Spring 2015 Math 233H schedule and homework from Stewart 7 (in the past, I've used Marsden & Tromba _Vector Calculus_ and may choose some material from there) due each Wednesday - the *ed problems are for you to ponder (assignments are always subject to change as necessary): *********************************************************************** *********************************************************************** ********************** The semester at a glance ********************* *********************************************************************** *********************************************************************** Week Sections Events 1/19 Intro, 12.1-12.3 Extra session Wed eve, MLKJr Mon! 1/26 12.4, 12.5, 12.6 Quiz #1, Extra session Wed eve 2/2 10.1, 13.1, 13.2 Extra session Wed eve, snow Mon! 2/9 13.3, 13.4 Quiz #2 cancelled, snow Mon!! 2/16 14.1, 14.2 Tue is Mon, Tue eve review! Exam 1: Wed 6pm 2/23 14.3, 14.4 No Mon class! 3/2 14.5, 14.6 Last day to drop with a "W": Mon 3/9 14.7, 14.8 Quiz #3: Mon in class! 3/23 15.1, 15.2 Extra session Wed eve 3/30 15.3, 10.3 Extra session Wed eve 4/6 15.4, 15.5* Exam 2: Wed 4/8 @ 6pm (LGRT 1334) 4/13 15.9* Extra session Wed eve, no Mon class! 4/20 16.1, 16.2 Wed is a UMass Mon, no Mon class! 4/27 16.3, 16.4 Extra session Wed eve! Thu is Mon! 5/4 FINAL EXAM Tue 8am (5/5) in our classroom (LGRT 202) The following sections of Stewart 7 will be (un)covered. Some* sections may not be lectured on, but you should read them on your own. This list may change as the semester evolves: 12.1 Three-dimensional coordinate systems 12.2 Vectors 12.3 The dot product 12.4 The cross product 12.5 Equations of lines and planes 12.6 Cylinders and quadric surfaces 10.1 Curves defined by parametric equations 13.1 Vector functions and space curves 13.2 Derivatives and integrals of vector functions 13.3 Arc length and curvature 13.4 Motion in space, velocity and acceleration 14.1 Functions of several variables 14.2 Limits and continuity 14.3 Partial derivatives 14.4 Tangent planes and linear approximations 14.5 The chain rule 14.6 Directional derivatives and the gradient vector 14.7 Maximum and minimum values 14.8 Lagrange multipliers 15.1 Double integrals over rectangles 15.2 Iterated integrals 15.3 Double integrals over general regions 10.3 Polar coordinates 15.4 Double integrals in polar coordinates 15.5 Applications of double integrals* 15.7 Triple Integrals 15.8 Triple Integrals in Cylindrical Coordinates 15.9 Triple Integrals in Spherical Coordinates* 16.1 Vector fields 16.2 Line integrals 16.3 The fundamental theorem for line integrals 16.4 Green's theorem *********************************************************************** *Turn****************************************************************** **in**************** HW problems and other details ******************** **on******************************************************************* *********************************************************************** 1/28 Consider a quadrilateral Q with vertices A,B,C,D. What can you say about the new quadrilateral P with vertices X=(A+B)/2, Y=(B+C)/2,Z=(C+D)/2,W=(D+A)/2 at the midpoints of Q? Read Chapter 12! *********************************************************************** 2/4 Chapter 12 Exercises: 3,4(a,b,c,d,f,h),5,6,7,8*,9*,11,12,14 For a tetrahedron (a pyramid with triangular base), show that the sum of the outer unit normal vectors weighted by the areas of the faces must vanish (also true for any polyhedron). [This turns out to be a "Discovery Project" problem in Chapter 12!] *********************************************************************** 2/11 Prove the "bac-cab" relation for cross product: a x (b x c) = b(a.c) - c(a.b) [so called because the parentheses go "in the back" - this is a more memorable way to state 12.4.11 Theorem 6]. Chapter 12 Exercises: 15,17,19,24,29,31,35,38* Chapter 12 Problems Plus: 1*,8* Explore the quadric surfaces in section 12.6 and try to figure out how some surfaces are really limiting cases of others: e.g. planes are limits of spheres (and many other surfaces), cylinders are limits of ellipsoids and hyperboloids, etc. Read Chapter 13! *********************************************************************** 2/18 For the helix X(t)= compute the velocity X'(t) and acceleration X"(t). Compute their dot product X'(t).X"(t) - could you have predicted this from the fact that the speed |X'(t)| is constant? [Hint: consider |X'(t)|^2=X'(t).X'(t) and use the product (Leibniz) rule. Cf. Chapter 13 Exercises 1,9] Chapter 13 Exercises: 2,5,6,9,11,18,21*,23*,24* Chapter 13 Problems Plus: 3[assume gravitational acceleration is <0,0,-g>, g=32feet/(sec)^2],6*[experiment with rope],7*[Math235] ****** Review @ 5:30PM on Tuesday 2/17 in 1334 LGRT ****** !!!!!!!!!!! Midterm #1 @ 6PM in 1322/1334 LGRT !!!!!!!!!!! *********************************************************************** 