Look out for typos! Homework 1: Review of Calc 1 and 2. Problem 1. Sketch the graphs of the following functions:

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1 Math 226 homeworks, Fall 2016 General Info All homeworks are due mostly on Tuesdays, occasionally on Thursdays, at the discussion section. No late submissions will be accepted. If you need to miss the class, make arrangements with the discussion section instructor for an early submission. Electronic submissions, by to the discussion section instructor, of typed or legible scanned/photgraphed/etc. handwritten papers are OK. On all the homework problems, you are welcome to collaborate with anybody and use any help you want, including computer, but I expect you to write everything yourself, to understand everything you wrote, and to make sure that what you wrote is correct. S14/3 means problem number 3 in the Spring 2014 final exam; F12/5 means problem number 5 in the Fall 2012 final exam The solutions to some of the old final exams are posted near the bottom of on our course web page. You are welcome to use them, but (a) look at the posted solution after you came up with your own solution (or got hopelessly stuck); (b) do not assume that all the posted solutions are correct. Look out for typos! Homework 1: Review of alc 1 and 2. Problem 1. Sketch the graphs of the following functions: y = x 2 + 1; y = 5 (x + 3) 2 ; y = x 2 + x + 1; y = x 1 ; y = ex ; y = e 2x ; y = ln(1 + x); y = ln(1 x); y = sin(sin 1 (x)); y = cos 1 (cos(x)). Problem 2. ompute the first two derivatives of the following functions: y = x 1 2x + 1 ; y = e x2 ; y = sin(cos(x)); y = x x ; y = (1 + x 2 ) cos(x). Problem 3. Write the equation in polar coordinates of the circle with center at (a, b) and radius R. How does the equation simplify if R 2 = a 2 + b 2? How does the corresponding circle look like? Problem 4. Sketch the following curves in artesian coordinates (a) (x 2 /4) + (y 2 /9) = 1, (x 2 /9)+(y 2 /4) = 1 (ellipses); (b) x 2 2y 2 = 1, y 2 2x 2 = 1 (hyperbolas); (c) y = (x 1) 2 +1, x = y 2 (parabolas). Homework 2: Vector operations and geometry. Problem 1. Write the vector parametric equation of the line that is perpendicular to the plane x y +2z = 12 and passes through the origin, and compute the coordinates of the point where the line intersects the plane. [The line is r(t) = t 1, 1, 2 and the point of intersection corresponds to t + t + 4t = 12, that is, 2, 2, 4.] Problem 2. onsider the vectors a = 1, 1, 1, b = 1, 1, 1, c = 1, 1, 1. ompute a ( b c), a ( b c), and ( a b) c.

2 2 I got 4, 2, 2, 0 = b c, and 0, 2, 2 = a + b, respectively. Problem 3. onsider three non-zero vectors a, b, c such that b c, b = c, and a b = a c. onfirm that a b = a c and therefore b = c 2 a c a 2 a. [Note that b and c are sides of a rhombus, with diagonals b ± c, and the diagonals in a rhombus are orthogonal; the assumptions also imply that a is parallel to b c. It then remains to write b c = x a, dot-multiply both sides by a, and solve for x.] Problem 4. onsider three distinct points A, B, on a circle such that the line segment [A, ] is a diameter of the circle. onfirm that the angle AB is a right angle. You need to confirm that AB B = 0. Put the origin at the center of the circle and write out the expression AB B using AB = OB OA, etc., and then use the definition of the dot product. A picture can help you keep track of all the angles and distances. Problem 5. (a) onsider the two lines defined by the vector parametric equations r 1 (t) = 1, 2, 3 + t 2, 3, 1 and r 2 (s) = 3, 1, 2 + s 1, 3, 2. onfirm that the lines are skew (that is, neither intersecting nor parallel) and compute the distance between the lines. (b) [The general version of part (a)] onsider two lines defined by the vector parametric equations r 1 (t) = a 1 + t d 1 and r 2 (s) = a 2 + s d 2. Determine the conditions on the vectors a 1, a 2, d 1 and d 2 so that the lines are skew and then compute the distance between the lines. onfirm that what you got in part (a) agrees with this general result. You are welcome to search for skew lines on the web and use the results. In particular, for skew lines, n = d 1 d 2 is not zero and is the normal vector for the two parallel planes that contain the two lines. The distance is then n ( a 1 a 2 ) ; having this distance no-zero is the condition for skewness. Homework 3: urves. Problem 1. S15/1. Problem 2. F14/1. Problem 3. F13/1, F13/2. Problem 4. S13/1. Problem 5. The main objective is to practice various vector operations, but it is also useful to keep in mind the words in italic [you also are welcome to use those words when searching the web]. As usual, the prime means derivative with respect to time; for the last part you will use three of those. onsider the right circular helix with fixed R > 0, ω 0, and b R. ompute r(t) = R cos(ωt), R sin(ωt), b t, t > 0, The velocity v(t) = r (t); The acceleration a(t) = r (t) = v (t); The unit tangent vector T (t) = v(t)/ v(t) ;

