Area Of A Surface Of Revolution Homework Stu Schwartz Calculus

The University of Sydney School of Mathematics and Statistics Tutorial 1 (Week 2) MATH2961: Linear Algebra and Vector Calculus (Advanced) Semester 1, 2014 Lecturers: James Parkinson and Ruibin Zhang Topics covered and aims In lectures last week:  The scalar (dot) product and the vector (cross) product.  The Cauchy-Schwarz inequality.  Equations of lines and planes.  Curves and the tangent vector.  Partial derivatives.  Tangent planes.  Continuity of and differentiability. After completing this tutorial sheet you will be able to:  Work with the dot product and the vector product.  Work with norms and inequalities.  Understand the geometry of the vector and dot products.  Understand the geometry lines, planes, and curves.  Compute partial derivatives and tangent planes.  Decide if a given function f : Rn → Rm is continuous at a point.  Decide if a given function f : Rn → Rm is differentiable at a point. Preparation questions to do before class 1. Let x = (1, −1, 0) and y = (−2, 1, −1) be vectors in R3 . (a) Compute x · y, kxk and kyk. Compare |x · y| and kxk kyk. (b) Compute the angle between x and y. (c) Compute the area of the parallelogram spanned by x and y. 2. Let x, y ∈ Rn . Prove the triangle inequality: kx + yk ≤ kxk + kyk. 3. Find a vector parallel to the line of intersection of the planes given by x − 2y + 5z = 2 and 3x − y + 5z = 3. 4. Find the tangent plane to f (x, y) = x2 −3 sin(x+y)−y 3 at the point (x, y) = (2, −2). c 2014 The University of Sydney Copyright 1 Questions to attempt in class 5. Let A(−1, 3, 1), B(2, −2, 0) and C(1, 1, −1) be three points in space. (a) Give an equation for the plane through A, B and C. (b) Write down an equation for the line perpendicular to the plane through A, B and C passing through A. 6. Let S be the graph of f (x, y) = x2 y 2 − 3xy 2 + x − 2. (a) Find the tangent plane of S at (x, y) = (1, 1). (b) Find the equation of the curve on S which lies directly above the unit circle in the xy-plane. Find the tangent line to this curve at (x, y, z) = (1, 0, −1). 7. A particle on a string is traveling according to the formula x(t) = (cos t, sin t, t2 ), t ∈ [0, ∞). At time t = 2π seconds the string breaks. Where is the particle 30 seconds later?  2  xy 2 8. Define f : R → R by f (x, y) = x2 + y 2  0 if (x, y) 6= (0, 0) if (x, y) = (0, 0). (a) Show that f is continuous everywhere. (b) Show that fx (0, 0) = fy (0, 0) = 0. (c) Is f differentiable at (0, 0)? 9. Let x and y be vectors in Rn . (a) Show that kx + yk2 + kx − yk2 = 2(kxk2 + kyk2 ). (b) Show that kx − yk2 = kxk2 + kyk2 − 2(x · y). Deduce that Pythagoras’ Theorem and the Cosine Rule are valid in Rn . Questions for extra practice 10. (a) Let x, y ∈ Rn . In class we showed that if n = 3 then the area A of the parallelogram spanned by x and y is kx × yk. For general n this formula is not available, because the cross product is only defined in R3 . But there is another formula: Let J = x y , an n × 2 matrix. Show that A= p det(J T J), where J T is the transpose of J. Hint: The area p of the parallelogram equals kxkkyk sin θ. Now show that kxkkyk sin θ = kxk2 kyk2 − (x · y)2 . (b) Use this formula to compute the area of the parallelogram spanned by the vectors x = (1, −1, 0, 2) and y = (2, 1, −1, 3) in R4 . 2  2 2  (x − y) 11. Let f (x, y) = x4 + y 2  1 if (x, y) 6= (0, 0) if (x, y) = (0, 0). (a) Show that f (x, y) approaches 1 as (x, y) → (0, 0) along any straight line. (b) Does lim(x,y)→(0,0) f (x, y) exist? Is f continuous at (x, y) = (0, 0)? 12. Check some of the computations we skipped in lectures. Specifically, check that for vectors x, y, z ∈ R3 we have: (a) x × y = −y × x  (b) z · (x × y) = det x y z , and hence x · (x × y) = y · (x × y) = 0 p (c) kx × yk = kxk2 kyk2 − (x · y)2 13. Show that the volume of the parallelepiped spanned by vectors x, y and z in R3 is  V = |z · (x × y)| = | det x y z |. 14. Let A(1, 0, 0), B(2, 1, 3), C(−1, 3, −1) and D(4, 4, 2) be four points in space. (a) Compute the area of the triangle ABC. −→ −→ −−→ (b) Find the volume of the parallelepiped spanned by AB, AC and AD. (c) Compute the volume of the pyramid ABCD. 15. Let a, b and c be vectors in R3 . (a) Show that (a × b) × c = (a · c)b − (b · c)a and a × (b × c) = (a · c)b − (a · b)c. (b) Prove the Jacobi identity: (a × b) × c + (b × c) × a + (c × a) × b = 0. 16. Let x1 , . . . , xk ∈ Rn be pairwise orthogonal. Prove that kx1 + · · · + xk k2 = kx1 k2 + · · · + kxk k2 . 17. Find the equation of the plane that contains the line (x, y, z) = (−1, 1, 2) + t(3, 2, 4) and is perpendicular the plane 2x + y − 3z = −4. 18. Show that kxk − kyk ≤ kx − yk 3 for all x, y ∈ Rn . Challenging problems 19. Prove Lagrange’s Identity: n n 1 XX (ai bj − aj bi )2 , kak kbk − (a · b) = 2 i=1 j=1 2 2 2 where a = (a1 , a2 , . . . , an ) and b = (b1 , b2 , . . . , bn ). Hence give another proof of the Cauchy-Schwarz inequality. 20. Let a, b ∈ R3 be nonzero orthogonal vectors. (a) Show that each vector x ∈ R3 can be written in exactly one way as a linear combination x = αa + βb + γ(a × b) with α, β, γ ∈ R. (b) Show that the following system of equations has a unique solution x ∈ R3 . x×a=b and 4 x · a = kak. The University of Sydney School of Mathematics and Statistics Tutorial 2 (Week 3) MATH2961: Linear Mathematics and Vector Calculus (Advanced) Semester 1, 2014 Lecturers: James Parkinson and Ruibin Zhang Topics covered and aims In lectures last week:  Partial and directional derivatives. Tangent planes to level surfaces using ∇f .  Higher order partials, The Mixed Derivatives Theorem.  Taylor polynomials of order 1 and 2 and the Hessian matrix Hf (a).  Taylor’s Theorem for functions f : Rn → R.  Critical points of a function f : Rn → R.  Quadratic forms, positive definite, negative definite, and indefinite matrices.  Second derivative test for functions f : Rn → R (to be continued this week). After completing this tutorial sheet you will be able to:  Compute tangent planes to implicitly defined surfaces and graphs of functions.  Find Taylor polys and understand how to create higher order approximations.  Be able to work with the remainder term.  Apply properties of the gradient vector to solve problems.  Find the critical points of a function f : Rn → R.  Apply the second derivatives test to functions f : Rn → R (more this week). Preparation questions to do before class 1. Compute the second order Taylor polynomial of f (x, y) = cos(2x + y − 1) + yex around (x, y) = (0, 1). Make a statement about the size of the remainder term. 2. Find the tangent plane to the surface z cos x − sin y + z 2 tan−1 x = 1 at (0, π, 1). 3. I’m skiing on a mountain with equation z = 100 − 2x2 − 3y 4 + x − 2y. My current location is (x, y) = (2, −1). Being adventurous I’d like to tackle the steepest slope down the mountain. Which way should I ski initially? What slope will I encounter? Questions to attempt in class 4. Find the equation of the tangent plane to the surface x3 − y 2 + xyz = 0 at the point (1, 1, 0). Do this in two ways: (a) by thinking of the surface as a level surface g(x, y, z) = 0, and (b) by thinking of the surface as the graph of a function z = f (x, y). c 2014 The University of Sydney Copyright 1 5. Find all points on the circle x2 + y 2 = 1 where the circle is tangent to a level curve of the function x2 + 6xy + y 2 . 6. Let f (x, y) = 2 cos(x + y 2 ) − y 2 + 3. (a) Find all critical points of f . (b) You should have found (0, 0) to be a critical point in part (a). Compute the second order Taylor polynomial of f around this critical point. (c) Use your answer from (b) to have a guess about the nature of the critical point (0, 0) (is it a local max, min, or saddle?). Now use Taylor’s Theorem to prove your claim. (d) Now apply the second derivatives test to get the same answer (hopefully!). 7. Find all critical points of f (x, y, z) = x2 +y 2 +z 2 +2xy +2yz −4z +x and determine their nature using the second derivatives test. 8. The temperature in Nemo’s tank at the point (x, y, z) is governed by the equation T (x, y, z) = x2 + 2y 2 − xyz. Nemo is at (x, y, z) = (1, 2, −1). It is way too cold! (T (1, 2, −1) = 11 degrees). Which way should he swim? Questions for extra practice 9. Let f : R2 → R. Decide what the 3rd order Taylor polynomial of f centered at (x, y) = (a, b) should be, and say what Taylor’s Theorem says about the remainder term (you are not asked to prove this - just make an educated guess). Use your formula to give a cubic approximation to f (x, y) = cos(2x + y − 1) + yex around the point (x, y) = (0, 1). 10. Find a point on the surface x2 /9 + y 2 /4 + z 2 = 1 at which the tangent plane is √ perpendicular to (1, 1, 3). Write down an equation of the tangent plane. 11. Find all critical points of the function f (x, y) = −xy + 12. 4 x + y2 . (a) Let f , g : R → Rn be differentiable. Prove that (f · g)′ = f ′ · g + f · g ′ . (b) Let f , g : R → R3 be differentiable. Prove that (f × g)′ = f ′ × g + f × g ′ .   