Fall 2010, Math 321 TA page

PART 1. Vector algebra: geometric concepts and index notation

• HOMEWORK 1 to 10 on PART 1

PART 2. Vector Calculus: curves, surfaces, volumes, gradient, divergence, curl, ...

Numbered exercises refer to the Vector Calculus notes on the Math 321 home page that you have been studying. There are no problems to hand in. You should work on the problems listed below in preparation for exams. `No pain, no gain'. If you get someone else to solve the problems for you, you are not learning. If you are not struggling with some of the material, you are not learning.

• Homework 11, Thur. 10/14/10 : Vector functions of one scalar variable, e.g. r(t), r(s), v(t). Vector notes sections 1.9, 1.10 (we will be skipping sections 1.11 and 1.12, this would be covered in a mechanics class like Physics 311) and Vector Calculus notes sections 2.1, 2.2.
  1. Review Chapter 13 in Thomas' Calculus (Vector Valued functions and motion in space) (Calculus 222) MOST OF THIS SHOULD BE a `REVIEW'!
  2. Concept of curve, parametric equations of curves r=r(t). Meaning of t. Concept of arclength often denoted "s". Concept of arclength parametrization r=r(s).
  3. What is the form of r=r(t) for (i) a straight line in 3D space, (ii) a parabola in 3D space, (iii) a circle in 3D space, (iv) an ellipse in 3D space?
  4. What is a closed curve? What can you say about r(t) for a closed curve?
  5. Must know and understand arclength ds = |dr| ≠ d|r| = dr, unit tangent T = dr/ds, unit normal N such that dT/ds= κ N = N/R, where κ=1/R is the curvature and R is the radius of curvature.
  6. Must know and understand relationships between velocity and acceleration and tangent and normal to the curve; curvature and radius of curvature.
  7. Given r(t) where t is time, how do you compute the velocity? the speed? the acceleration? the unit tangent? the unit normal to the curve? the curvature? the radius of curvature? Calculate all those quantities for (i) a line, (ii) a circle in 3D space.
  8. Given r₀, r₁, ..., r_N, sketch the curve r(t) defined such that r(t)=(1-u) r_n-1 + u r_n for n-1 ≤ t < n, with u = t-n+1 and n=1,..., N.
  9. Quadratic Bézier curve: Given the radius vectors r₀, r₁, r₂ of points P₀, P₁, P₂, respectively, define points A₀, A₁ and P such that they divide the segments P₀P₁, P₁P₂ and A₀A₁, respectively, into equal ratios, that is P₀A₀/P₀P₁ = P₁A₁ / P₁P₂ = A₀P / A₀A₁ ≡ t, with 0 ≤ t ≤ 1. Derive the equation that expresses r(t), the radius vector of P, in terms of r₀, r₁, r₂. Make a sketch showing all 6 points and r(t) for t=1/3 and another for t=1/2 (for example).
  10. What kind of abstract mathematician was Pierre Bézier ?! Did he use these curves to torture Math 321 students?
  11. Cubic Bézier curve: Given r₀, r₁, r₂, r₃, consider the curve defined by r(t) = (1-t)³ r₀ + 3 (1-t)² t r₁ + 3 (1-t) t² r₂ + t³ r₃ with 0 ≤ t ≤ 1.
      • Expand (a+b)³. Now let a=(1-t) and b=t so (a+b)=1. The polynomials (1-t)³, 3 (1-t)² t, 3 (1-t) t², t³ are the Bernstein polynomials of degree 3. PLOT them for t=0 to 1.
      • Show that (1-t)³ + 3 (1-t) t² + t³= 1- 3 (1-t)² t   (don't do that the stupid way).
      Show that the curve passes through points P₀ and P₃ (easy) but does not pass through P₁ and P₂ (less easy) in general (e.g. pick r₀=r₂=r₃ and use the Bernstein polynomial results above). Calculate the directions of the tangents to the curve at points P₀ and P₃, interpret your results. Show that r(t) can be computed with the following recursive algorithm: ( de Casteljau's algorithm )
      1. a₀ = (1-t) r₀ + t r₁;   a₁ = (1-t) r₁ + t r₂;   a₂ = (1-t) r₂ + t r₃;  
      2. b₀ = (1-t) a₀ + t a₁;   b₁ = (1-t) a₁ + t a₂;  
      3. r = (1-t) b₀ + t b₁;
  12. What can you say about v(t) if v . dv/dt = 0 for all t?
  13. Let r=r(t) represent an ellipse in a 2D plane through the origin O which is outside of the ellipse. The center C of the ellipse is such that the vector OC is NOT parallel to the major or minor axes of the ellipse. Make a sketch. Indicate all points on your sketch where r . dr/dt = 0.
  14. Solve dv/dt = a where a is a constant.
  15. Solve d²r/dt² = a where a is a constant. What kind of curve is r(t)?
  16. Solve da/dt = a × b where b is a constant. Is a constant? Can a be constant? Make clean sketches that shed light on this vector differential equation. Write this vector diff eq in index notation with respect to a fixed cartesian basis. Write the index equations in matrix-vector notation. What kind of curve is described by the head of a(t) when its tail is at a fixed point?
  17. Suppose you know a and da/dt. Derive expressions for d|a|/dt and dâ/dt in terms of a and da/dt. (Recall that |a| is the magnitude of vector a and â=a/|a| is its (unit) direction vector). Do your equations say what you expect them to say? Make sketches to interpret your results.
  18. A+: Bead on a wire: a bead of mass m is launched with initial speed v₀ along a fixed wire. What is the force on the wire according to Newton's law F=ma if there is no friction and no external force acting on the bead (no gravity, etc.)? Formulate and solve the problem. OK, maybe you are not a mechanical engineer, but kids play with toys of that kind: beads on wire frames. Make a sketch and clearly define everything you need to formulate the problem.