3 The unit normal vector N(t) = T (t) T (t) and confirm that N(t) T (t) = 0 for all t > 0; The binormal vector B(t) = T (t) N(t); [once again, confirming that B T = B N = 0 is a good idea]; The curvature v(t) a(t) κ(t) = ; v(t) 3 The torsion τ(t) = r (t) ( v(t) a(t) ) v(t) a(t) 2. The computations are long but straightforward; you are welcome to find answers/solutions on the web or in other books. It helps to keep track of units, with R measured in units of length, ω measured in units of inverse time (also known as frequency), and b measures in units of speed. In the end, you should get both κ and τ independent of t, and this turns out to characterize the right circular helix uniquely: there are no other (reasonable) curves having both curvature and torsion constant. I got κ = Rω 2 /(b 2 + R 2 ω 2 ), τ = bω/(b 2 + R 2 ω 2 ). Note that the circle can be considered a particular case, with b = 0, and then you get κ = 1/R and τ = 0. If you set R = 0, then you get κ = 0 and τ = ω/b. Does this make sense? Homework 4: Functions of several variables. Problem 1. F15/1, S12/1, F11/1. Problem 2. S15/2, F14/2, S14/2, S12/2. Problem 3. F13/3, S13/7. Problem 4. Let a, b, c be three vectors such that a ( b c) 0. Draw a picture of each of the following sets: t a + (1 t) b, 0 t 1 [should be the line segment connecting the tips of a and b]; s a + t b + (1 s t) c, 0 s, t 1, s + t 1 [should be the triangle with vertices at the tips of a, b, c]; s a + t b + u c, 0 s, t, u 1, s + t + u 1 [should be the tetrahedron spanned by a, b, c]. Problem 5. Assume that the equalities xy + yz + zw = 3, x 2 + y 2 + z 2 + w 2 = 4 define a function x = x(z, w). onfirm [or deny and then let me know right away] that ( ) x = 1. w (1,1,1,1) Keep in mind that y = y(z, w) is another function, and as a bonus you will see that ( y/ w) z (1,1,1,1) = 0. Homework 5: Surfaces. Problem 1. F14/3, F12/9a. Problem 2. S14/1. z 3

4 4 Problem 3. F11/1. Problem 4. For every possible value of c (, + ), name the surface defined by the equation x 2 + y 2 z 2 = c. Problem 5. At the basic level, this is just a computational drill, but it is possible to take this problem much further. You are welcome to search for the solutions on the web (use the words below that are in italic). arrying out all the computations correctly to the end is a BIG accomplishment. If the computations become unbearable, you are welcome to give up, with little or no penalty. onsider the surface defined by the vector parametric equation r(u, v) = x(u, v), y(u, v), z(u, v). Define the normal vector N(u, v) = r u (u, v) r v (u, v). In this problem we will work with two particular surfaces: the sphere of radius R > 0, given by r(u, v) = R cos(u) sin(v), R sin(u) sin(v), R cos(v) 0 u < 2π, 0 v π; the pseudosphere of radius R > 0, given by r(u, v) = R cos(u) sin(v), R sin(u) sin(v), R ( cos(v) + ln(tan(v/2)) ) ; 0 u < 2π, 0 < v < π; there is indeed a natural log of the tangent in the third coordinate. (1) The first fundamental form of a surface is the expression where I = E(u, v)du 2 + 2F (u, v)du dv + G(u, v)dv 2, E = r u 2, F = r u r v, G = r v 2. (a) onfirm that, for every smooth surface, N 2 = EG F 2. (b) ompute the first fundamental form for the sphere and the pseudosphere. (2) The second fundamental form of a surface is the expression where and II = L(u, v)du 2 + 2M(u, v)du dv + N(u, v)dv 2, L = r uu ˆn, M = r uv ˆn, N = r vv ˆn, ˆn = N N. [Please do not confuse the functions N and N] ompute the second fundamental form for the sphere and the pseudosphere. (3) The Gaussian curvature of the surface is the function K(u, v) = L(u, v)n(u, v) M 2 (u, v) E(u, v)g(u, v) F 2 (u, v). onfirm that K = 1/R 2 for the sphere and K = 1/R 2 for the pseudosphere. In other words, the sphere and pseudosphere are two remarkable surfaces that have constant Gaussian curvature. Homework 6: Optimization. Problem 1. F15/2, S15/3. Problem 2. F14/4, S14/3.