a c be a 2 × 2 real symmetric matrix. Show that: 13. Let A = c b (a) A is positive definite if and only if det(A) > 0 and a > 0. (b) A is negative definite if and only if det(A) > 0 and a < 0. (c) A is indefinite if and only if det(A) < 0. This checks the ‘quick version’ of the 2nd derivatives test for functions f : R2 → R. 2 14. Suppose that f : R2 → R has continuous second order partial derivaties. Suppose that z = f (u, v) and that u = x + y and v = x − y. Show that ∂z ∂z  ∂z 2  ∂z 2 = − ∂x ∂y ∂u ∂v and ∂ 2z ∂ 2z ∂ 2z = − . ∂x∂y ∂u2 ∂v 2 15. A function f : RN → R is radial if f (x) depends only on kxk, i.e., if f (x) = g(kxk), where g(r) is some function, which we shall assume to be differentiable. Show that if f is radial then ∇f (x) is a scalar multiple of x. Challenging problems 16. Let f : R2 → R and assume that the mixed partial derivatives fxy and fyx both exist in a disc around (a, b). Let h, k > 0, and let ∆(h, k) = f (a + h, b + k) − f (a, b + k) − f (a + h, b) + f (a, b). (a) Write ∆(h, k) as F (a + h) − F (a) where F (x) = f (x, b + k) − f (x, b). Use the Mean Value Theorem twice to prove that ∆(h, k) = fxy (α, β) hk where α = α(h, k) and β = β(h, k) with (α, β) → (a, b) as (h, k) → (0, 0). (b) Argue analogously to show that ∆(h, k) = fyx (γ, δ) hk where γ = γ(h, k) and δ = δ(h, k) with (γ, δ) → (a, b) as (h, k) → (0, 0). (c) Deduce that if fxy and fyx are continuous at (a, b) then fxy (a, b) = fyx (a, b). 3 The University of Sydney School of Mathematics and Statistics Tutorial 3 (week 4) MATH2961: Linear Algebra and Vector Calculus (Advanced) Semester 1, 2014 Lecturers: James Parkinson and Ruibin Zhang Topics covered and aims In lectures last week:  Second derivative test for functions f : Rn → R.  Extreme Value Theorem.  Global maximum and minimum values.  Double integrals and iterated integrals. After completing this tutorial sheet you will be able to:  Apply the second derivatives test to functions f : Rn → R.  Find global maximum and minimum values of functions.  Use optimisation techniques to solve practical problems.  Analyse complicated functions f : R2 → R.  Sketch the domain of integration for double integrals.  Change the order of integration in a double integral. Preparation questions to do before class 1. Let f (x, y) = x3 − x2 y + y. (a) Find all critical points of f , and use the second derivatives test to classify them as local maxima, local minima, or saddle points. (b) Briefly explain why f (x, y) attains a global maximum and global minimum on the unit disc D = {(x, y) ∈ R2 | x2 + y 2 ≤ 1}. (c) Find the global maximum and minimum values attained by f on D. 2. Find the maxima and minima of f (x, y) = (1 + x)y on the square  S = (x, y) ∈ R2 | −1/2 ≤ x, y ≤ 1/2 . 3. Sketch the domain of integration of the following iterated integrals, and then write the integrals as iterated integrals in the reverse order: (a) Z 1 Z 0 1+x  f (x, y) dy dx; 1−x c 2014 The University of Sydney Copyright (b) Z 1 Z 0 1 1 y2  f (x, y) dx dy. Questions to attempt in class 4. Find all critical points of f (x, y) = 4 x + 2 y − xy and determine their nature. 5. Find all critical points of f (x, y) = x3 − 3x2 y + y 3 and determine their nature. Also try to do so if the second derivatives test is inconclusive. 6. Explain why f (x, y) = 6x + 2y 3 attains a global maximum and global minimum on the set D = {(x, y) ∈ R2 | 4x2 + y 2 ≤ 1}. Find the global maximum and global minimum values. 7. Determine why it is difficult to integrate in the order given. Then compute the integrals. (a) Z 1 Z 0 1 e y −x2  dx dy; (b) Z 1 Z 0 x 1p  1 − y 2 dy dx. 8. Find the equation of the plane that passes through (1, 2, 3) and cuts off the smallest volume in the first octant. Questions for extra practice 9. Let D = {(x, y) ∈ R2 | x2 + y 2 ≤ 1} and let f (x, y) = x2 + y 2 − x − y + 1. Explain why f attains a global maximum and minimum on D. Find these values. 10. Sketch the domain of integration of the following double integrals: Z 1 Z y 2 Z √2 Z √4−2y2   f (x, y) dx dy; (a) (b) f (x, y) dx dy. √ −1 0 0 − 4−2y 2 11. Find the maximum and minimum values of f (x, y, z) = x2 − y 2 + z 2 − z on D = {(x, y, z) | x2 + y 2 + z 2 ≤ 1 and z ≥ 0}. Challenging problems 12. Let A be an n × n real matrix. The induced matrix norm kAk of A is defined by kAk := max{kAxk such that x ∈ Rn and kxk ≤ 1}. This maximum is attained because f (x) = kAxk is continuous, and the unit ball kxk ≤ 1 is closed and bounded.   2 1 (a) Let A = . Compute the induced matrix norm of A. −1 0 p (b) With A as in (a), calculate λmax (AT A), and verify that kAk = λmax (AT A). p (c) Now prove the formula kAk = λmax (AT A) for arbitrary real matrices. 2 13. Find the maximum value of f (x1 , . . . , xn ) = x1 x2 · · · xn on the set S = {(x1 , . . . , xn ) ∈ Rn : x1 + · · · + xn = 1 and x1 , . . . , xn ≥ 0}. Use this to derive the Arithmetic Mean-Geometric Mean Inequality: (a1 a2 · · · an )1/n ≤ a1 + · · · + an , n which is valid for a1 , . . . , an ≥ 0. 14. Let (x1 , y1 ), (x2 , y2 ), . . . , (xn , yn ) ∈ R2 be given data points. We want to find the equation of the straight line y = mx + b which fits the data best, where ‘best’ means that m and b are chosen to minimise d21 + · · · + d2n , where di = yi − (mxi + b) is the discrepancy between the data point (xi , yi ) and the line y = mx + b. Find m and b. (This is the method of least squares from statistics. You may assume that the minimum is attained). 15. Functions f : R2 → R can display some pretty complex phenomena. In this question you investigate a function which appears to have a local maximum at (0, 0) if one travels along straight lines through the origin, yet appears to have a local minimum at (0, 0) if one travels along x = y 2 .  6 2 2x2 y − x2 − y 2 + 4x y if (x, y) 6= (0, 0) (x4 + y 2 )2 Define f : R2 → R by f (x, y) =  0 if (x, y) = (0, 0). (a) Show that f (x, y) is continuous for all (x, y) ∈ R2 . (b) Let v = (a, b) be an arbitrary unit vector, and define gv : R → R by gv (t) = f (at, bt) for all t ∈ R. Thus gv (t) gives the values of f along the line x = tv, and for all t ∈ R we have gv (t) = 2a2 bt3 − t2 + 4a6 b2 t4 . (a4 t2 + b2 )2 Show that for each v the function gv (t) has a local maximum at t = 0. (c) Compute h(t) = f (t, t2 ) and observe that it has a local minimum at t = 0. (d) Try to visualise what is happening... 3 The University of Sydney School of Mathematics and Statistics Tutorial 4 (week 5) MATH2961: Linear Algebra and Vector Calculus (Advanced) Semester 1, 2014 Lecturers: James Parkinson and Ruibin Zhang Topics covered and aims In lectures last week:     Change of variables for double integrals. Arc length, and integrals of scalar valued functions along curves. Integrals of vector valued functions along curves. Green’s Theorem. After completing this tutorial sheet you will be able to:  Apply the transformation formula (change of variables) for double integrals.  Parametrise basic curves.  Calculate arc length  Compute integrals of scalar valued functions along curves  Compute integrals of vector fields along curves.  Know the statement of Green’s Theorem, and be able to apply it. Preparation questions to do before class 1. Let P be the parallelogram in R2 spanned by a = (2, 3) and b = (3, 5). Evaluate ZZ (x + xy) dA P by making an appropriate change of variables. 2. Let C be the triangle with vertices (0, 0), (1, 0) and (1, 2) oriented counterclockwise. Evaluate the following integrals. Does Green’s Theorem help with any of them? Z Z (x − y) ds x2 y dx − x dy (b) (a) C C Questions to attempt in class 3. Let C be the ellipse x2 /a2 + y 2 /b2 = 1, traversed once counterclockwise. Calculate Z f · ds, where f (x, y) = (x + y, y − x). C Do the calculation directly, and then by applying Green’s Theorem. 4. Let D = {(x, y) ∈ R2 | 1 ≤ x2 + y 2 ≤ 16 and x, y ≥ 0}. Evaluate ZZ  sin(x2 + y 2 ) + 30xy 4 dA by using a suitable change of variables. D c 2014 The University of Sydney Copyright 1 5. Sketch the curve y 2 = x3 in R2 and compute its length between (1, 1) and (1, −1). 6. Let D = {(x, y) ∈ R2 | x2 y 2 − (x2 + y 2 )3 ≥ 0 and x, y ≥ 0}, and let C = ∂D be the boundary of D, oriented counterclockwise. (a) Find a parametrisation of C. Hint: Use polar coordinates. (b) Compute the area of D. (c) Prove that the length of C is equal to one eighth of the circumference of 4x2 +y 2 = 4. Questions for extra practice 7. Let C be the boundary of the region x2 ≤ y ≤ x, 0 ≤ x ≤ 1, with positive orientation. Compute Z (y 3 , x3 + 3xy 2 ) · ds. C Do the calculation directly, and then by applying Green’s Theorem. 8. Let C be the lower arc of the circle x2 + y 2 = 2 between (1, 1) and (−1, 1), oriented such that (1, 1) is the starting point and (−1, 1) the endpoint. Z Z 3 (x − y) dx + (x + y) dy. x y ds (b) Compute (a) Compute C C 9. Compute the area of the cardioid 0 ≤ r ≤ 1 − cos θ, where θ ∈ [0, 2π]. 10. Use the transformation formula to evaluate ZZ 2 2 e−9x −16y dx dy, where E is the ellipse 9x2 + 16y 2 ≤ 36. E 2 2 11. Let S be the surface the curve √ z = (x − 1) + y . Let C be √ √ on S lying above the part of the semicircle y = 1 − x2 between (1, 0) and (1/ 2, 1/ 2). Orient C so that (0, 0, 1) is the starting point. Compute Z Z f · ds, f ds and C C where f (x, y, z) = xy + yz + x2 + y 2 − 1 − 2y and f (x, y, z) = (2x, xy, z 2 ). 12. Compute Z C y dx + z dy + x dz where C is given by (t, t2 , 2t), t ∈ [0, 1]. 13. Write down a parametrisation of the following curves: (a) The ellipse 9x2 + 4y 2 = 18 (b) The curve x2 − xy + y 2 = 4. (c) The intersection of x2 + y 2 = z 2 and x2 + y 2 + z 2 = 16 in the lower half space. 2 14. Let C be a curve in R3 , and let γ : [a, b] → R3 be a parametrisation of C. Suppose that the vector field f : R3 → R3 has the property that f (γ(t)) is parallel to γ ′ (t) for each t ∈ [a, b]. Show that Z Z C f · ds = C kf kds. Challenging problems 15. Let T be the triangle with vertices (0, 0), (1, 0), and (0, 1). Compute the integral ZZ (x2 − y 2 ) cos(x2 + 6xy + y 2 ) dxdy. T Hint: Diagonalise the quadratic form x2 + 6xy + y 2 . 