• Homework 12, Tue. 10/19/10 : Vector Calculus , Sections 1.1, 1.2: curves and integrals along curves (a.k.a. `line integrals').
  1. Wait! isn't an integral the ``area under a curve?'' how can we have an integral along a curve? what does that mean?
  2. Study each line of section 1.2.
  3. Exercises 1.2. 4, 5, 6 Visualize the problems! What is dr? what is F? what is F . dr ?
  4. A curve is parametrized by the cartesian x coordinate, i.e. r=r(x) = x e_x + y(x) e_y + z(x) e_z with a ≤ x ≤ b, what is the x-integral for the length of that curve? Derive the formula from first principles, don't just look it up elsewhere.
  5. A planar curve is defined as u=u(v), where u is distance of point P to the origin O, and v is the angle between the line OP and a reference direction. Derive a v-integral for (i) the length of the curve, (ii) the area swept by the radius vector (without looking them up elsewhere).

• Homework 13, Thur. 10/21/10 : Vector Calculus , Sections 1.1, 1.2.
  • Homework 12 again + Exercise 1.2.1, 2 but on your own this time, without looking at your class notes.

• Homework 14, Tue. 10/26/10 : Vector Calculus , Sections 1.3, 1.4: Surfaces, surface element, surface line element, surface integrals
  1. Study sections 1.3, 1.4. (do not simply `look at' or `overlook'). It may help to review sect. 16.6 in Thomas' Calculus.
  2. Exercises 1.3. 1, 2, 3, 5.
     In #5, you are expected to deduce the meaning of `outer radius' R and `inner radius' a from the equations that are given. They may not be what you think, but they are what the equations say.
  3. Exercises 1.4. 1, 2, 3, 4. Visualize the problems! What is dS? what is r . dS ?

  4. A+ (Advanced: i.e. only try if previous problems are done and `too easy') Curves on surfaces. Given points A and B on a sphere of radius R (centered at O), can you explain why the curve that provides the shortest distance between A and B on the sphere, satisfies the equation d²r/ds² = r/R² where s is arclength along the curve? What is the geometric meaning of d²r/ds²? (Homework 11). Can you prove that the shortest distance between two points on a sphere is the arc of great circle through those 2 points? Don't compute too much, think.
  5. How do we represent surfaces in the real world? Example: Icosahedral grid for a sphere

