Sept , 17, 23, 29, 37, 41, 45, 47, , 5, 13, 17, 19, 29, 33. Exam Sept 26. Covers Sept 30-Oct 4.

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1 MATH 23, FALL 2013 Text: Calculus, Early Transcendentals or Multivariable Calculus, 7th edition, Stewart, Brooks/Cole. We will cover chapters 12 through 16, so the multivariable volume will be fine. WebAssign will be an optional part of the course. The computer generates problems and guides you to the solution. We will not keep track of your performance on WebAssign. If you purchase your text at the Lehigh Bookstore, part of the price is WebAssign access. You may purchase the text on-line if you do not wish to use WebAssign. Homework and quizzes: There are Suggested Problems, which are odd-numbered problems in the text, which you are expected to work but not hand in. Your quizzes during recitation will consist of problems very similar to Suggested Problems, as will a portion of your major exams. You will be able to ask for help on these problems during recitations and office hours. There are also Hand-in Problems, usually six per week. These will be graded. Each of your quizzes and weekly HW assignments will be worth 20 points. They will all be part of the same category of your grade. Your three lowest scores on these will be dropped, and the remaining scores averaged onto a 100-point basis. Half-credit is given on quizzes for just handing them in. Late HWs are not accepted. If you are unable to hand in a HW, get someone else to hand it in for you, or hand it in early. If you miss a quiz for a very good reason which your instructor considers worthy of an Excused Absence, you will receive the average score made by your class on that quiz. You may work together on HW problems, but when it comes to writing it up, you must do it yourself without copying. Solutions to the HW will be posted on CourseSite shortly after their due date. Exams and grades: There will be midterm exams from 4:10 to 5:10 on Sept 26 and Oct 31. If you must miss one of these, arrange it with your professor in advance. A makeup exam will be given approximately one week after the regular exam. The date of the final exam has not yet been set. It will cover the entire course, with an emphasis on the later material. Do not plan to leave Lehigh in December until the date of your final exam is known. Your basic grade is computed out of 500 points (100 for HW/Quiz, 100 for each midterm, and 200 for the final exam). However, if a student does poorly on one midterm exam, that exam may have its component toward your grade divided 1

2 2 by 2, if that helps your final average. For example, if your scores are (HW&Q,MT1,MT2,Final) = (90,30,90,180), your average would be 78%, but dividing MT1 by 2 yields ( )/450 = 83.3%, which would be your grade. Decisions about possible scaling of grades will be made at the end of the course. You are guaranteed that if your final average is at least 90 (resp. 80, 70, 60), your course grade will be at least A (resp B, C, D ). Calculators and computers: Some of the homework problems will require you to use a calculator or the computer algebra system Maple, which is available on most campus computers. However, calculators are not allowed on quizzes or exams. Attendance: Attendance in lecture and recitation is required. Students missing an excessive number of classes will receive a Section 3 report from the instructor, at which time the student will have to explain the absences to the Dean of Students. Accommodations for Students with Disabilities: If you have a disability for which you are requesting accommodation, please contact both your professor and the Office of Academic Support Services, University Center C212 ( ) as early as possible in the semester. You must have documentation from the Academic Support Services office before accommodation can be granted. Weekly schedule and suggested problems for Math 23 Aug , 9, 13, 17, 27, , 13, 25, 29, , 7, 19, 23, 27, 43, 47, 53. Sept , 9, 13, 17, 31, , 5, 7, 13, 21, 25, 31, 45, , 5, 21-28, 35. Sept , 7, 13, 21-26, 31(M), , 7, 11, 19, 29(M), 35, , 5, 13, 17, 23, 31, 47. Sept , 11, 15, 17a, 19, 23, 25, , 9, 15, 19, 29, 33, 35, , 13, 17, 31, 39.