16. Let S denote the part of the strip 2 ≤ x + y ≤ 3 in the first quadrant. Evaluate ZZ x − y  exp dx dy by using a suitable transformation. x+y S 17. Let C be a simple curve in Rn with parametrisations α : [a, b] → Rn and β : [c, d] → Rn . Assume that h : [a, b] → [c, d] is bijective and differentiable with α = β◦h. Let f : Rn → R and f : Rn → Rn . Show that Z b f (α(t))kα′ (t)kdt = a Z b a Z d f (β(t))kβ ′ (t)kdt, and c ′ f (α(t)) · α (t) dt = ± Z d c f (β(t)) · β ′ (t) dt with the + sign if α and β have the same orientation, and the − sign if α and β have opposite orientations. 18. Compute ZZ e−(x DR 2 +y 2 ) dA, where DR = {(x, y) | x2 + y 2 ≤ R2 }. Use your answer to prove that Z ∞ 2 e−x dx = −∞ 3 √ π. The University of Sydney School of Mathematics and Statistics Tutorial 5 (week 6) MATH2961: Linear Algebra and Vector Calculus (Advanced) Semester 1, 2014 Lecturers: James Parkinson and Ruibin Zhang Topics covered and aims In lectures last week:  Green’s Theorem, and area formula (and the planimeter).  Divergence Theorem in the plane.  Circulation along curves and flux across curves.  Interpretations of divergence and curl.  Setting up triple integrals.  Change of variables for triple integrals. After completing this tutorial sheet you will be able to:  Apply Green’s Theorem in various contexts.  Use Green’s Theorem to compute areas.  Be able to sketch simple vector fields in R2 .  Understand the geometric significance of divergence and curl in the plane.  Compute circulation along curves and flux across curves; and relate to div and curl.  Set up triple integrals, and be able to apply the transformation formula. (more examples next week) Preparation questions to do before class 1. Calculate the divergence and curl, and sketch the following vector fields. (a) f (x, y) = (−y, x) (b) f (x, y) = (−x, −y) 2. Let C be the circle with radius R centred at the origin, oriented counterclockwise. (a) Compute the circulation of f along C, where f is as in Question 1(a). (b) Compute the flux of f across C, where f is as in Question 1(b). 3. Let D be the domain in R3 lying above z = x2 and below z = 1 − x2 − y 2 . Determine the limits in the iterated integral Z Z Z ZZZ   f (x, y, z) dx dy dz = f (x, y, z) dz dy dx. D 4. Let D be the region of R3 which lies outside the cylinder x2 + y 2 = 1, inside the sphere x2 + y 2 + z 2 = 4, and satisfies x, y, z ≥ 0. Calculate the integral ZZZ z dV. D c 2014 The University of Sydney Copyright 1 Questions to do in class 5. Let C be the unit circle x2 + y 2 = 1, traversed once counterclockwise. Evaluate Z   −y 3 + sin(xy) + xy cos(xy) dx + x3 + x2 cos(xy) dy. C 6. Let P be the parallelepiped spanned by (1, 1, 0), (1, −1, 0) and (0, 0, 1). (a) Write down a formula for a linear transformation g : R3 → R3 which maps the unit cube [0, 1] × [0, 1] × [0, 1] onto the parallelepiped P. ZZZ 2 2 2 (b) Calculate the triple integral (x2 − y 2 )zex +y +z dxdydz. P 7. Let C be any smooth simple closed curve enclosing the origin, oriented counterclockwise. Use Green’s Theorem to compute Z 2 x y dx − x3 dy . (x2 + y 2 )2 C Note that since (0, 0) is enclosed by C, and since this is a singular point of the vector field, you cannot apply Green’s Theorem on the entire region enclosed by C. 8. Find the area enclosed by the hypocycloid given by x = a cos3 θ and y = a sin3 θ, where a > 0 is a constant and 0 ≤ θ ≤ 2π. 9. (a) Compute the circulation of the vector field f (x, y) = (y+ln(x2 +1), cos4 y) along the circle of radius r centred at (a, b) ∈ R2 , where the circle is oriented counterclockwise. (b) Find the flux of the vector field f = (2x, −3y) across the boundary of the ellipse x2 + y 2 /42 ≤ 1 (where the boundary is given the positive orientation). Questions for further practice 10. Determine the limits in the iterated integral such that Z Z Z Z 1 Z √1−x2 Z 1   f (x, y, z) dy dz dx = √ √ −1 − 1−x2 x2 +y 2   f (x, y, z) dz dy dx. 11. Calculate the divergence and curl, and sketch the following vector fields. (b) f (x, y) = (x2 , −y) (a) f (x, y) = (x + y, 0) 12. Determine the flux of f (x, y) =  x , y x2 +y 2 x2 +y 2 (a) the circle x2 + y 2 = r2 for r > 0;  across the curves: (b) the circle (x − 2)2 + (y − 1)2 = 1; (c) an arbitrary closed smooth curve winding around (0, 0) once anticlockwise; (d) an arbitrary closed piecewise smooth curve not enclosing (0, 0). 13. Sketch the integration region of the iterated integral 2 Z 1 3Z 4Z x 2 0 f (x, y, z) dzdydx. Challenging questions 14. The quadrifolium curve is given by (x2 + y 2 )3 = (x2 − y 2 )2 . Convert this to polar form and hence sketch the curve. Compute the enclosed area. 15. Compute the area of D = {(x, y) ∈ R2 | x3 + y 3 ≤ 3xy, x ≥ 0, y ≥ 0}. The boundary curve with equation x3 + y 3 = 3xy is the Folium of Descartes. 16. Let f : R2 → R2 be a differentiable vector field in R2 . Show that if curlf = 0 then f is path independent. That is, Z Z f · ds = f · ds C1 C2 whenever C1 and C2 are smooth curves with the same start and end points. 17. Let C be the lower arc of the circle x2 + y 2 = 2 between (1, 1) and (−1, 1), oriented such that (1, 1) is the starting point and (−1, 1) the endpoint. Compute Z (x − y) dx + (yesin y − 2x) dy. C 18. In this question you use Green’s Theorem to prove Cauchy’s Theorem. This illustrates how vector calculus can be applied to complex analysis. This question is mainly for students doing MATH2962 (who will see Cauchy’s Theorem later in the semester). (a) A function f : C → C is called differentiable at α = a + ib if the limit f ′ (α) = lim z→α f (z) − f (α) z−α exists. Write z = x + iy and f (z) = P (x, y) + Q(x, y)i where P : R2 → R and Q : R2 → R. Then the above limit as a 2-variable limit (x, y) → (a, b). Considering paths of approach along y = b, and along x = a, show that if f is differentiable then ∂Q ∂P = ∂x ∂y and ∂P ∂Q =− . ∂y ∂x These formulae are called the Cauchy-Riemann equations. (b) Let C be a smooth oriented curve in C with orientation preserving parametrisation γ(t) : [a, b] → C. The contour integral of f (z) along C is defined by Z Z b f (γ(t))γ ′ (t) dt. f (z) dz = C a Note the similarity with the definition of the integral of a vector field along a curve. Indeed, show that Z Z Z f (z) dz = P dx − Qdy + i Qdx + P dy, C C C where the integrals on the right are integrals of vector fields along curves. (c) Let C be a simple closed piecewise smooth curve in the complex plane C. Prove Cauchy’s Theorem: If f is differentiable, then Z f (z) dz = 0. C Hint: Apply Green’s Theorem, and then use the Cauchy-Riemann equations. 3 The University of Sydney School of Mathematics and Statistics Tutorial 6 (week 7) MATH2961: Linear Algebra and Vector Calculus (Advanced) Semester 1, 2014 Lecturers: James Parkinson and Ruibin Zhang Topics covered and aims In lectures last week:      Parametrisations of surfaces. The normal vector to a parametrised surface. Surface area and integrals of scalar and vector valued functions across surfaces. Vector fields in R3 ; divergence and curl. Stokes’ Theorem. After completing this tutorial sheet you will be able to:     Write down parametrisations of surfaces. Understand what it means for a surface to be oriented. Compute surface areas and various surface integrals. Apply Stokes’ Theorem. Questions to to before class 1. Let S be the portion of the plane 3x + y − z = 0 lying Z Z over the square with vertices (0, 0), (y + z) dS. (1, 0), (1, 1) and (0, 1) in the xy-plane. Evaluate S 2. Let f (x, y, z) = (z, x − 3xz, 2x), and S the hemisphere xZ2Z+ y 2 + z 2 = 1, z ≥ 0 oriented curl f · dS. upwards. Use Stokes’ Theorem to calculate the integral S 3. Compute the flux of the vector field f (x, y, z) = (2y, xy + z, 3z) upwards across the part of the plane 3x − z = 2 within the cylinder x2 + y 2 = 4. That is, compute ZZ (2y, xy + z, 3z) · dS. S Questions to do in class 4. Find parametrisations for the following surfaces. In each case give a sketch, showing the normal vectors given by your parametrisation. (a) The plane containing the points a = (0, 1, 1), b = (1, 0, 2) and c = (1, 3, 1). (b) The part of the cylinder with axis the z-axis and radius R between z = ±1. 5. Let the top half S of a torus be parametrised by x = (2 + cos θ) cos ϕ, y = (2 + cos θ) sin ϕ, z = sin θ, where θ ∈ [0, π] and ϕ ∈ [0, 2π]. We orient the surface S upwards. Given the vector field f (x, y, z) = (−y, x, z 2 ), compute both of the integrals appearing in Stokes’ Theorem. c 2014 The University of Sydney Copyright 1 6. Calculate the integral in Question 2 in two other ways: (a) By a direct calculation. (b) By using Stokes’ Theorem to replace the surface by a simpler surface with the same boundary as S. 7. Evaluate the integral of (x2 + y 2 )z over the top half of the sphere of radius R > 0 with centre at (0, 0, 0). Questions for extra practice 8. The intersection of the paraboloid z = x2 + y 2 and the cylinder (x − 1)2 + (y − 1)2 = 1 is a curve C in R3 . Orient C counterclockwise when viewed from above. Compute Z f · ds, where f (x, y, z) = (x, z, 2y), C directly, and by Stokes’ Theorem. 9. Let D be the unit cube [0, 1] × [0, 1] × [0, 1], and let S be the surface consisting of all faces of D except for the top of D, with outward orientation. Compute ZZ (∇ × f ) · dS S x2 where f (x, y, z) = (−2y + ze , x − sin(y 2 ), ex 10. Using Stokes’ Theorem, evaluate Z 2 −y+z 2 ). 