• Homework 15, Thur. 10/28/10 : Vector Calculus , Sections 1.5, 1.6, 1.7: Volumes, parametrizations, Jacobian determinant, curvilinear/generalized coordinates, mappings, change of variables.
  1. Study sections 1.5, 1.6, 1.7. It may help to review sect. 15.7 in Thomas' Calculus. (except that they use the `American mathematician' notation for spherical coords, not ISO 31-11 . Come to think of it, that `American Mathematician' was probably George Thomas!).
  2. Exercises 1.5. 1, 2, 3, 4
  3. (Simplified version of 1.6.1) Calculate the coordinate displacement vectors ∂r/∂r, ∂r/∂θ, ∂r/∂φ for spherical coords using both the hybrid approach and the direct geometric approach as sketched in class. What are r̂, θ̂ and φ̂ in terms of x̂, ŷ and ẑ? Are the coordinates orthogonal? (what does that mean?) Are there coordinate singularities? (what does that mean?) Calculate dS in terms of θ and φ for a spherical surface of radius R. Calculate dV, the volume element in physical space, in terms of r, θ, φ. Calculate the surface area and the volume of a sphere from those results.
  4. Use a hybrid or geometric approach to calculate ∂r̂/∂r, ∂r̂/∂θ, ∂r̂/∂φ, ∂θ̂/∂r, ∂θ̂/∂θ, ∂θ̂/∂φ, ∂φ̂/∂r,∂φ̂/∂θ, ∂φ̂/∂φ. ( θ̂ is θ-hat with the hat not showing well :-( Make sketches.
  5. (rewrite of 1.6.2) A curve in the (x,y) plane is specified in polar coordinates as ρ=ρ(φ) where x=ρ cos φ and y=ρ sin φ. Sketch. Write the 2D radius vector r in terms of the polar coordinates ρ and φ. Find the coordinate displacement vectors ∂r/∂&rho, and ∂r/∂φ using a hybrid approach and a direct geometric approach. Make a sketch showing x,y, ρ, φ as well as the direction vectors x̂, ŷ, ρ̂ and φ̂. Derive an explicit φ integral for the length of the curve in terms of the function &rho(φ).
  6. Exercises 1.6. 3, 4
  7. Exercises 1.7. review change of variables in multiple integrals in your prerequisite course on multivariable calculus and study section 1.7 first, then work on 1, 2, 4 (nice and easy!), 5 (a classic!), 6

• Homework 16, Tue. 11/2/10 : Vector Calculus , Sections 2.1, 2.2: Rate of change of a scalar field along a curve: f(r(t)), f(r(s)) , Directional derivatives, Gradient, Geometric meaning of gradient, gradient of spherical functions, gradient in cylindrical and spherical coordinates .
  1. Study lecture notes, study problems solved in class and in the notes. There are some Extra notes on GRADIENT .
  2. Concept of an isosurface and isocurve (or contour line) (e.g. isobars, isotherms, isopycnals, isobaths, ...)
  3. Consider f(r)= exp(-[(x-x₀)²+(y-y₀)²+(z-z₀)²]/a²), where x₀, y₀, z₀, a are constants. Calculate ∇f, using both cartesian coordinates and a direct geometric approach. Do you get the same result?
  4. Express ∇[f(r) g(r)] in terms of ∇f and ∇g.
  5. Consider f(r)= x exp(-[x²+y²+z²]/a²), where a is constant. Calculate ∇f. Express ∇f in a compact form. [Hint: use the product rule and compute easy gradients]
  6. Consider f(r)= z exp(-[x²+y²]/a²), where a is constant. Calculate ∇f. Express ∇f in a compact form. [Hint: see previous hint]
  7. Use a direct geometric approach to calculate ∇r, ∇θ and ∇φ where r is distance to the origin, θ is the polar angle (co-latitude) and φ is the azimuthal angle (longitude). Check your results with the expression for the gradient in spherical coords that was derived in class (and in the Extra notes on GRADIENT ). Verify the chain rule for gradients: ∇f(r,θ,φ) = ∂f/∂r ∇r + ∂f/∂θ ∇θ + ∂f/∂φ ∇φ. Describe in words/sketch the isosurfaces of r, θ and φ.
     Can you derive all these formulas? How about these formulas?. you should be able to now if you understood homework 15 and 16. (NOTE: don't trust everything you read on wikipedia or the internet in general, unfortunately. In particular ∇ ("del") is defined in spherical coordinates. )
  8. Consider f(r) = r^-1 sin θ cos φ , where r, θ, φ are spherical coordinates. Calculate ∇f. (OK, this is a bit of a tricky one...)