2/25 Recall the special case of "bac-cab" when a = c is a unit vector: the parentheses can go in either place, and the vector a x b x a is projection of b to the plane perpendicular to a (we checked this algebraically for a = i, but the general case follows from the right- hand rule). Use this to show that the curvature vector of any (not necessarily unit speed) parametrized curve X(t) is K(t) = X'(t) x X"(t) x X'(t)/|X'(t)|^4 and thus its length (the curvature function) is k(t) = |K(t)| = |X'(t) x X"(t)|/|X'(t)|^3. [Hint: expand acceleration into tangent and normal parts.] [Remark: from the special baa-aab case of bac-cab when a.a=1, we get a x b x a = b - a(a.b), which subtracts from b the component of b along a, i.e. it projects b to the plane perpendicular to a.] [Note: this problem can be turned in next Monday in class!] Read Chapter 14! *********************************************************************** 3/4 Chapter 14 Exercises: 2,4,5,7,8,10 Chapter 14 Problems Plus: 4 *We observed in class that the function f(x,y) = xy has this property: through each point on its graph pass 2 distinct straight lines which entirely lie on this graph. What functions have 3 such lines?* *********************************************************************** 3/11 Chapter 14 Exercises: 13,16,19,21[compare mixed 2nd partials], 25,28,31* *In class we discussed f(x,y)=2xy/(x^2+y^2). Explain why there is no single limit value L for f(x,y) as tends to <0,0>, but that instead, the closed interval I=[-1,1] is a "limit set" in the follwing sense: for any neighborhood V of I in R, we can find a neigborhood U of <0,0> in R^2 with f(U) contained in V.* Chapter 14 Problems Plus: 4,5* *********************************************************************** *************************** SPRING BREAK ****************************** *********************************************************************** 3/25 Chapter 14 Exercises: 11,33,34,35,37,40* Chapter 14 Problems Plus: 6* *In class we discussed that a function f(x,y) with partials f_x and f_y defined at =<0,0> might fail to be differtiable at <0,0> (i.e. fail to have good linear approximation). Try to write a formula for such a function (perhaps cosult Wikipedia, or other sources - and if you do, please cite your sources).* *Similarly, we discussed Clairut's Theorem about the equality of of mixed partials. Try to write a formula for a function f(x,y) for which f_xy and f_yx are both defined at <0,0>, but are NOT equal (of course, by the hypotheses of Clairut's Theorem, f_xy and f_yx, as functions of , can't both be continuous at <0,0>).* We explored the chain rule f_t = f_x x_t + f_y y_t for the t-derivative of the composite function f(x(t),y(t)) of a curve in a plane domain D followed by a function f(x,y) on D (using arrows: I \subset R --> D \subset R^2 -f-> R), but ran out of time at the end to check: if the curve is a contour (or level set) for f, then \grad f is perpendicular to the curve, i.e. its dot-product (grad f). with the velocity must vanish. Please finish the argument [hint: since f=const on the contour, f_t=?] Start reading Chapter 15! *********************************************************************** 4/1 Chapter 14 Exercises: 44,47,49,50,52 In class (and at last Wed eve session) we discussed the "egg carton" surface, the graph of f(x,y)=(sin x)(sin y). Since f(x,y) is a "periodic" function (it has the same values if either x or y are translated by 2\pi) we can completely describe its behavior on a "fundamental domain" such as the square [-\pi,\pi]\times[-\pi,\pi] R^2. We found all the critical points for f(x,y) on this domain and conjectured which were local maxima, minima and saddles. Please work out the details, and check our claim that #maxima+#minima-#saddles=0 for this example. What happens to this (signed) count of critical points if we use a bigger domain, like [-m\pi,m\pi]\times[-n\pi,n\pi] for some integers m,n>0? (My egg carton was square and held 36 eggs, in which case m=? and n=?) [For more on this sort of thing, please google "Morse Theory" - and (while you're at it) see if you can find the classic short animated film "Peaks, Pits and Passes"!?!] Continue reading Chapter 15!! *********************************************************************** 4/8 Remark on the very last question ("My egg carton was square and held 36 eggs, in which case m=? and n=?"): some of you figured out that for each [-\pi,\pi]\times[-\pi,\pi] square, there were 2 minima (the eggs nestle there) among the 8 critical points, so we'd need 18 of these to hold 36 eggs - how could that be (as 18 is not a perfect square)?! [Hint: in each square, there's a pair of smaller squares (rotated 45 degrees), each with half the area, and each with half as many critical points - in particular 1 minimum - as in the original square; my egg carton is really a 6 x 6 array of those smaller squares.] Chapter 14 Exercises: 