2xyz dx+x2 z dy +x2 y dz where C is the intersection C of the cylinder x2 + y 2 = 2y with the plane y = z. Orient C anti-clockwise when viewed from the top. 11. Compute the integral in Question 5 in a third way. 12. [2009 Exam Question] Let S be the part of the sphere x2 + y 2 + z 2 = 52 between the planes z = −4 and z = 4, oriented with outward pointing normal. Let f : R3 → R3 be the vector field f (x, y, z) = (yz + 2x, −xz + 4y 2 , z 4 − 2). (a) Draw a careful sketch of S. Identify the boundary ∂S of S, and indicate the positive orientation on ∂S induced by the outward orientation on S. (b) Compute curl f , and find the value of the surface integral ZZ (curl f ) · n dS. S There are at least 4 ways to calculate this integral... 13. Recall from MATH1903 that the surface area of the solid of revolution formed by rotating f (x) ≥ 0 (a ≤ x ≤ b) about the x-axis (not including any end caps) is Z b p f (x) 1 + f ′ (x)2 dx. 2π a Deduce this formula from the general surface area formula 2 ZZ 1 dS. S 14. Find parametrisations for the following surfaces. In each case give a sketch, showing normal vector of your parametrisation. (a) A circular paraboloid with axis the x-axis containing (0, 0, 0) and (1, 1, 0). (b) The top half of the ellipsoid with axes lengths 2, 3 and 5. (c) The triangular region in R3 with vertices (2, 0, 0), (0, 1, 0) and (0, 0, 4). 15. Let f be the vector field f (x, y, z) = (cos(xz) + sin(ex ), x + esin y , ex 2 +y ), and let S be the upper half of the sphere x2 + y 2 + z 2 = 1 with outward pointing orientation. Make a clever application of Stokes’ Theorem to compute ZZ curl f · dS. S 16. Let R be the region in R3 determined by x2 + y 2 ≤ z ≤ 18 − x2 − y 2 . A vector field is given by f (x, y, z) = (0, 0, z 2 ). Compute the flux out of R. 17. Let R be the unit cube 0 ≤ x, y, z ≤ Z Z1 with outer pointing unit normal vectors n, and let 2 2 2 f · n dS. f (x, y, z) = (x , y , z ). Compute ∂R 18. Let 0 < r < R be given. A torus can be represented by the parametrisation  g(ϕ, θ) = (R + r cos θ) cos ϕ, (R + r cos θ) sin ϕ, r sin θ with ϕ, θ ∈ [0, 2π]. Compute the surface area of the torus. 19. Suppose that x(t) = x(t)i + z(t)k, a ≤ t ≤ b, is a parametrisation of a curve C in the right half of the xz-plane. Let S be the surface obtained by revolving C about the z-axis. (a) Write down a parametrisation of S. (b) Derive a formula for the surface area of S. Test this using Question 18. Questions on material covered in Week 7 20. Which of the following vector fields are conservative? For the vector fields f that are conservative find a potential function g : R3 → R (that is, f = ∇g). (a) f (x, y, z) = (yz + z sin x, xz, xy − cos x) (b) f (x, y, z) = (x3 − zy, sin(xy), ex ) (c) f (x, y, z) = (ez cos y, −ez x sin y, ez x cos y − cos z) 2 2 (d) f (x, y, z) = (ez xy, sin y + ez , ex ) 2 2 2 21. Let R be the unit Z Z cube [0, 1] × [0, 1] × [0, 1] and let f (x, y, z) = (x , y , z ). Last week f · n dS directly (where ∂R has outward pointing normal, as usual). you computed ∂R Now make the computation using the Gauss’ Divergence Theorem. 3 22. Let f = (2xzex 2 +y 2 − 2x sin(x2 ), 2yzex 2 +y 2 − cos y, ex 2 +y 2 ). Compute Z C f · ds where C is the curve with parametrisation γ(t) = (t2 + cos(2πt2 ), sin(πt2 ), t3 − t + cos(πt)), t ∈ [0, 1]. 23. Let R be the bounded solid region in R3 bounded by the parabolic ridge z = 1 − x2 and the planes z = 0, y = 0, and z + y = 2. Let S = ∂R be the boundary surface of R with outward pointing orientation. Use Gauss’ Divergence Theorem to compute the surface integral ZZ S f · dS, where f (x, y, z) = (x2 + y 3 , y + z 2 − x, 4z). 24. Let R be a solid region in R3 with boundary surface ∂R oriented outwards (as usual). Let f, g : R3 → R be twice differentiable. Show that ZZZ ZZ  f ∇2 g + ∇f · ∇g dV, (f ∇g) · dS = R ∂R where ∇2 g = ∇ · (∇g). 25. (a) Let R be a smoothly bounded region of R3 with outward pointing unit normals n. Let g : R → R be differentiable. Using the divergence theorem, show that ZZZ ZZ ∇g dV. gn dS = R ∂R (The integral of the vector function on the left hand side is defined componentwise) (b) Use part (a) to derive Archimedes law of buoyancy: “The force of buoyancy acting on a body R submerged in a fluid equals the weight of the fluid displaced by R”. Suggestion: Let S0 be the part of ∂R submerged in the fluid. Suppose that the fluid has constant mass density ρ. Set up coordinates so that x3 = 0 is the surface of the fluid with x3 pointing downward. If p denotes the pressure then the buoyant RR force on R is B = − S0 pn dS. Using that the pressure is p = ρgx3 (g is the gravitational constant), show that B = −gρV e3 , where V is the volume of the submerged part of R, and e3 = (0, 0, 1). 26. [2009 Exam Question] For r ≥ 0 let Sr = {x ∈ R3 | kxk = r} be the sphere of radius r centered at the origin. Let f : R3 → R be a differentiable function and suppose that ∇f (x) = g(x)x for all x ∈ R3 , where g : R3 → R is some function. (a) Show that f is constant on each sphere Sr . Hint: Let a, b be points on Sr , and let γ(t) be a curve lying on Sr joining a and b. (b) Let h(r) denote the constant value of f (x) for x ∈ Sr , so that f (x) = h(kxk). Assuming that h : (0, ∞) → R is differentiable, show that for x 6= 0, kxkg(x) = h′ (kxk). (c) Hence show that if Br = {x ∈ R3 | kxk ≤ r} is the ball of radius r, then ZZZ div(∇f ) dV = 4πr2 h′ (r). Br 4 27. Let R be the region in R3 determined by x2 + y 2 ≤ z ≤ 18 − x2 − y 2 . A vector field is given by f (x, y, z) = (0, 0, z 2 ). Calculate ZZ f · dS, S where S is the boundary surface of R by using the Divergence Theorem. Some more triple integrals 28. Let a, b > 0 be constants. Let D be the region Z Z Z above the cone z = b 2 2 2 2 z dxdydz. the sphere x + y + z = a . Calculate p x2 + y 2 and inside D 29. Let D be the region in R3 bounded by the cylinder x2 + y 2 = a2 , the xy-plane, the xz-plane, and the plane z = y. Calculate the integral of f (x, y, z) = yz over D. 30. Compute the integral of the function f (x, y, z) = by x2 + y 2 + z 2 ≤ 4z and z ≥ 1. p x2 + y 2 + z 2 over the domain D given 31. Let R > 1 and α ∈ R, and let D be the region between the spheres of radius 1 and R centred at the origin 0 in R3 . Evaluate (with care when α = 3/2) ZZZ 1 dxdydz. 2 2 2 α D (x + y + z ) 32. Use the fact that Z ∞ 2 e−u du = √ π to compute −∞ ZZZ e−(2x 2 +3y 2 +2z 2 −2xy+2yz) dxdydz. R3 Hint: Write the exponent as xT Ax with A symmetric and orthogonally diagonalise. 33. Let D be the unit ball in R3 . Let a ∈ R3 be a fixed nonzero vector. Compute the integral ZZZ cos(a · x) dxdydz. D Hint: Make a change of variables into spherical coordinates with the vector a playing the role of the z-axis. 5 The University of Sydney School of Mathematics and Statistics Tutorial 11 (Week 12) MATH2961: Linear Algebra and Vector Calculus (Advanced) Semester 1, 2014 Web Page: http://www.maths.usyd.edu.au/u/UG/IM/MATH2961/ Lecturers: James Parkinson and Ruibin Zhang Questions marked with * are more difficult questions. Material covered (1) image and kernel of linear operators (2) complements of subspaces (3) change of basis and transition matrices Outcomes After completing this tutorial you should (1) find the basis of kernel and image (2) determine complements of subspaces (3) be able to find transition matrices for a change of basis (4) use kernel and image to prove properties of linear operators Questions to complete during the tutorial 1. A linear operator from R5 → R4 is given by the matrix   2 4 1 2 6 1 2 1 0 1   −1 −2 −2 3 6  . 1 2 −1 5 12 (a) By reducing the matrix to reduced row-echelon form, find a basis of the image and the kernel of the linear operator. (b) What is the rank of the given matrix.         1 0 0 1 −2 1 3 2 3       0 , 1 , 0 2. Let T : R → R be defined by T (x) = x for all x ∈ R . Let 2 1 −3 0 0 1     1 −2 be the standard basis in R3 and let , a new basis in R2 . 2 1 (a) Find the transition matrix from the standard basis to the new basis in R2 . (b) Find the matrix representation of T relative to the standard basis in R3 and the new basis in R2 . c 2014 The University of Sydney Copyright 1 3. Consider the subspace V := span(et , e−t ) in the space of functions from R to R. It is given that (et , e−t ) is a basis of V . (a) Show that (cosh t, sinh t) is a basis of V as well. (b) Compute the transition matrix from the basis (et , e−t ) to the basis (cosh t, sinh t). 4. Let V be a finite-dimensional vector space and T : V → V a linear operator. For each positive integer n let T n : V → V be defined recursively by T n (v) = T (T n−1 (v)) for all v ∈ V , where T 0 := I is the identity. Note: The questions below require to prove inclusions or equality of sets. This is usually done as follows: Let A, B be sets. To show that A ⊆ B we take x ∈ A, and show that x ∈ B as well. The argument used needs to be valid for every choice of x ∈ B. To show the equality A = B we often firs show that A ⊆ B, and then B ⊆ A. (a) Show that ker T n ⊆ ker T n+1 for all n ≥ 0. (b) Show that if ker T n = ker T n+1 for some n ≥ 0 then ker T k = ker T n for all k ≥ n. (c) Show that im T n+1 ⊆ im T n for all n ≥ 0. (d) Show that if im T n+1 = im T n for some n ≥ 0 then im T k = im T n for all k ≥ n. By considering dimensions, show that im T n = im T n+1 must hold for some n ≤ dim V . Extra questions for further practice       1 1 1 5. Consider the vectors v 1 = 0, v 2 = 1 and v 3 = 1 in Z32 . 1 0 1 3 (a) Show that (v 1 , v 2 , v 3 ) is a basis of Z2 . (b) Determine the transition matrix for the change of basis from (v 1 , v 2 , v 3 ) to the standard basis. (c) Determine the transition matrix for the change of basis from the standard basis to (v 1 , v 2 , v 3 ).       −1 1 −7 3       1 , 0 , 5  another 6. Let (e1 , e2 , e3 ) be the standard basis of R and let b := 1 2 2 basis. (a) Compute the transition matrices from the basis b to the standard basis and vice versa.   −2 (b) A vector is given by v =  4  with respect to the standard basis. Find the 11 coordinate vector of v with respect to the basis b. (c) Suppose that the matrix representation of T : b is  −1 1  B = 0 −2 0 0 R3 → R3 with respect to the basis  0 1 −1 Find its matrix representation with respect to the standard basis. 2   2  7. The vector v = 1 spans a one dimensional subspace U in Z3 . Find a basis of a 1 complement of U in Z33 . 8. Let A ∈ Mat(n × n; R) be a symmetric matrix. In vector calculus the dot product and the Euclidean norm was introduced. (a) Show that kAxk2 = x · (AT Ax) for all x ∈ Rn . (b) If A is symmetric, show that ker A = ker A2 . (c) Hence show that Rn = im A ⊕ ker A. 9. Let A1 ∈ Mat(` × `; F), A2 ∈ Mat(p × p; F) and B ∈ Mat(` × p, F). Consider the block matrix   A1 B A := ∈ Mat((` + p) × (` + p); F), 0 A2 where 0 is the p × ` zero matrix. (a) If A1 and A2 are invertible, show that  −1  −1 A1 −A−1 BA 1 2 = A−1 . 0 A−1 2 You need to think about how to multiply block matrices and be careful with the order of multiplication since matrix multiplication is not commutative. (b) Prove that A is invertible if and only if A1 and A2 are invertible. 10. Suppose that V is a finite dimensional vector space over C with dim V = n and T : V → V linear. (a) Fix a vector v ∈ V \ {0}. Explain why there are a0 , . . . , an ∈ C, not all equal to zero, so that a0 v + a1 T v + a2 T 2 v + · · · + an T n v = 0. (b) Replace T k v by λk in the polynomial in part (a). By the fundamental theorem of algebra that polynomial can be written as a product of linear factors p(λ) = (λ − λ1 )(λ − λ2 ) · · · (λ − λm ) for some 1 ≤ m ≤ n Replacing λ by T , show that p(T )v := (T − λ1 I)(T − λ2 I) · · · (T − λm I)v = 0. Explain why (T − λk I) is not injective for some k = 1, . . . , m. 3 (1) The University of Sydney School of Mathematics and Statistics Tutorial 12 (Week 13) MATH2961: Linear Algebra and Vector Calculus (Advanced) Semester 1, 2014 Web Page: http://www.maths.usyd.edu.au/u/UG/IM/MATH2961/ Lecturers: James Parkinson and Ruibin Zhang Questions marked with * are more difficult questions. Material covered (1) determinants (2) inner product spaces (3) Gram-Schmidt orthogonalisation (4) unitary matrices Outcomes After completing this tutorial you should (1) should be able to compute determinants using Laplace’s formula or using row operations in a general field. (2) be able to work with inner products in a theoretical and practical setting (3) be able to perform the Gram-Schmidt process for a given inner product. (4) be able to work with unitary matrices Questions to complete during the tutorial 1. Use row operations to calculate the determinant of   1 5 11 2 2 11 −6 8   −3 0 −452 6  −3 −16 −4 13   4 2 0 2. Let A ∈ Mat(3 × 3; Z5 ) be given by A = 1 3 1 . Compute the determinant of A. 2 2 4 3. Let A ∈ Mat(n × n, C) be Hermitian, that is, A¯T = A. The dot product of u = (x1 , . . . , xn ) and v = (y1 , . . . , yn ) in Cn is defined by u · v = uT v¯ = x1 y¯1 + · · · + xn y¯n (note the complex conjugates). The dot product is an inner product on Cn . (a) Prove that u · (Av) = (Au) · v. (b) Prove that all eigenvalues of A are real. (c) Suppose that u and v are eigenvectors corresponding to distinct eigenvalues of A. Show that u · v = 0 (d) Show that ker A2 = ker A. c 2014 The University of Sydney Copyright 1 4. Let V be a vector space over R or C and h· , ·i : V × V → V a map satisfying the following properties: (i) hv, vi ≥ 0 with equality if and only if v = 0; (ii) v 7→ hv, wi is linear for every w ∈ V ; (iii) hv, wi = hw, vi for all v, w ∈ V . p Define kvk := hv, vi. Then hv, wi is called inner product in V and kvk the induced norm. (a) Prove that ¯ vi + µ hu, λv + µwi = λhu, ¯hu, wi for all scalars λ, µ and all u, v, w ∈ V . (b) Prove that kλvk = |λ|kvk for all scalars λ and all v ∈ V . (c) Show that |hv, wi| ≤ kvkkwk (Cauchy-Schwarz inequality) Hint: Consider the discriminant of the polynomial p(t) = kv − thv, wiwk2 ≥ 0 with real coefficients which has at most one zero. (d) Use the Cauchy-Schwarz inequality to prove that kv +wk ≤ kvk+kwk in a similar manner as done for the Euclidean norm in Cn . 5. For vectors x = (x1 , x2 ) and y = (y1 , y2 ) in R2 define hx, yi := 2x1 y1 + x1 y2 + x2 y1 + x2 y2 p and the induced norm kxk = hx, xi. (a) Show that hx, yi is an inner product on R2 . (b) Show that the vectors x = (2, 1) and y = (3, −5) are orthogonal with respect to the given inner product. Are they orthogonal with respect to the usual dot product on R2 ? (c) Compute ke1 k and ke2 k, where e1 , e2 are the standard basis vectors (the norms are to be computed with respect to the norm induced by the given inner product). (d) Find orthonormal vectors w1 , w2 such that w1 = λe1 . Orthonormal means the vectors are orthogonal, and their length is one. (This is the first step in the Gram-Schmidt orthogonalisation process.) Extra questions for further practice 6. Let V be an inner product space over R or C. Prove the generalised “Pythagoras Theorem”: if v1 , v2 , . . . , vn are pairwise orthogonal then kv1 + v2 + · · · + vn k2 = kv1 k2 + kv2 k2 + · · · + kvn k2 . 7. (a) Suppose that { vi : 1 ≤ i ≤ n} is an orthonormal set of vectors in the inner product space V , and let v = λ1 v1 + λ2 v2 + · · · + λn vn . Show that λi = hv, vi i for each i = 1, . . . , n.           5 1 1 1 1 −2 1 1  1  1 −1 1 −1 1          (b) Write   4  as a linear combination of 2 1, 2 −1 2  1  2 −1. −1 1 −1 −1 1 2 8. Suppose that U is a subspace of V . We define its orthogonal complement by U ⊥ = {v ∈ V : hx, vi = 0 for all x ∈ U }. (a) Prove that U ⊥ is a subspace of V . (b) Prove that U ⊕ U ⊥ is a direct sum. 9. A matrix A ∈ Mat(n × n, C) is called unitary if A¯T = A−1 . Prove that A is unitary if and only if its columns form an orthonormal basis of Cn (relative to the standard complex dot product). 10. Let A be an n × m matrix and B an m × p matrix (over the field F). Show that that the rank of BA is less than or equal to the minimum of the ranks of A and B. 11. Suppose that A, B ∈ Mat(n × n, F) and AB = BA. (a) Show that for each λ ∈ F the λ-eigenspace of A is B-invariant. *(b) Suppose A and B are symmetric matrices. Show that there is a basis of eigenvectors so that A and B are diagonal with respect to that basis. 3 The University of Sydney School of Mathematics and Statistics Tutorial 7 (Week 8) MATH2961: Linear Algebra and Vector Calculus (Advanced) Semester 1, 2014 Web Page: http://www.maths.usyd.edu.au/u/UG/IM/MATH2961/ Lecturers: James Parkinson and Ruibin Zhang Questions marked with * are more difficult questions. Material covered (1) General Fields and examples (2) Genearl vector space axioms (3) Proofs using the field and vector space axioms (4) Revision of matrices Outcomes After completing this tutorial you should (1) be familiar with the field axioms and vector space axioms (2) be able to prove simple properties of fields and vector spaces from the given axioms (3) be able to verify the field axioms and vector space axioms for specific examples. Questions to complete during the tutorial 1. Let F be a field. (a) According to one of the axioms, there is an element 0 ∈ F such that a + 0 = a for all a ∈ F. Assume that z ∈ F is such that a + z = a for all a ∈ F. Show that z = 0. (In mathematical parlance, F has a unique zero element.) (b) Assume that z ∈ F are such that a + z = a for some a ∈ F. Again show that z = 0. (c) Show that a0 = 0 for all a ∈ F. (d) If x, y ∈ F and xy = 0, prove that x = 0 or y = 0. 2. Let p ≥ 1 be an integer and Zp := {0, 1, 2, . . . , p − 1}. Define addition and multiplication in Zp in the following way: • perform addition/multiplication as usual; • then take the remainder after dividing by p. (a) Show that Zp is a field if p is prime. (b) Which field axiom fails if p is not prime? c 2014 The University of Sydney Copyright 1 coefficients in Z5 :    1 1 4 3 2 B = 3 1 1 3 4 2 0 3. Consider the following matrices with  2 3 A = 0 4 4 1 Matrix multiplication is defined in a similar way as in first year linear algebra except that we use multiplication in Z5 rather than multiplication in R. (a) Compute AB (b) Determine A−1 if it exists. (c) Reduce A to row-echelon form. 4. Let X be a set and F a field. (a) Verify that FX = { functions : X → F} is a vector space with respect to pointwise addition and scalar multiplication. (b) For which X might we identify RX with the following vector spaces: (i) R2 ? (ii) R3 ? (iii) R? (iv) {0}? (v) the space of polynomials with real coefficients of degrees ≤ n? (vi) the space of formal power series with real coefficients? 5. Which of the following are subspaces of R2 ? (a) {(x, y) | y = 3x} (b) (c) { (x, y) | y = 3x + 1 } { (x, y) | y ≤ 3x } Extra questions for further practice 6. The element 1 ∈ F satisfying a1 = a for all a ∈ F is called the identity element. Show that the identity element is unique. 7. Let F be a field and let a0 , a1 , . . . , an ∈ F not all equal to zero. Using induction on n, show that the equation a0 + a1 x + a2 x 2 + · · · + an x n = 0 has at most n solutions in F. 8. Consider the set of matrices F :=    α β : α, β ∈ R . −β α (a) Show that AB = BA for all A, B ∈ F. 2 (b) Show that every A ∈ F \ {0} is invertible and compute A−1 . (c) Show that F is a field. (d) Show that F can be identified with C. (e) What form of matrix in F corresponds to the modulus-argument form of a complex number. Comment on the geometric significance. 