CALCULUS REVIS ITED PART 2 A Self-study Course

Size: px

Start display at page:

Download "CALCULUS REVIS ITED PART 2 A Self-study Course"

Roland Page
5 years ago
Views:

1 CALCULUS REVIS ITED PART 2 A Self-study Course STUDY GUIDE Block 3 Partial Derivatives Herbert I. Gross Senior Lecturer Center for Advanced Engineering Study Massachusetts Institute of Technology

2 by Massachusetts Institute of Technology Cambridge, Massachusetts All rights reserved. No part of this book may be reproduced in any form or by any means without permission in writing from the Center for Advanced Engineering Study, M.I.T.

3 CONTENTS Studv Guide Pretest Unit 2: An Introduction to Partial Derivatives Unit 3: Differentiability and the Gradient Unit 4: The Directional Derivative in n-dimensional Vector Spaces (Optional) Unit 5: The Chain Rule, Part 1 Unit 6: The Chain Rule, Part 2 Unit 7: More on Derivatives of Integrals Unit 8: The Total Differential Quiz Solutions Pretest Unit 2: An Introduction to Partial Derivatives Unit 3: Differentiability and the Gradient Unit 4: The Directional Derivative in n-dimensional Vector Spaces (Optional) Unit 5: The Chain Rule, Part 1 Unit 6: The Chain Rule, Part 2 Unit 7: More on Derivatives of Integrals Unit 8: The Total Differential Quiz iii

4 I BLOCK 3: PARTIAL DERIVATIVES

5 Pretest 1. Let w = f (x,y) = 2Xy 2, (x,y) # (0.0). Show that x +Y lim f(x,y) depends on the path by which (x,y) approaches (x,y)+(o,o) 2. Find the equation of the plane which is tangent to the surface x4 + y 6 z + xyz5 = 3 at ~ ~, ~, ~ ~. I. Suppose w depends on r but not on 8, say w = h(r), and that h is a twice-differentiable function of r. Determine -a'w + -a ;, expressed ax a~ in terms of r. rn,. L 4. Find the equation of the curve C if C passes through the origin and has its slope at each point (x,y) given by Given that g(y) = i1 b dx where y > b > -1, determine gl(y).

6 Unit '1: Functions of More Than One Variable 1. Lecture 3.010

7 2. Read Supplementary Notes, Chapter Read Thomas, Section (Optional) Read Thomas, Sections and (These sections will help you feel more at home with equations of surfaces. The idea is that just as the graphs of functions of a single variable are curves in the xy-plane, the graphs of functions of two real variables are surfaces in space. Except for any peace-of-mind that you get in feeling at home with the various equations, it should be noted that we can survive the remainder of this course without recourse to accurate graphs just as was the case in functions of a single real variable.) 5. Exercises : Define as the Minkowski metric. That is, if -x = (xl,...,xn), then IlxIl - = max~lxll...i xnil. a. Show that 1. 1k1) >/ 0 for all x 2- IIx + yll 4 lkll + Ikll 3. IlaxII = lal IkII - and llxll - = 0 if and only if - x = - 0. b. Compute -x *~,llxllr and llrll (where we are still using the Minkowski metric) if 5 = (2,4,1) and 4~ = (4,4,5). From this conclude that it need not be true that ~ -X - ~ 6I l l ~ l l Ilu_ll. Mimic the proof of the corresponding 1-dimensional case to prove lim 1i m that if - x and -a belong to E" and x-rg f (x) = L1 while x,a g (5) = L2, then lim [f (x) + g(x)i - x-rg = L1 + L2

8 a. Using the Minkowski metric, suppose E>O is given; find 6 such that for this choice of 6 b. Interpret the answer in (a) geometrically and explain why the same value of 6 as in (a) would have sufficed had we used the Euclidean metric rather than the Minkowski metric (L) Let -x = (xl,x2,x3,x4) and let -1 = (1,1,1,1). Define f by 3 2 f(x) = x12 + 2x2 + x3 + x4. Prove that f is continuous at = Let f, -x and such that -1 be as in Exercise For a given E>O, find (L) Let f be defined by a. b. c. Is f continuous at (0,0)? lim lim Compute both lim limf(x,y)l. y+q [x+o f (x,y)l and x+o ry+(, Investigate the behaviour of in more detail by introducing polar coordinates.

9 Let f be defined by a. Show that lim f(x,y) depends on the path by which (xiy) (x,y)+(o,o) approaches (0,O. 1i m b. Compute (x,y)+(o,o) f (x,y) if (x,y) approaches (0,O) along the 71 ray t3 = - 4. c. Show that if (x,y) approaches (0,O) either along the x-axis or the y-axis then lim f(x,y) = 0. Define g by r l. a. Show that g is not continuous at (0,O). b. Show that lim g(x,y) = g(o,o) if (x,y) is allowed to (x,y)+(o,o) approach (0,O) along either axis. Let the function :E~-+E be continuous. Prove that f cannot be 1-1. Comment The following two exercises are optional. They may be omitted without loss of continuity to our present discussion. Their main purpose is to supply the interested reader with a few clues as to how analytic proofs are carried out in n-dimensional vector spaces (with n greater than three) using the ordinary properties of real number arithmetic.

10 lock 3: Partial Derivatives Let -a and b belong to E 4. Prove that our definition of -a = b_ is an equivalence relation because a the fact that "ordinary" equality is an equivalence relation on the set of real numbers. Let a, 2 and -c be elements of E 4. With the dot product as defined in our supplementary notes, prove that

11 MIT OpenCourseWare Resource: Calculus Revisited: Multivariable Calculus Prof. Herbert Gross The following may not correspond to a particular course on MIT OpenCourseWare, but has been provided by the author as an individual learning resource. For information about citing these materials or our Terms of Use, visit:

The Double Sum as an Iterated Integral

Unit 2: The Double Sum as an Iterated Integral 1. Overview In Part 1 of our course, we showed that there was a relationship between a certain infinite sum known as the definite integral and the inverse