3 3 Sept , 17, 23, 29, 37, 41, 45, 47, , 5, 13, 17, 19, 29, 33. Exam Sept 26. Covers Sept 30-Oct , 5, 9, 13, 23, 27, 29, 33, , 9, 13, 17, 23, 25, 29, 41, , 9, 13, 19, 29, 31, 35, 41, 51. Oct 7-Oct , 5, 11, 15, 21, 27, , 3, 5, 11, 13, , 9, 11, 13, 15, 17, 19, 23, 27, 35. Oct 16-Oct , 9, 13, 15, 17, 21, 25, 31, 43, 47, , 9, 13, 15, 17, 21, 25, 31, 39. Oct 21-Oct , 5, 9, 11, 15, 21, , 5, 9, 11, 21, , 7, 9, 13, 19, 33, 37, 41, 45. Oct 28-Nov , 5, 9, 15, 17, 19, 21, , 5, 7, 15, 17, 21, 25, 35, 39. Exam Oct 31. Covers Nov 4-Nov (start) , 5, 11-14, 21, 23, 29-32, , 7, 9, 11, 21, 33, , 7, 9. Nov 11-Nov , 15, 19, 31, , 7, 9, 13, 17, 21, , 3, 5, 9, 11, 12, 13, 19, 31. Nov 18-Nov , 5, 13-18, 19, 23, 25, 33, 35, 39, 41, , 9, 13, 19, 23, 27. Nov 25-Nov (start). Dec , review , 5, 7, 11a, 17, , 3, 7, 11, 17.

4 4 Math 23 hand-in HW problems, Fall a. Show that the points P (1, 2, 3), Q(4, 5, 2), and R(0, 0, 0) are the vertices of a right triangle by finding the lengths of the sides of the triangle. Which side is the hypotenuse? 12.1-b. Is the equation x 2 + y 2 + z 2 4x + 4y + 6z + 20 = 0 an equation for a sphere? If so, find its center and radius. If not, explain why not a. Find the vectors of norm 2 parallel to 3j + 2k b. Find the vector that makes an angle of 5π/6 with the positive x-axis and has norm a. Find the angle between the vectors 3i j 2k and i + 2j 3k b. Find the scalar and vector projections of b = 2, 1, 1 onto a = 4, 3, a. Find a vector N that is perpendicular to the plane determined by the points P (1, 1, 4), Q(2, 0, 1), and R(0, 2, 3), and find the area of the triangle P QR b. Prove that for any vectors a and b, a b 2 +(a b) 2 = a 2 b a. Find the point where lines L 1 and L 2 intersect, and find the cosine of the angle between L 1 and L 2. L 1 : x = 3 + t y = 1 t z = 5 + 2t L 2 : x = 1 y = 4 + u z = 2 + u 12.5-b. Find an equation for the plane that passes through the points P (0, 1, 1), Q(1, 0, 1), and R(1, 3, 1) c. Find the angle between the planes 5(x 1) 3(y + 2) + 2z = 0 and x + 3(y 1) + 2(z + 4) = a Identify and sketch the surface 4x 2 +9z 2 8x 4y = 0, indicating any significant points a. The cylinders x 2 +y 2 = 5 2 and y 2 +z 2 = 3 2 intersect in two identical closed curves, one with x > 0 and one with x < 0. Sketch the cylinders and the curves. Let C be the curve with x > 0; it looks like a slightly deformed circle. Write a parametrization of C in terms of θ in the y-z plane. (In Section 3.3, you will be asked to compute the length of C.) 13.2-a. Find the unit tangent vector to the curve r(t) = t sin t, t cos t, 2t at t = π/2.