• Homework 17, Thur. 11/4/10 : Vector Calculus , Sections 2.3, 2.4: Divergence, curl, vector identities You should know how to rederive vector identities quickly using vector notation as well as index notation in cartesian coords.
  1. In order to develop intuition about divergence and curl, it is useful to compute div and curl for some simple 2D vector fields. You should sketch the fields and compute their div and curl. Here x, y are the usual cartesian coordinates in the orthogonal directions x̂ and ŷ respectively. (ρ and φ are the cylindrical (polar) coordinates defined by x= ρ cos φ and y=ρ sin φ. You're on your own for the direction vectors ρ̂ and φ̂. SKETCH v(r). CALCULATE ∇.v and ∇×v. DIGEST the results, how do they fit with your sketches?
     1. v= x x̂ + yŷ = ρ ρ̂.
     2. v= x x̂.
     3. v= y x̂.
     4. v= x x̂ - y ŷ.
     5. v= y x̂ - x ŷ = ρ φ̂.
  2. Exercises 2.4. All. Several of these exercises have been done in earlier sections and/or in class.
  3. Swiftly calculate ∇ x (ω x r) where ω is constant and r is the radius vector.
  4. Calculate (φ̂∂/∂φ) . (u r̂) where u=u(r,θ,φ) and r,θ,φ are our usual spherical coordinates.
  5. Calculate (φ̂∂/∂φ) . (u ρ̂) where u=u(ρ,φ,z) and ρ,φ,z are cylindrical ccoordinates with ρ² = x² +y².
  6. What is ∇ in cylindrical coordinates (ρ,φ,z)? What are ∇.v and ∇×v in cylindrical coordinates?
  7. Write the vector fields v=v(r)= (ẑ×r) and B=B(r) = v/|v|² in cartesian coords AND in cylindrical coords. SKETCH THE VECTOR FIELDS. Calculate ∇.v, ∇×v, ∇.B and ∇×B using both cartesian and cylindrical coords. Calculate these quantities using vector identities also. (So three different methods!).
  8. If ∇²= ∇.∇ (i.e. Laplacian = del dot del) expand out ∇²(f g) where f=f(r) and g=g(r) are arbitrary scalar fields (twice continuously differentiable).

• Homework 18, Tue. 11/9/10 : Vector Calculus , Sections 3.1 -3.5. Fundamental Theorem of Calculus (FTC). Multi-D versions of the theorem.
  Discussed in class today: 3.1, 3.2 + a little extra on FTC for line integrals and a nice application of formula (52), 3.3, then 3.5 formula (103) and (104) then to (106) and back to (84). Divergence and curl forms of Green's theorem.
  1. What's going on in eqns (73) and (74)?
  2. What is the `Fundamental Theorem of Calculus'? State the theorem and justify it.
  3. Prove/derive (81) and/or (83) and/or (84).
  4. What's the connection, if any, between (85) and (84)?
  5. State the divergence theorem (Gauss) in complete mathematical terms (i.e. do not write a novel about it, write the theorem mathematically and define all its ingredients)
  6. State the curl theorem (Stokes) in complete mathematical terms (i.e. do not write a novel about it, write the theorem mathematically and define all its ingredients)
  7. Exercises 3.7. 1, 2, 3, 4 (with n = ω ), 5--11.

• EXAM 2 is this THURSDAY NOV 11, 11:00-12:15 in 1257 Comp Sci (last names A to F), or in 121 Psych (G-Z).

• Homework 19, Tue. 11/16/10 : Vector Calculus , Curl and divergence forms of Green's theorem. Interpretation of curl and div.
  • Study equations (85), (86), (87) in the notes. (study= read, think about, analyze, understand, learn, apply).
  • Study equations (106), (111), (112) in the notes. (study= read, think about, analyze, understand, learn, apply).
  • Study equations (113)--(120) in the notes. (study= read, think about, analyze, understand, learn, apply).
  • Exercises 3.7. 1, 2, 3, 4 (with n = ω ), 5--11. #3 was discussed ad nauseum in lecture today.

• Homework 20, Thur. 11/18/10 : Vector Calculus ,Fundamental Theorems of Vector Calculus.
  • Section 3.7: the Peng Qi Theorem (also known as Fundamental Theorem of Calculus) equation (113). Digest it. Deduce the divergence theorem from it, then also the `gradient' theorem, the `curl' theorem, the `directional derivative' theorem (eqn (110)), etc.
  • Explain why eqn. (113) implies eqn. (110).
  • Explain why eqn. (110) implies eqn. (113).
  • Explain why eqn. (113) implies eqns (115) and (118).
  • Exercises 3.7. 5, 6, 7, 9, 10, 11 were essentially solved in lecture today, but all that needs to be digested by you. (Should take less than 20 years, but more than 20 minutes!).
  • Show that the net flux of the curl of a vector field over a closed surface is zero. Huh?! What is even the question? write it in math form.
  • Show that ∯_S n × ∇f   dS = 0 for any (sufficiently differentiable) scalar field f and closed surface S, where n is the unit outward normal to the surface. (well OK, not quite any closed surface, not this kind of crazy surface, which you can read more about here . The surface S must be orientable .)
  • Suppose S is the outer surface of a cylinder (without the end disks), like a sheet of paper rolled up into a cylinder. If we orient S such that its normal is pointing outward, what is the boundary of S and how should it be oriented for Stokes theorem?
  • Consider the vector field v =v(r) = ∮_C dr' × (r- r')/|r- r'|³, where C is a closed curve. Show that ∇×v =0 and ∇∙v=0, except on C.