MATH 241, Spring 2014: Calculus III

News

• Matlab Project 2, due May 13 in the discussion.
  Note: I will drop the lowest of the two Matlab projects. So you don't have to do this if you are satisfied with the score of the first project.
  Hints:
  Problem 1: See Examples for 12.3 for finding r(t), the distance, and plotting the trajectory.
  See Examples for 13.9 for a Lagrange multiplier example.
  Problem 2: See Examples for 14.5 for plotting a surface over a vertically simple region using ezsurfvs, and plotting a surface over a region given in polar coordinates using ezsurfpol. This also shows how to compute the volumes.
• Exam 4 was on Wednesday, May 7. It covered chapter 15 (except 15.7)
  Solution of Exam 4, m-file
  • Topics for exam 4 (what I covered during review on Fri, May 2)
  • Practice problems (problems I did in class on Fri, May 2)
    solutions
• Practice problems for exam 3 (with solutions)
• Exam 3 was on Wednesday, April 23. It covered sections 14.1-14.9.
  Solution of Exam 3, m-file
• Exam 2 was on Monday, March 31. It covered sections 13.1-13.9.
  Solution of Exam 2, m-file
• What you should do now:
  Do the recommended problems for sections 11.1, 11.2., 11.3, 11.4, 11.5, 11.6, 12.1, 12.2, 12.3, 12.4, 12.5, 12.6, 13.1, 13.2, 13.3, 13.7, 13.4, 13.5, 13.6, 13.8, 13.9, 14.1, 14.2, 14.3, 14.4, 14.5, 14.6, 14.7, 14.8, 14.9, 15.1, 15.2, 15.3, 15.4, 15.5, 15.6, 15.8
• Practice problems for Exam 1, Solutions
• Exam 1 was on Wednesday, February 26. It covered Chapter 11 and 12.
  Topics:
  • dot product, projection pr_ab, decomposition b=b_par+b_orth
  • cross product, determinant of 2×2 and 3×3 matrix, area of parallelogram & triangle, volume of parallelepiped & tetrahedron
  • lines: parametric form, symmetric form, for a point P find distance to line & closest point on line
  • planes: equation of plane, for a point P find distance to plane & closest point on line; intersection of planes
  • vector valued functions. Position r(t), velocity v(t), speed V(t), acceleration a(t). For given acceleration a(t) and initial position r(0) & velocity v(0) find position r(t).
  • curves: length, decomposition a(t) = a_TT + a_NN with a_T=V'(t), a_N=κV² and curvature κ
• For the Matlab homeworks you can work together in teams of up to three students. Teams of four or more students will receive zero points. All team members must have the same TA.
  Note the instructions for preparing Matlab homeworks above.
  Your solutions should look like the examples below in "What I did in class". All the Matlab code you need for the homework is explained in these examples. You should also read the Matlab instructions for MATH241.
• Matlab Project 1: Due date extended to Thu, Feb. 20 in the discussion session. (The homework was posted and announced by e-mail on Monday, Feb. 3)
  You have to download the m-files from the Matlab instructions page and put them in the same directory (folder) as your other m-files.
  Problem 1(b): The parellograms must have the same size and shape as the one from (a). There are twelve of them, and they form the surface of a convex region. In 1(c) you have to find the volume of this region.
  Problem 1(c): The volume of a tetrahedron with vertices (0,0,0), a, b, c is given by det(M)/6 where M is the matrix with rows a, b, c (see dot product, cross product, determinants).
  Problem 2(b):
  First find the closest point in the plane of the triangle. If this point is inside the triangle you are done.
  If this point is outside of the triangle: Find the closest points on each of the three lines given by the sides of the triangle. Find the distance to those points (but points outside of the triangle can't be used). Also check the distance to the three vertices of the triangle.
• MATH241 Tutoring has started, see schedule.
  MATH241 Matlab Tutoring has started, see schedule.
• What I did in class: (in order to run the m-files below, you first have to download the m-files from the Matlab instructions)
  • 11.1, 11.2: vectors, norm of vector, sphere, addition of vectors, multiplication by scalars, unit vector,
    P+t·(Q-P) describes line through points P and Q
    example, m-file
  • 11.3: dot product: finding angle between two vectors, proj_ab projection of one vector onto another vector, decomposing vector b = b_par + b_orth where b_par is parallel to a, b_orth is orthogonal on a
    examples, m-file
  • 11.4: area of parallelogram, 2-dimensional case: determinant, 3-dimensional case: cross product,
    Read Dot product, cross product, determinants.
    examples, m-file