9. Which of the following are subspaces of the space V of n × n matrices with entries from a field F? (a) {A ∈ V | A is symmetric} (b) {A ∈ V | AB = BA}, for any fixed B ∈ V (c) { A ∈ V | det A = 0 }. Challenge questions (optional) 10. Let Z2 = {0, 1} be the two-element  0  T := 1 0 field and let  0 1 0 1 ∈ Mat(3 × 3, Z2 ). 1 0 Let E = {aT 2 + bT + cI : a, b, c ∈ Z2 } where I is the 3 × 3 identity matrix over Z2 . Addition and multiplication of matrices is defined in the usual way but with addition and multiplication in Z2 . (a) How many elements does the set E have? (b) Show that T 3 = T + I. *(c) Prove that E is a field. 3 Axioms of a field A field F is a set with two operations, namely (x, y) 7→ x + y (addition) (x, y ) 7→ x · y (multiplication) It has two special elements, 0 and 1 such that for all x, y , z ∈ F (F1) (F2) (F3) (F4) (F5) (F6) (F7) (F8) (F9) x + y = y + x (commutative law of addition) (x + y) + z = x + (y + z) (associative law of addition) x + 0 = x (neutral element for addition) for every x ∈ F there exists y ∈ F with x + y = 0 (existence of additive inverse) x · y = y · x (commutative law of multiplication) (x · y ) · z = x · (y · z) (associative law of multiplication) 1 · x = x (neutral element for multiplication) for every x ∈ F \ {0} there exists y ∈ F with x · y = 1 (existence of multiplicative inverse) x · (y + z) = x · y + x · z (distributive law) Axioms of a vector space V over a field F A vector space over the field F is a set V with two operations: (u, v) 7→ u + v (λ, v) 7→ λu for u, v ∈ V (addition) for λ ∈ F and u ∈ V (multiplication by scalars) There is one special element 0 ∈ V , called the zero vector such that for all u, v , w ∈ V and λ, µ ∈ F (V1) (u + v) + w = u + (v + w ) (Associative law) (V2) u + v = v + u (Commutative law) (V3) u + 0 = u (neutral element for addition) (V4) For every u ∈ V there exists v ∈ V such that u + v = 0 (existence of additive inverse) (V5) 1v = v (multiplication by scalar 1) (V6) λ(µu) = (λµ)u (associative law) (V7) (µ + λ)u = µu + λu (Distributive law) (V8) λ(u + v ) = λu + λv (Distributive law) 4 The University of Sydney School of Mathematics and Statistics Tutorial 8 (Week 9) MATH2961: Linear Algebra and Vector Calculus (Advanced) Semester 1, 2014 Web Page: http://www.maths.usyd.edu.au/u/UG/IM/MATH2961/ Lecturers: James Parkinson and Ruibin Zhang Questions marked with * are more difficult questions. Material covered (1) Spans and linear combinations (2) Bases (3) Revision of row reductions from first year. Outcomes After completing this tutorial you should (1) be able to determine the spans and subspaces (2) be able to write proofs (3) be able to use concepts from first year linear algebra in the context of abstract linear algebra. (4) be able to prove linear independence in various contexts. Questions to complete during the tutorial 1. Consider the following family of vectors in Z33 .     1 0 2 , 1 , 1 2   1 1 2 Decide whether or not the family is linearly independent in Z33 . 2. Recall that if M is a subset of a vector space V then span(M ), the subspace of V spanned by M , is the set of all linear combinations of elements of M . (a) Let V = R3 and let W be the subspace spanned by the family     1 −2 M = 4 , −7 . 2 2 Find the Cartesian equation of span(M ), and deduce that it is a plane through the origin. (b) Let now V = R4 and consider the family   1  4    M =  2  , −1   −2 −7   .  2  4 Find a system of two linear equations in four unknowns (the coordinates) having span(M ) as its solution set. c 2014 The University of Sydney Copyright 1 3. Let c be a constant. Show that cos(t + c) ∈ span(cos t, sin t). 4. Suppose that (v1 , v2 , v3 ) is linearly independent in the vector space V over the field F, and that w = 2v1 + 5v2 − 2v3 . Prove that (w, v2 , v3 ) is linearly independent. 5. In first year linear algebra you learnt how to find the reduced row-echelon form of a matrix with coefficients in R. The same method of Gaussian elimination applies to matrices with coefficients in an arbitrary field. Consider the following matrix with entries in Z5 :   2 3 2 4 3 3 2 3 0 2  A= 4 4 3 1 3 2 3 2 4 3 Find the reduced row-echelon form of A as a matrix with coefficients in Z5 . 6. Consider the family of functions uk (t) = cos kt, k ∈ N, t ∈ R. Prove that the family of functions on R is linearly independent. Hint: Multiply the condition to test for linear independence by cos mt over the interval (−π, π) and use the identity 2 cos a cos b = cos(a + b) + cos(a − b) (check!). Extra questions for further practice 7. Consider a system of three linear equations a11 x1 + a12 x2 + a13 x3 = b1 , a21 x1 + a22 x2 + a23 x3 = b2 , a31 x1 + a32 x2 + a33 x3 = b3 . with coefficients in R. (a) Geometrically explain the two alternatives: • For every b = (b1 , b2 , b3 ) the system has precisely one solution. • Given b = (b1 , b2 , b3 ) the system has either no solution or infinitely many solutions. Hint: Note that each equation describes a plane in R3 . (b) Prove that the system has a unique solution for every b = (b1 , b2 , b3 ) if and only if the vectors ai = (a1i , a2i , a3i ), i = 1, 2, 3, are linearly independent. (c) Consider the linear system x1 + 2x2 + 5x3 = 4, 2x2 − x2 + 4x3 = 2 −4x1 + 7x2 − 2x3 = 2 Find its solutions. Find right hand sides, so that it does not have any solutions. 8. (a) Let A be an n × n matrix over a field F and let λ be an arbitrary element of F . As learnt in first year linear algebra the λ-eigenspace of A is defined to be the set of all v ∈ Fn such that Av = λv. Prove that the λ-eigenspace is a subspace of Fn .   2 1 1 (b) Find a basis for the 1-eigenspace of 1 2 1 (where the field is R). 1 1 2 9. Find a basis for the space of matrices Mat(m × n, F) over a field F. 2 10. Let V be a vector space over a field F. (a) If (Vi )i∈I is a family of subspaces of V , show that ∩i∈I Vi is a subspace as well S (b) Let V1 , . . . , Vn be subspaces of V . Show that span( ni=1 Vi ) = V1 + V2 + · · · + Vn . 11. Determine whether or not the following two subspaces of R3 are the same:         1 2 1 2 span  2  , 4 and span 2 ,  4  . −1 1 4 −5 3 The University of Sydney School of Mathematics and Statistics Tutorial 9 (Week 10) MATH2961: Linear Algebra and Vector Calculus (Advanced) Semester 1, 2014 Web Page: http://www.maths.usyd.edu.au/u/UG/IM/MATH2961/ Lecturers: James Parkinson and Ruibin Zhang Questions marked with * are more difficult questions. Material covered (1) Dimension of a vector space (2) Basic facts on linear operators (3) Image and kernel of a linear operator (4) Isomorphisms Outcomes After completing this tutorial you should (1) understand basic properties of linear operators in a variety of contexts; (2) be able to determine kernel and image for simple linear operators; (3) be able to check that a linear operator is an isomorphism; (4) understand what the dimension of a vector space is. Questions to complete during the tutorial 1. Let V, W be vector spaces. The kernel (or null space) of a linear operator T : V → W is the set ker T = {v ∈ V : T v = 0} ⊆ V . Its image is im T := {T v : v ∈ V } ⊆ W . Find the kernel and image of the linear map from R3 to R3 given by the matrix   2 3 −3 A := 1 −1 4  3 2 1 2. Let R2 [X] be the vector space of all polynomials of degree at most two with coefficients in R. (a) Write down the coordinate vector of p(X) = 2 + X − 5X 2 with respect to the basis (1, X, X 2 ) of R2 [X]. (b) Show that (1, 2 − X, X − X 2 ) is a basis for R2 [X]. (c) Find the coordinate vector of X 2 with respect to the basis (1, 2 − X, X − X 2 ). 3. We can consider R to be a vector space over Q. What is the dimension of this vector space? Hint: Use that R is not countable, but Q is countable (see the end of this sheet). 4. Suppose that T : V → W is a linear operator between the vector spaces V and W . Further assume that dim W < ∞ and that T is surjective. (a) Prove that there exists a linear operator S : W → V such that T S = idW is the identity map on W . c 2014 The University of Sydney Copyright 1 (b) Does (a) imply that T is invertible. Give a proof or a counter example. 5. Consider the linear system of differential equations x˙ 1 (t) = a11 x1 (t) + a12 x2 (t) + · · · + a1n xn (t) x˙ 2 (t) = a21 x1 (t) + a22 x2 (t) + · · · + a2n xn (t) .. . x˙ n (t) = an1 x1 (t) + an2 x2 (t) + · · · + ann xn (t) with continuous coefficients aij : R → R. The system can be written in vector form as ˙ x(t) = A(t)x(t) with x(t) = (x1 (t), . . . , xn (t)) and A(t) = [aij ]1≤i,j≤n . (a) Show that the set of solutions of the above system of differential equations forms a vector space. (b) In the theory of differenital equations it is shown that the above linear system has a unique solution for every initial vector x(0) = x0 ∈ Rn . Use this fact to show that the space of solutions is isomorphic to Rn and therefore has dimension n. (c) Consider the n-th order linear differential equation y (n) (t) + an−1 (t)y (n−1) (t) + · · · + a0 (t)y(t) = 0. Prove that the space of solutions is n-dimensional. Hint: Turn the equation into a first order system of the form above by setting x1 (t) = y(t), x2 (t) = x˙ 1 (t), . . . , xn (t) := x˙ n−1 (t) and identify the matrix associated with the system. Extra questions for further practice 6. Let V be a vector space and U , W subspaces of V . We define U + W := {u + w : u ∈ U, w ∈ W }, U × W := {(u, w) : u ∈ U, w ∈ W }. We know from lectures that U + W is a subspace of V . Clearly U × W is a vector space if we define addition and multiplication by scalars componentwise, that is, (u1 , w1 ) + (u2 , w2 ) := (u1 + u2 , w1 + w2 ) and λ(u, w) := (λu, λw). (The space U × W is often called the exterior direct sum.) We define a map U × W → V by T (u, w) := u + w. (a) Show that T is linear. (b) Show that ker T = {(u, −u) : u ∈ U ∩ V }. (c) We know from lectures that U ∩ W is a subspace of V . Show that ker T is isomorphic to U ∩V. (d) Suppose that U and W are finite dimensional. (i) Show that dim(U × W ) = dim U + dim W . (ii) Use the dimension formula for linear operators to show that dim(U + W ) = dim U + dim W − dim(U ∩ W ). Countable and uncountable sets (optional) This section is optional and for your interest only. The material will not be tested. 