55,56,59,61,64,66* Chapter 14 Problems Plus: 1,8* *********************************************************************** 4/15 Chapter 15 Exercises: Take a break and enjoy the sunshine... Chapter 15 Problems Plus: ... and warm weather!!! Finish reading Chapter 15 and start reading Chapter 16! Not a problem, but a little project for before Wednesday's class: How would you measure the volume of potato? OK, you could be clever, like Archimedes 2300 years ago, and find instead the displaced volume of fluid. But suppose you were on a rocky desert planet, with no fluids (like water or even sand) at your disposal, and you had to figure it out by slicing and dicing. The practical aspect of the project is this: can you (collectively) please procure (perhaps from a DC, or a farmer or grocer) about 1 potato per person so that we can do some experiments on Wednesday afternoon? I can supply the knives, but remember (or beware!) it was the Ides of March - and not the Ides of April - when Caesar's generals sliced and diced him! *********************************************************************** 4/22 Thanks to those of you who DID procure a potato today! Unfortunately, there weren't enough to go around; nor did I have the sharp knives or measuring tools that we might use to carry out the experiment I had in mind carefully, and so it became a "thought" experimemt in class. But we heard some good methods (I particularly liked the idea of mashing the potato: in order to get around the lack of fluids on our desert island, we turn the potato into a fluid). So here's part 1 of this weekend's homework (for those of you with a potato, please use your potato; for those without, please use a fruit substitute, something about the same size): (i) procure (e.g. borrow) length and volume measuring devices with metric units (please stick to CGS: cm for length, and (cm)^3=cc=ml for volume); (ii) first try to estimate carefully the volume of your potato (or fruit substitute) by a method like the Archimedes displacement method - if instead you want to use the Matt mashing method, do step (iii) before (ii); (iii) slice your potato (or fruit substitute) into 1 cm thick slabs (home fries?), then slice those again into 1 (cm)^2 cross-section strips (freedom fries?) whose lengths you can measure to estimate the volume; you could go even further and dice those into 1 (cm)^3 cubes (tater tots?) and simply count, but (as we saw today in class) that makes it hard to reassemble into a potato (the boundary "skin" help, just like with a puzzle). How close are your two ways of measuring volume? Keep notes and photographic evidence of your experiment! (You may work alone or in teams of at most 3.) *Think about what would account for the difference - besides measurement errors, there's a boundary effect - what feature of the boundary is being measured?* Chapter 15 Exercises: 1,2,3,4 Chapter 15 Problems Plus: 1,2* Continue reading Chapter 16!! *********************************************************************** 4/29 [We made the next stuff due "infinitely" later.... ;-] Chapter 15 Exercises: 5,7,8,10,11,15,49* Chapter 15 Problems Plus: 4*,5*,6* *Consider the solid region defined by drilling a cylindrical hole through the center of a ball <--??--> /| |\ || || \| |/ <----??----> so the resulting height is 2H (the "cored apple" described in class). What is its volume?* Chapter 16 Exercises: 1,2,7,9,11,12,18,20 Chapter 16 Problems Plus: 1*,3,5 In the last two classes (and the evening session between) we discussed grad, curl and div [since I cannot easily make a "nabla" in ASCII, I'll use these monosyllables instead] for smooth functions and vectorfields on R^3. Recall that the geometric interpretation of grad f is the vectorfield whose direction is that for which f increases most rapidly and whose length is the rate of increase. Another way to say this is the "flow" along grad f is (locally) how we translate as f increases. (If v=v(X) is a vectorfield, where X is position and t is time, then v has flow F(t,X)=X+tv(X)+O(t^2) which gives the path of a point starting at X(0) moving to X(t)=F(t,X) after time t - this involves solving a differential equation, but for small time we can approximate the flow by X(t)=F(t,X)=X+tv(X) - the point has initial velocity X'(0)=v(X).) We also saw curl v gives the tendency of the flow of v to rotate: it measures the circulation (path integral) of w around a loop, as made precise with Stokes Theorem: The double integral of (curl w).N over any surface (with unit normal N) equals the path integral of w around the boundary loop (the right-hand rule gives the correct sign: if the thumb points along N, then the fingers indicate the way we follow the loop). We checked (using the the equality of mixed 2nd partials) the important identity: curl(grad f)=0 and we saw how this means curl v is the "obstruction" to v being grad f: path integrate v along