  • 11.4: volume of parallelepiped, determinant of 3 by 3 matrix, orientation of vectors a, b, c
    11.5: lines in space: parametric form, symmetric form, finding distance of point to line using (i) projection, (ii) cross product
    examples, m-file
  • 11.6: planes in space: given point P₀ and normal vector N we get equation N₁x+N₂y+N₃z=D; distance of plane to (0,0,0); distance of plane to point P; intersection of two planes
    examples, m-file
  • 12.1, 12.2, 12.3: Vector-valued functions, limits, continuity, derivatives, integrals, velocity, acceleration, Newton's law
    Typo in class: It should have been 0 = t - g·t²/2 = t·(1 - t·g/2), hence 1 - t·g/2 = 0 and t = 2/g, therefore distance = 2/g:
    examples, m-file
    Run the animated versions ex6a.m, ex6b.m in Matlab
  • 12.4: curve vs. parametrization, smooth curve, piecewise smooth curve, length of curve
    examples, m-file
    Run the animated versions ex7a.m, ex7b.m in Matlab
  • 12.5, 12.6: unit tangent vector, normal vector, tangential and normal component of acceleration, curvature
    Read Curves: Length, Tangent and Normal Vector, Curvature and do the problems. Solutions of the problems
    examples, m-file
  • Review of 12.5, 12.6: formulas for a_N and curvature κ=a_N/V², binormal vector, osculating circle
    13.1: Functions of several variables: function f(x,y), domain, graph of function, level curves of function
    examples, m-file
  • 13.1: Graph and level curves for function f(x,y) of 2 variables, level sets (surfaces) for function f(x,y,z) of 3 variables
    13.2: Limit of f(x,y) as (x,y) tends to (x₀,y₀), Definition: For any given z-tolerance ε>0 we can find (x,y) tolerance δ>0; continuity of a function f(x,y)
    examples, m-file
  • 13.2: Review of limits, continuity
    13.3: Partial derivatives
    examples, m-file
  • Review for exam (Ch. 11, 12)
    13.3, 13.7: Partial derivatives, tangent vectors to graph, tangent plane, differentiability
    examples, m-file
  • 13.7, 13.5, 13.6, 13.4: Review: tangent plane, differentiability, directional derivative D_af(x₀,y₀)
    For differentiable function f:
    gradient g:=grad f(x₀,y₀), D_af(x₀,y₀) = g·a
    directional derivative is maximal for a=g/||g||, is minimal for a=-g/||g||, is zero for a orthogonal on g
    Chain rule: For F(t)=f(x(t),y(t)) there holds F'(t) = f_x(x₀,y₀)x'(t₀) + f_y(x₀,y₀)y'(t₀) = g·r'(t₀)
    examples, m-file
  • Examples for chain rule: F(t)=f(X(t),Y(t)), F(u,v)=f(X(u,v),Y(u,v)), F(t) = f(X(t),Y(t),Z(t)); gradient vector for function f(x,y,z), tangent plane to level surface
    examples, m-file
  • 13.8: relative maximum/mininum, gradient in relative extremum, critical points, recipe for finding maximum and minimum of f(x,y) over a region R
    examples, m-file
  • 13.8: Second partials test
    examples, m-file
  • 13.9: Lagrange multipliers
    example for f(x,y), m-file
    example for f(x,y,z), m-file
  • 14.1: Double integrals
    examples, m-file
  • 14.2: Double integrals in polar coordinates
    examples, m-file
  • Exam 2:
    solution, m-file
  • 14.3: Surface area
    14.4: Triple integrals
    examples, m-file
  • 14.5: Cylindrical coordinates
    examples, m-file
  • 14.6: Spherical coordinates
    14.7: center of mass (aka center of gravity)
    examples, m-file
  • 14.8: Change of variables
    examples, m-file
  • 14.9: Parametrized surfaces
    examples, m-file
  • 14.9: Area of parametrized surfaces
    examples, m-file
  • 15.1: Vector fields, conservative vector fields, curl of a vector field, finding the potential of a vector field
    examples, m-file
    curl and divergence of a 2D vector field, m-file
    Example where curl F is zero, yet there exists no potential, m-file
    
  • 15.2: "Work integral" for vector field and oriented curve:   dr = r' dt
    15.3: Work integral for conservative vector fields for oriented curve from A to B
    "Fundamental Theorem of Line Integrals": ∫_Cgrad f·dr = f(B)-f(A)
    examples, m-file
  • 15:2: Line integral for scalar function f:   ds = ||r'||dt
    15.4: Green's theorem:   ∫∫_R(F_2x-F_1y)dA = ∫_C(F₁,F₂)·(dx,dy) (interpretation as curl and work integral  ∫_CF·dr )
    Green's theorem: example for curl and div version of the theorem , m-file
  • 15.4: Green's theorem:   ∫∫_R(G_1x+G_2y)dA = ∫_C(G₁,G₂)·(dy,-dx) (interpretation as divergence and flux integral  ∫_CG·n ds )
    15.5: surface integral for scalar f:   dS = ||r_u×r_v||du dv
    15.6: flux surface integral for vector field F:     n dS = ±(r_u×r_v)du dv (use "-" if n and r_u×r_v point in opposite directions)
    15.8: divergence theorem:   ∫∫∫_Ddiv F dV = ∫∫_ΣF·n dS