2 By definition, a set X is finite if there exists n ∈ N and a bijection ϕ : {1, . . . , n} → X. “Counting” the elements of a set is equivalent to giving such a bijection. The set X is countable if there exists a bijection ϕ : N → X. One can show that this is equivalent to the existence of a surjective map ψ : N → X. There are a number of examples: • By definition, N is countable. The bijective map ϕ we can take is ϕ(n) = n. • The set Z of all integers is countable. We enumerate Z by starting at zero, going to 1, then −1, then 2, then −2, and so on, as shown below: −5 −4 −3 −2 −1 0 1 2 3 4 5 • The set of rational numbers Q, that is, the set of fractions m/n with m ∈ Z and n ∈ N \ {0} is also countable. This is not quite as obvious as for the integers. To show the countability of Q we use that every ordered pair (m, n) represents a fraction m/n. This representation is not unique as for instance (1, 2) and (2, 4) represent the same fraction 1/2 = 2/4. However, if we show that the number of pairs (m, n) is countable, then we have also shown that also Q is countable. To enumerate these pairs of numbers we place them on an array and enumerate the pairs on the array by following the arrows as shown below: (−4, 1) (−3, 1) (−2, 1) (−1, 1) (0, 1) (1, 1) (2, 1) (3, 1) (4, 1) (−4, 2) (−3, 2) (−2, 2) (−1, 2) (0, 2) (1, 2) (2, 2) (3, 2) (4, 2) (−4, 3) (−3, 3) (−2, 3) (−1, 3) (0, 3) (1, 3) (2, 3) (3, 3) (4, 3) (−4, 4) (−3, 4) (−2, 4) (−1, 4) (0, 4) (1, 4) (2, 4) (3, 4) (4, 4) (−4, 5) (−3, 5) (−2, 5) (−1, 5) (0, 5) (1, 5) (2, 5) (3, 5) (4, 5) • The argument above also shows that the union of any countable number of countable sets is countable. We enumerate each set and arrange its elements in a row of an array. Then we enumerate the elements of the array in a similar fashion as above. • In contrast to the rational numbers, the irrational numbers are not countable. We can see this using an argument due to Georg Cantor (1845–1918). It is sufficient to show that the numbers, including the rational numbers, in the interval (0, 1] are uncountable. As the rational numbers are countable as proved already, the irrational numbers are uncountable. To prove that the numbers in (0, 1] are uncountable we use the fact that every number in (0, 1] has a unique representation as an infinite decimal expansion. By infinite we mean that infinitely many digits are non-zero. For the unique representation it is important to assume the decimal expansion is infinite. For instance 1.0000000000 · · · = 0.99999999999 · · · 3 are two different decimal representation of 1, but the infinite one is unique. Similarly, 0.25000000000 · · · = 0.249999999999 · · · are two representations of 1/4. The infinite expansion is unique. If we assume that the set of numbers in the interval (0, 1] are countable, then we can enumerate them as a1 , a2 , . . . . Let akj , j ∈ N by the digits in the decimal expansion of ak . We can write these digits in as an infinite array as shown below. a1,1 a1,2 a1,3 a1,4 a1,5 a1,6 a1,7 a1,8 · · · a2,1 a2,2 a2,3 a2,4 a2,5 a2,6 a2,7 a2,8 · · · a3,1 a3,2 a3,3 a3,4 a3,5 a3,6 a3,7 a3,8 · · · a4,1 a4,2 a4,3 a4,4 a4,5 a4,6 a4,7 a4,8 · · · a5,1 a5,2 a5,3 a5,4 a5,5 a5,6 a5,7 a5,8 · · · a6,1 a6,2 a6,3 a6,4 a6,5 a6,6 a6,7 a6,8 · · · a7,1 a7,2 a7,3 a7,4 a7,5 a7,6 a7,7 a7,8 · · · .. . .. . .. . From our assumption every possible sequence of digits appears precisely once. We then form a sequence by looking at the “diagonal sequence” a1,1 , a2,2 , a3.3 , . . . . For each k ∈ N we choose bk ∈ {1, . . . , 9} such that bk 6= akk . We claim that the sequence b1 , b2 , . . . does not appear as a row in the above array. Indeed, the sequence does not appear as a row by construction, because by choosing the k-th row we find that bk 6= ak,k . As bk 6= 0 for all k ∈ N we can form the infinite decimal expansion x = 0.b1 b2 b3 b4 b5 b6 · · · ∈ (0, 1], which is not one of the ak ’s. This is a contradiction because we listed all possible decimal expansions. Hence (0, 1] consists of uncountably many numbers as claimed. 4 The University of Sydney School of Mathematics and Statistics Tutorial 10 (Week 11) MATH2961: Linear Algebra and Vector Calculus (Advanced) Semester 1, 2014 Web Page: http://www.maths.usyd.edu.au/u/UG/IM/MATH2961/ Lecturers: James Parkinson and Ruibin Zhang Questions marked with * are more difficult questions. Material covered (1) The matrix of a linear transformation (2) Image and kernel of a linear operator (3) The rank-nullity theorem dim(im(T )) + dim(ker(T )) = dim(V ). Outcomes After completing this tutorial you should (1) to be able to find the matrix representing a given linear operator in a variety of contexts (2) work with images and kernels with and without the rank-nullity theorem Questions to complete during the tutorial 1. In the plane a linear transformation T : R2 → R2 is given by reflecting vectors at the line x = y. Find the matrix representation of T with respect to the standard basis. 2. Let V = span(cos t, sin t) be a subspace of all functions on R. Define the linear operator by Du := u0 (the derivative of u) for all u ∈ V . (a) Show that D : V → V , that is, im D ⊆ V . (b) Find the matrix representation of that linear operator with respect to the basis (cos t, sin t). 3. Denote by Fn [X] the space of polynomials of degree at most n. Define a linear operator T : Fn [X] → Fn+1 [X] by setting T (a0 + a1 X + · · · + an X n ) := a0 X + an a1 2 a2 3 X + X + ··· + X n+1 2 3 n+1 (integration of polynomials). Find the matrix representation of T with respect to the basis (1, . . . , X n ) in Fn [X] and (1, . . . , X n+1 ) in Fn+1 [X]. 4. Consider the matrix H with entries in Z2  1 H := 0 0 given by  0 0 1 0 1 1 1 0 1 1 0 1 . 0 1 1 1 1 0 The columns of H are all non-zero vectors in Z32 . c 2014 The University of Sydney Copyright 1 (a) Explain why dim(ker(H)) = 4. Find a basis of ker(H) in the form         ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗         ∗ ∗ ∗ ∗                v1 =  1 , v 2 = 0 , v 3 = 0 , v 4 = 0 0 0 1 0         0 1 0 0 1 0 0 0   (b) Form the matrix M := v 1 v 2 v 3 v 4 with columns v i , i = 1, . . . , 4, from part (a). M H We know that Z42 −→ Z72 −→ Z32 . Show that ker(H) = im(M ).   1 0   1    (c) Why is the vector v =  0 not in the image of M ? 1   0 0 (d) Information encoded in binary format can be represented by a vector with components in Z2 . When transmitted (over the Internet for instance) it is sent in blocks, usually of length 8 (one byte). For simplicity we assume that the blocks have length four. Such a block is represented by a vector u ∈ Z42 . Rather than u, the vector M u is transmitted. Note that the last four components of M u are the components of u. Assume that the vector v from part (c) is received, and that during transmission one error occured. Explain why v = M u + ei , where ei ∈ Z72 is one of the standard basis vectors and determine the vector u originally transmitted. 5. Consider the plane U given by 2x − 3y + 5z = 0 is a two dimensional subspace of R3 . Find two different complementing subspaces. Extra questions for further practice 6. Suppose that V, W are vector spaces over F and T : V → W is linear. Let U be a complement of ker T in V . Prove that U is isomorphic to im T . 7. Consider the plane x + 3y + 2z = 0 in R3 . The linear operator T : R3 → R3 is the reflection at the given plane. (a) Find the matrix representation of T with respect to the new basis       2 3 1  0  , −1 , 3 −1 0 2 (b) Find the matrix representation with respect to the original basis (the standard basis). 8. The linear operator T : Mat(2 × 2; C) is given by   a b T := a + d c d (the trace of the matrix). (a) Show that T is linear. 2 (b) Determine the dimension of ker T and give a basis of ker T . (c) Find the basis for a complement to ker T . (d) Putting together the basis for ker T and the complement found in the previous parts we get a basis for Mat(2 × 2; C). find the matrix representation of T with respect to that basis. 9. Suppose that S, T : V → V is a linear operator such that ST = T S. Show that im T and ker T are S-invariant, that is, S : im T → im T and S : ker T → ker T . 10. Let V be a vector space over a field F. (a) Suppose that P ∈ L(V, V ) is such that P 2 := P ◦ P = P . Show that V = im P + ker P and im P ∩ ker P = {0}. (We say this is a direct sum and write V = im P ⊕ ker P ) (b) Suppose that dim V < ∞ and that U is a subspace of V . Show that there exists a linear operator P ∈ L(V, V ) with P 2 := P ◦ P = P such that U = im P . 3