any loop and find that its circulation must vanish, then recover f (up to constant). What about the divergence div w of a vectorfield w? At the very end of class we hurried to see it measures the volume expansion under the w-flow X(t)=X+tw(X)+O(t^2): If we write X(t)=, w= and ignore O(t^2) terms in what follows, then the chain rule gives dX(t)=dX+tdv= where the "..." are the analogous compnents involvling w2 and w3. Using the "^" here to denote the way we multiply the differentials used to compute area or volume (skew symmtric like cross product or determinant) we get (after some algebra) volume element dx^dy^dz(t)=dx^dy^dz(1+t(w1_x + w2_y + w3_z)) since lots of terms cancel or are O(t^2). Therefore at t=0 d/dt[dx^dy^dz(t)] = (w1_x + w2_y + w3_z) = div w The analog of Stokes Theorem for divergence is sometimes called the Gauss Law or Divergence Theorem: The triple integral of div w over a region of R^3 equals the double integral of w.N over the boundary surface with (outerward) unit normal N. The latter is called the "flux" of w across the boundary surface and if this flux vanishes for all closed surfaces, then w is "volume preserving" and (equivalently) has div w = 0. For example, if you apply this to position vector field w=X with div X = 3 (recall we did that calculation in class) then you can quickly see that the area of the unit sphere is exactly 3 times the volume of the unit ball (4\pi = 3 (4\pi/3)) or more generally for the area and volume for the sphere and ball of radius R. Analogous to the important identity above, you should check: div(curl v)=0 but the vanishing of div w doesn't guarantee that w = curl v. (That leads to deeper features of vectorfields - like harmonic functions and forms - which may be treated in Math 425 or 704, and are the foundation of "Calculus on Manifolds" or "Differential Topology"! *********************************************************************** See you all at 8AM Tuesday 5 May 2015 in our regular classroom for the Math 233H final exam (mainly on chapters 15 and 16)... ... and have a great summer (and beyond)!!!!!!!!!!!!!!!!!!!! *********************************************************************** *********************************************************************** *********************************************************************** ******************** Old HW problems (M & T, etc.) ******************** **************************** from S02 ******************************* *********************************************************************** *********************************************************************** *********************************************************************** Take a triangle in the plane with vertices P, Q and R. The midpoints of its edges form a new triangle with vertices (P+Q)/2, (Q+R)/2 and (R+P)/2. What happens when we iterate this process? To what point in the plane do the successive triangles limit to? And what is the limiting shape of these triangles? The answer to the analogous questions for a quadrilateral is related to the "parallelogram conjecture" mentioned in class. What happens for a general n-gon? Read Chapter 1 of M & T 1.1#8,14,16,22,26 1.2#8,12(for 8 only),16,26 1.3#3,4,5,17,26,30 1.4#2,6,14(may be useful at Jiffy-Lube ;) Consider the outer normals to the triangular faces a tetrahedron, each with length equal to the area of the corresponding face; use vector methods to show the sum these normal vectors is zero. Read Chapter 2 of M & T 2.1#2,10,18,30,32 2.2#9 2.3#1,13,8,14,20 2.4#12,18 2.5#4,8,12,14,20,24 2.6#3,6b,16c Review problems from Chapter 2 of M & T Read Chapter 3 of M & T 3.1#2,4,8,16 3.2#4,7* 3.3#4,10,18,24,27,36*,42 3.4#10,12 Here is what the topographic map is for: find a contour which loops around and encloses an interesting region of Amherst; highlight this contour; now try to locate all critical points inside (the maxima should be easy, minima are probably under water, and the saddles are tricky); use the relation #maxima + #minima - #saddles = 1 to ensure that you have found all the saddles! Read Chapter 4 of M & T 4.1#6,7,14,18 4.2#7,12,13,14*,15* 4.3#3,7,9,18* 4.4#2,8,13,21,26 Read Chapter 5 of M & T 5.1#1,5 5.2#2,3,4 5.3#1,11,12,16 5.4#1,10,14 5.6#3,12,15,24 Read Chapter 6 of M & T 6.1#1,2 6.2#1,2,6,7,8,16 6.3#4,8,13,14,17 Read Chapter 7 of M & T 7.1#11 7.2#6,14 These are sample topics from a previous 233H final exam about which you may wish to think: a) derivatives and integrals of the function f(x,y) = cos((x+y)/(x-y)) over various domains in the (x,y) plane; b) the change of variables T: R^2 -> R^2 defined by T(u,v) = (x(u,v), y(u,v)) where x(u,v) = e^u cos v [only u in the exponent] y(u,v) = e^u sin v; c) properties of the solid region defined by drilling a cylindrical hole of radius 8 through the center of a ball of <--16--> /| |\ || || \| |/ <----20----> radius 10 (this was the "cored apple" described in class).