5 13.2-b. Find r(π/2) if r (t) = cos(2t), sin(2t), t 2 and r(0) = 1, 1, a. Find the length of the curve C in problem 13.1-a. Use a calculator or Maple to evaluate the definite integral. Compare your answer with the circumference of the circle bounding the smaller cylinder b. Find the curvature of the circular helix r(t) = 2 cos ti+2 sin tj+ tk. It will have the same value at every point c. For the curve r(t) = t, 2t, t 2, find T(t) and T (t). Show that these are orthogonal for all t. Compute T, N, and B when t = a. Find the velocity, acceleration, and speed of a particle with position function r(t) = 1, 1 t, 1 + 2t t 2. Identify the curve b. A projectile is fired with an initial speed of 98 m/s and an angle of elevation 30 o. Find the range of the projectile, the maximum height reached, and the speed at impact c. Find the tangential and normal components of acceleration if r(t) = t, 2t, t a. Sketch the level curves of the function f(x, y) = x 3 y corresponding to the values c = 1, 0, 1, a. Let f(x, y) = (x y 4 )/(x 3 y 4 ). Determine whether or not f has a limit at (1, 1) b. Let f(x, y) = (x y)/(1 e x y ) wherever the denominator is nonzero. Explain how to define f(x, y) at other points in such a way that the expanded function is continuous at every point of the plane. Justify your answer a. Compute the partial derivatives f x y sin t dt. x 2 f and y 14.3-b. Use implicit differentiation to compute z x xy + y sin z = 1. 5 where f(x, y) = and z y where x a. Find the equation of the tangent plane of the surface z = x ln (x + y 2 ) at (1, e 1, 1) b. Find the differential of w = x. z 2 +xy 14.5-a. u = xe yz, x = r 2 s, y = s 2 t and z = t 2 r. Compute u (r, s, t) = ( 1, 2, 1)., u, u r s t at

6 b. Assume xy + y cos z = z 2. Use the chain rule to compute z x and z y a. Find the normal line of the ellipsoid 4x 2 + 4y 2 + z 2 = 12 at (1, 1, 2) b. Find the directional derivative of f(x, y) = direction v = i + 3j. xy x 2 +y 2 at (1, 2) in the 14.7-a. Assume x, y, z 0 and x + y + z = 3. Find the minimum value of f(x, y, z) = x 2 + y 2 + z b. A box without lid has surface area 12. Find the dimensions of the box with largest volume a. Use the method of Lagrange multipliers to find the maximum value of f(x, y) = 3x + 2y subject to x 2 + y 2 = b. Let C be the curve of the intersection of x + y + z = 1 and z = 4x 2 + 4y 2. Find the points on C that are nearest to and farthest from the origin a. Let R = [0, π] [0, π]. Use a Riemann sum with m = n = 4 to estimate the value of (sin x + cos y) da. Take the sample R points to be upper left corners of the rectangles b. Use the definition (limit of Riemann sum) to compute (x + y) da R where R = [0, 1] [0, 1] a. Compute the iterated integral 1 1 (x + y) x y 2 dx dy b. Compute the double integral y sin x da where D = [ π, π] D 1+x 2 +y 2 [0, 1] a. Let D R 2 be the region bounded by x = 0, x = 2, x = y 2, y = 2 + x. Compute D x2 y 2 da b. Let D R 2 be the region bounded by y = 0, y = sin (x 2 ), x = 0 and x = π. Compute the double integral D x da a. Compute the area of the region enclosed by one loop of the rose r = 2 cos 4θ b. Find the volume of the region above the paraboloid z = x 2 + y 2 and inside the sphere x 2 + y 2 + z 2 = 2.

7 15.5-a. A lamina occupies the upper half of the unit disk x 2 + y 2 1 and the density at any point is proportional to the square of its distance to the origin. Find the center of mass b. A lamina occupies the lower half of the unit disk and the density at any point is equal to its distance to x-axis. Find the moment of inertia I a. Find the area of the paraboloid z = x 2 + y 2 cut off by the plane z = b. Find the area of the part of the sphere x 2 + y 2 + z 2 = 2z that lies inside the paraboloid z = x 2 + y a. Compute the triple integral E e z y dv with E = {1 y 2, y x 2, 0 z xy} b. Let E be the tetrahedron bounded by the the plane x+y+z = 2, x = 0, y = 0 and z = 0 with density ρ(x, y, z) = x + y + z. Find the center of mass of E a. Find the volume of the solid that lies outside the cylinder x 2 + y 2 = 4 and inside the sphere x 2 + y 2 + z 2 = b. Find the mass of the unit ball x 2 + y 2 + z 2 1 if the density at any point is equal to its distance to the x-axis a. Compute the integral a a 2 x 2 0 a 2 x 2 y 2 +y a 2 x 2 a 2 +z 2 ) dz dy dx. 2 x 2 y 2(x b. Find the volume of the solid that lies within the unit sphere x 2 + y 2 + z 2 = 1, above the xy-plane and below the cone z = x2 + y a. Let F (x, y, z) = x + yz. Find the gradient vector field of f at the point (6, 3, 1) b. A particle moves in a velocity field V(x, y) = y 2, x 2. If it is at position (3, 1) at time t = 2, estimate its location at time t = a. Evaluate C x2 y ds, where C is the top half of the circle x 2 +y 2 = 1, traced counterclockwise b. Evaluate C F dr where F(x, y) = ey i sin(πx)j and C is the triangle with vertices (1, 0), (0, 1), and ( 1, 0) traversed counterclockwise.