5 5 Problem 3. F13/4, S13/3. Problem 4. F12/6, F12/7. Homework 7: Optimization. Problem 1. F15/3, S15/4. Problem 2. F14/5, S14/4. Problem 3. S13/4, F12/4. Problem 4. S12/3, F11/3. Homework 8: Double integrals. Problem 1. F15/4, S15/5 Problem 2. F14/6, S14/5 Problem 3. F13/6, S13/5 Problem 4. S12/5. Problem 5. By considering a double integral and switching to polar coordinates, confirm that e x2 dx = π. Solution: By denoting the value of the integral by I, we see that 2π + I 2 x = e 2 y 2 da = e r2 rdr dθ = π. R Homework 9: Triple integrals. Problem 1. F15/5, F11/4. Problem 2. S14/6. Problem 3. F13/5, S13/6. Problem 4. F12/8, S12/6, S12/8. Homework 10: url and Divergence. The notations,,, and 2 = mean, respectively, gradient, divergence, curl, and the Laplacian. Problem 1. Verify the following identities: (fg) = g f + f g, (f F ) = F f + f F, (f F ) = ( f) F + f F. Problems 2. Verify the following identities: ( f) = 0, ( F ) = 0, ( F G) = G ( F ) F ( G), ( f g) = 0. Problem 3. Let r = x, y, z, r = r = x 2 + y 2 + z 2, and let h = h(t), t 0, be a twice continuously differentiable function. Then h(r) is a scalar field. onfirm that Problem 4. h(r) = h (r) r r, 2 h(r) = h (r) + 2h (r). r

6 6 (a) onfirm that, in cylindrical coordinates, 2 f = 1 ( r f ) r r r + 1 r 2 2 f θ f z 2. (b) onfirm that, in spherical coordinates, 2 f = 1 ( r 2 f ) 1 2 f + r 2 r r r 2 sin 2 φ θ r 2 sin φ φ ( sin φ f ). φ Homework 11: Line integrals. Problem 1. F15/6, S15/6. Problem 2. F14/7, F14/8, S14/7. Problem 3. F13/8, F13/9, S13/2. Problem 4. F12/2, S12/9. Problem 5. F11/5, F11/6. Homework 12: Surface integrals. Problem 1. F15/7. Problem 2. S14/8. Problem 3. S13/8. Problem 4. F12/9b. Homework 13: General integration. Note that problem F15/8 can be done two ways: (1) by Stokes (as intended; you can also follow up with Green, instead of computing the path integral directly); (2) by Gauss (close up the surface and note that divergence of the curl is zero, meaning that the flux through S is the same as the flux through the disk in the (x, z) plane; you will need to compute the curl, but the subsequent computations are straightforward. Accordingly, try to think of alternative solutions for all other problems in this homework. Problem 1. F15/8, S15/7, S15/8, S15/9. Problem 2. F14/9, F14/10, S14/9. Problem 3. F13/7, F13/10, S13/9, S13/10. Problem 4. S12/4, F11/7, F11/8. Problem 5. F12/1, F12/3, F12/5, F12/10. Problem 6. (a) onsider the vector field in the plane F (x, y) = y x 2 + y, x. 2 x 2 + y 2 onfirm that F d r = 0

