations:
Homework Exercises 1 1 Find the complete solutions (if any!) to each of the following systems of simultaneous equations: (i) x 4y + 3z = 2 3x 11y + 13z = 3 2x 9y + 2z = 7 x 2y + 6z = 2 (ii) x 4y + 3z =
More information3 Stability and Lyapunov Functions
CDS140a Nonlinear Systems: Local Theory 02/01/2011 3 Stability and Lyapunov Functions 3.1 Lyapunov Stability Denition: An equilibrium point x 0 of (1) is stable if for all ɛ > 0, there exists a δ > 0 such
More informationDifferential Topology Final Exam With Solutions
Differential Topology Final Exam With Solutions Instructor: W. D. Gillam Date: Friday, May 20, 2016, 13:00 (1) Let X be a subset of R n, Y a subset of R m. Give the definitions of... (a) smooth function
More informationMTH 2032 SemesterII
MTH 202 SemesterII 201011 Linear Algebra Worked Examples Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education December 28, 2011 ii Contents Table of Contents
More informationContents. 2.1 Vectors in R n. Linear Algebra (part 2) : Vector Spaces (by Evan Dummit, 2017, v. 2.50) 2 Vector Spaces
Linear Algebra (part 2) : Vector Spaces (by Evan Dummit, 2017, v 250) Contents 2 Vector Spaces 1 21 Vectors in R n 1 22 The Formal Denition of a Vector Space 4 23 Subspaces 6 24 Linear Combinations and
More informationReal Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi
Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.
More informationCOMP 558 lecture 18 Nov. 15, 2010
Least squares We have seen several least squares problems thus far, and we will see more in the upcoming lectures. For this reason it is good to have a more general picture of these problems and how to
More information