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1 Calculus on the wall mastermathmentor.com presents Stu Schwartz
Helping students learn … and teachers teach21. Infinite SeriesCreated by:Stu SchwartzUn-narrated VersionGraphics:Apple Grapher: Version 2.3Math Type: Version 6.7Intaglio: 2.9.5aFathom: Version 2.11

2 Calculating PiWe know that the circumference C of a circle is calculated by multiplying π times the diameter. So π is the ratio of the circumference to the diameter. But how do we calculate the value of π without this geometric approach?Circumferencediameter

3 Vocabulary Terms used in this slide show: • Sequence • p-series test
• Recursive formula • Harmonic series• Explicit formula • Alternating series test• Convergent sequence • Telescoping series• Divergent sequence • Integral test• Series • Direct comparison test• Infinite series • Limit comparison test• Convergent series • Ratio test• Divergent series • Absolutely convergent• Convergence test • Conditionally convergent• nth term test • Root test• Geometric test

4 SequencesWe define a sequence as a set of numbers that has an identified first member, 2nd member, 3rd member, and so on. We use subscript notation to denote sequences: a1, a2, a3, etc. Sequences are rarely made up of random numbers; they usually have a pattern to them.What is the 4th and 5th members of the sequence: 1, 2, 4, … ?If you said that the 4th member is 8 and the 5th member is 16, the pattern is: to get the next term, we double the previous term.If you said that the 4th member is 7 and the 5th member is 11, the pattern is: we add 1 to the first member, add 2 to the 2nd member and thus add 3 to the 3rd member.The problem with giving the terms of a sequence is that there is an assumption that the student can see the pattern of the terms. Sometimes the pattern isn’t obvious, even if many terms are given. For instance, finding the next term in the sequence 1, 2, 3/2, 2/3, 5/24 … is quite difficult, and yet when you see the answer in the next slide, it makes perfect sense.

5 Defining SequencesRather than give the terms of a sequence, we give a formula for the nth term an. There are two ways to do so: recursively and explictly.Recursive: we give the first term a1 and the formula for the nth term an in terms of a function of the (n - 1)st term, an-1 .Explicit: we give a formula for the nth term an in terms of n.Recursive: a1 = 1, an = 2an-1The 5th term would be 16.Advantage: pattern easy to see.Disadvantage: to find the 10th term, we need the 9th which needs the 8th, …For the sequence: 1, 2, 4, 8, …Explicit: an = (1/6)n3 - (1/2)n2 + (4/3)The 5th term would be 15.Advantage: easy to find the 10th term.Disadvantage: formula is not obvious.In this course, students rarely will be asked to generate a formula for the nth term from the terms themselves. However, if they are given the nth term formula, which is usually given explicitly, they should be able to find any term. For instance, the sequence 1, 2, 3/2, 2/3, 5/24, … is defined as an = n2/n!. The 6th term is 36/720 = 1/20.

6 Convergent and Divergent Sequences
Sequences either converge (are convergent) or diverge (are divergent).Convergent sequences: the limit of the nth term as n approaches ∞ existsDivergent sequences: the limit of the nth term as n approaches ∞ does not exist

7 Using L’Hospital’s Rule
Whenever you are asked about the convergence or divergence of a sequence when given an, it is best to write out a few terms of the sequence to get a sense of it. Even then, you can get fooled. To be sure, you can use L’Hospital’s rule to find the limit of an as n approaches infinity, and if L’Hospital’s rule cannot be used, use simple logic.

8 SeriesWe define a series as the sum of the members of a sequence starting at the first term and ending at the nth term:We define an infinite series as the sum of the members of a sequence starting at the first term and never ending.The remainder of this slide show is centered whether an infinite series converges (has a limit). A sequence converges if its nth term an has a limit as n approaches infinity. A series converges if the sum of its terms has a limit as n approaches infinity. Determine whether each of the following series converges.

9 Convergence TestsA convergence test is a procedure to determine whether an infinite series given the formula for an is convergent. Some convergence tests can be done by inspection while others involve a bit of work that will need to be shown.The nth term testIf the nth term does not converge to zero, the series must diverge.If the nth term does converge to zero, the series can converge.the sequence termsthe partial series

10 Geometric Series & Convergence
A series in the form of is a geometric series.The Geometric testIf |r| ≥ 1, the series diverges.If |r| < 1, the series converges to a/(1 – r)The figure is a square of side 8 and the midpoints of the square are vertices of an inscribed square with the pattern continuing forever. Show that the sum of the areas and perimeters are convergent.

11 The p-Series and Harmonic Series
A series in the form of where p > 0, is a p-series.The p-series test:If 0 < p ≤ 1, the series diverges.If p > 1, the series converges.p is a positive constant:the sequence termsthe partial seriesAny series in the form of∑ [(c/an + b)] is called an harmonic series and is divergent.

12 More on the Harmonic Series 1/n
It seems counter-intuitive that is divergent. Here’s a simple proof:It takes 31 books for the overhang to be 2 books long, 227 books for the overhang to be to be 3 books long, 1,674 books for the tower to be 4 books long, and over 272 million books for the overhang to be 10 books long. Try it with a deck of cards.Deck of52 cardsThe length of the overhang is where n is the number of books. This is the harmonic series and thus theoretically, it will balance with an infinite number of books.⅙…Suppose we stack identical books of length 1 so that the top book overhangs the book below it by ½, which overhangs the book below it by ¼, which overhangs the book below it by ⅙, etc. This structure, called the Leaning Tower of Lire will (just barely) balance.⅛1/10

13 Alternating SeriesAn alternating series is one whose terms alternate in signs.The Alternating series test:Error in an alternating seriesIn a convergent alternating series, the error in approximating the value of the series using N terms is the (N + 1)st term .

14 Telescoping SeriesHowever, it is possible that we subtract one divergent series from another, our answer converges.A telescoping series is an alternating series in the form of If this passes the nth term test, the series converges. Expand the expression.If we take a divergent series and subtract a divergent series, we can get another divergent series.

15 The Integral TestIf the integral test shows convergence of a series, the value of the integral is not the value of the series. It is merely an indicator that the series converges.

16 Comparison TestsIf there is a deviation from the forms already studied, the tests cannot be used.The Direct Comparison test:

17 The Limit Comparison Test

18 The Ratio TestThe Ratio test:

19 Failure of the Ratio Test
You may wonder: if the ratio test is so versatile, why do we need all of the other tests? Why not just apply the ratio test immediately?The ratio test will always be inconclusive with a series in the form of a polynomial over a polynomial or polynomials under radicals.∑an absolutely convergent:∑an converges and ∑|an | converges∑an conditionally convergent:∑an converges and ∑|an | diverges

20 The Root Test The Root test:
The following numbers are written from smallest to largest:n20, 20n, n!, nn.While an expression of a smaller expression over a larger expression could converge, an expression of a larger expression over a smaller expression diverges.The Root test:

21 Series Convergence/Divergence Flowchart
nth term testNoYesGeometric testNoNop-series testYesNoAlternating series testYesIs the series telescoping?

22 More Tests in Order of Usefulness
Ratio testYesNoLimit Comparison testYesNoRoot testYesNoIntegral testYesNo

23 Vocabulary Do you understand each term? • Sequence • p-series test
• Recursive formula • Harmonic series• Explicit formula • Alternating series test• Convergent sequence • Telescoping series• Divergent sequence • Integral test• Series • Direct comparison test• Infinite series • Limit comparison test• Convergent series • Ratio test• Divergent series • Absolutely convergent• Convergence test • Conditionally convergent• nth term test • Root test• Geometric test

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