7 for every simple closed curve that does not go around the origin. Then confirm that F d r = 2π for every circle centered at the origin, and then deduce that F d r = 2π for every simple closed curve that goes around the origin. [Note that the integral is not defined if the curve passes through the origin because F is not defined at the origin.] (b) onsider the vector field in space F = r, r = x, y, z, r = r. r3 onfirm that F = 0 away from the origin, and therefore F ds = 0 for every nice surface that does not enclose the origin. Next, confirm that F ds = 4π for every sphere centered at the origin, and therefore F ds = 4π S S S for every nice surface that encloses the origin. [Note that the integral is not defined if the surface S passes through the origin because F is not defined at the origin. You should also notice that, by Problem 3 in Homework 10, F = (r 1 )]. 7 Homework 14: Review for the final exam, part 1. Problem 1. onsider four points in R 3 : P (1, 1, 1), Q( 1, 0, 2), R(1, 1, 1), S(1 + a, 0, 1 2a), where a is a real number. (1) ompute the coordinates of P Q, P R, P S. (2) ompute the value of a so that the angle RP S is the right angle. (3) ompute the area of the triangle P QR. (4) Write the vector parametric equation of the line that passes through the point P and is perpendicular to the line QR, and compute the coordinates of the point where the two lines intersect. (5) ompute the equation of the plane containing the points P, Q, R. (6) ompute the volume of the parallelepiped spanned by the vectors P Q, P R, P S (it will be a function of a). (7) For what value of a will the point S lie in the same plane as the points P, Q, R? (8) Write the vector parametric equation of the line that passes through the point (1, 0, 1) and is perpendicular to the plane through points P, Q, R, and compute the coordinates of the point of intersection of the line and the plane.

8 8 Problem 2. A particle moves so that its position at time t is given by the vector function r(t) = 1 t 2, t 3, 1 + t 2, t 0. ompute: (1) oordinates of the particle at time t = 1; (2) Velocity of the particle for t 0; (3) Speed of the particle for t 0; (4) Acceleration of the particle for t 0; (5) Vector parametric equation of the tangent line to the trajectory at (0, 1, 2); (6) The coordinates of the point of intersection of the trajectory with the plane z x = 2; (7) Distance traveled (arc length) from t = 0 to t = 1. Problem 3. (1) onsider a function f(x, y) = 2x 2 xy + y 2 x + y 1. ompute the gradient of f. ompute the rate of change of f at (1, 1) in the direction toward the origin. Is the function increasing or decreasing in that direction? Determine the direction of most rapid decrease of f at (1, 1) and compute the rate of change of the function in that direction. Equation of the tangent plane to the surface z = f(x, y) at the point (1, 1, 1). (2) Here is a drill on partial derivatives. onsider the function ( 1 V (t, x; s, y) = 2π(1 e 2(t s) ) exp (x ) ye (t s) ) 2. 2(1 e 2(t s) ) Just in case, exp(a) = e a is the usual exponential function. onfirm that the function satisfies V t = 2 V x + x V + V, t > s, 2 x and V s + 2 V y y V = 0, s < t. 2 y For those who want to know: the function V is the probability density function of a famous random process, and the above equations are known as the forward and backward Kolmogorov equations, respectively. There is a general result saying that the probability density of certain random processes must satisfy equations like this. So this is not a random exercise (no pun intended...) Problem 4. Evaluate the following integrals: (1) (2y 2 + 2xz)dx + 4xydy + x 2 dz, : x(t) = cos t, y(t) = sin t, z(t) = t, 0 t 2π. (2) y 2 dx + x 2 dy, where is the boundary of the rectangle with vertices (1, 0), (3, 0), (3, 2), (1, 2), oriented counterclockwise. (3) ydx zdy + ydz, where is the ellipse x 2 + y 2 = 1; 3x + 4y + z = 12 oriented counterclockwise as seen from the point (0, 0, 1000). Problem 4. ompute the following quantities using a suitable integral: (1) The mass of the curve shaped as a helix r(t) = cos t, sin t, t, t [0, 2π], if the density at every point is the square of the distance of the point to the origin. (2) The area between x-axis and the curve r(t) = t sin t, 1 cos t, t [0, 2π]. (3) The average distance to the (x, y) plane of the points on the hemisphere z = 1 x 2 y 2.

9 (4) The flux of the vector field F (x, y, z) = x, y, z through the lateral surface of the cylinder x 2 + y 2 = 1, z [0, 2]. Problem 5. Let G be the parallelogram with vertices (0, 0), (3, 0), (4, 1), (1, 1), and f, a continuous function. Write the limits in the iterated integrals below: f da = fdx dy = fdy dx = frdr dθ = fdθ rdr. G Problem 6. (a) Write the equation of the unit sphere in cylindrical coordinates. (b) Write the equation of the cylinder x 2 + y 2 = R 2, 0 < z < 1, in spherical coordinates. Homework 15: Review for the final exam, part 2. Solve the most recent final exam (Spring 2016) under the actual exam setting (2 uninterrupted hours, no notes, etc.) Use the formula sheet you would like to use on the actual exam. Then bring your questions/comments/suggestions (together with the solved exam) to the discussion on Tuesday. 9