8 c. A wire of uniform density consists of a quarter circle in the first quadrant connecting the points (1, 0) and (0, 1), and a line segment from (0, 1) to ( 1, 0). Find its center of mass a. Let F(x, y) = y 2 i + (2xy + cos(2y))j. Determine whether or not F is a conservative vector field. If it is, find a function f such that F = f b. Use Theorem 2 to evaluate C F dr where F = (e2y 2xy)i + (2xe 2y x 2 + 1)j and r(t) = te t i + (1 + t)j, 0 t a. Use Green s Theorem to evaluate C (3x2 + y) dx + (2x + y 3 ) dy, where C is the positively oriented circle x 2 + y 2 = a b. Find the work done by the force F(x, y) = (x 2 y 3 )i + (x 2 + y 2 )j in moving an object around the circle x 2 + y 2 = 1 in the counterclockwise direction c. What is the largest possible value of C y3 dx + (3x x 3 ) dy out of all simple closed curves C in the plane? 16.5-a. Find the curl and the divergence of the vector field F(x, y, z) = xyzi + xzj + zk b. Let r = xi + yj + zk and r = r. Prove that div(r 2 r) = 5r 2. (You may use the result of Exercise 25.) 16.6-a. Find a parametric representation for the part of the plane z = x + 2 that lies inside the cylinder x 2 + y 2 = b. Find the equation of the tangent plane to the surface x = u, y = u 2 v, z = u 3 v 2 at the point (2, 4, 8) c. Find the area of the surface 3z 2 = (x + y) 3 with x + y 2, x 0, y a. Evaluate the surface integral xy ds, where S is the portion S of the plane x + 2y + 3z = 6 in the first octant b. Evaluate the surface integral z ds, where S is the surface S y = z 2 x, 0 x 1, 0 z c. Calculate the flux of the vector field F = xi + yj across the sphere x 2 + y 2 + z 2 = a a. Use Stokes Theorem to evaluate curl F ds where F = S xzi + yzj + xyz 2 k and S is the cap of the paraboloid z = 5 x 2 y 2 above the plane z = 3, oriented upward.

9 16.8-b. Use Stokes Theorem to evaluate F dr where F = zi zj + C (x 2 y 2 )k and C consists of the three line segments that bound the plane z = 8 4x 2y, oriented counterclockwise when viewed from above c. Verify Stokes Theorem for F = 3yi + 3xj + z 4 k with S the portion of the ellipsoid 2x 2 + 2y 2 + z 2 = 1 that lies above the plane z = 1/ 2, by computing both the line integral and the surface integral a. Use the Divergence Theorem to compute the flux of the field F = x 2 i + y 2 j + z 2 k out of the box 0 x 1, 0 y 2, 0 y b. Verify the Divergence Theorem for the field F = 3x, 3y for the ball of radius 1 centered at the origin c. Use the Divergence Theorem to find the flux of the field F = y, x z, y across the surface S consisting of the hemisphere x 2 +y 2 +z 2 = 1, z 0, together with the disk z = 0, x 2 +y

Math 233. Practice Problems Chapter 15. i j k

Math 233. Practice Problems Chapter 15. i j k Math 233. Practice Problems hapter 15 1. ompute the curl and divergence of the vector field F given by F (4 cos(x 2 ) 2y)i + (4 sin(y 2 ) + 6x)j + (6x 2 y 6x + 4e 3z )k olution: The curl of F is computed