CHAPTER 1: Vectors.

• Homework 1, Wed. 1/20/2010 : STUDY SECTIONS 1.1, 1.2 focusing on geometric aspects
  • Who was Gibbs? When and where did he live? What did he do in science and math?
  • What is Gibbs notation? How do we denote a vector? the magnitude of a vector? the direction of a vector?
  • What are the units of a unit vector? What is the length of a unit vector? in what units of length?
  • What is a scalar ? Can a scalar be a function of time and/or position, or is it always a constant?
  • What is the `radius vector' r of a point P? How do we express it in cartesian, cylindrical and spherical coordinates? You should be able to derive all the formulas `from scratch'. It's not hard if you understand the coordinates, learn to visualize properly (often by taking the appropriate orthogonal projection onto a plane like the x, y plane or the ρ, z plane), and ... you know your basic trig!
  • Make a simple 3D perspective sketch showing all 3 sets of coordinates without just looking them up in your notes or a book
  • What are the cylindrical and spherical coordinates for the points whose cartesian coordinates are (0,0,1), (0,-1,0), (1,1,1)?
  • If we want a unique set of coordinates for each point in 3D space, what range of values do we need for x, y, z, ρ, r, φ, θ ?
  • Express ρ and r in terms of x, y, z . Express x, y, z in terms of the cylindrical coords ρ, φ, z . Express x, y, z in terms of spherical coordinates r, θ, φ .
  • Express the direction vectors in the directions ρ and r (that is `rho-hat' and `r-hat' or e_ρ and e_r) in terms of the cartesian direction vectors.
  • Wikipedia on spherical coordinates (pretty good today... also contains cylindrical coords)
  • Visualize/sketch the 8 properties (1)--(8) of vector addition and multiplication by a real number.
  • What is a vector basis? Is a basis vector a unit vector? Are basis vectors perpendicular to each other?
  • Points A, B and C have cartesian coordinates (A₁,A₂,A₃), (B₁,B₂,B₃) and (C₁,C₂,C₃), respectively. What are the coordinates of the midpoint M of the segment BC? What is the equation of the line that passes through A and M?
  • Exercises: 1.2. 1 , 5, 6, 7, 8. (In 1.2.1(i) both α and β are between 0 and 1)

• Homework 2, Fri. 1/22/2010 : STUDY SECTIONS 1.1, 1.2 of the vector notes , focusing your study on geometric aspects
  • Visualize --- i.e. make a mental picture and one or more clean sketches --- the 8 properties (1)--(8) of vector addition and vector scaling ( i.e. multiplication by a real number) for displacements in 3D space.
  • Do you understand reference points and basis vectors? Points A, B and C have cartesian coordinates (A₁,A₂,A₃), (B₁,B₂,B₃) and (C₁,C₂,C₃), respectively. Write an explicit expression for the vector OA. What are the cartesian coordinates of the midpoint M of the segment BC? What is the equation of the line that passes through A and M? What are the cartesian coordinates of the midpoint N of segment AC? (yep, AC, not AB as in class) What is the equation of the line that passes through B and N? What are the cartesian coordinates of the intersection of AM and BN? Solved in class, today, 1/22/2010 Try to solve this problem from scratch, on your own, without looking at your class notes. Then compare to/study in-class solution. Refine/trim the solution. Try to repeat using different notation, different reference points, ...
  • Show/Explain that the solution we did in class also solves problem 1.2. 5, 6 and 7 of the vector notes! Wait! it also helps a lot for 1.2.8, wow! good stuff.
  • What is the vector equation of a line? What does that equation look like when expressed in terms of cartesian coordinates x, y, z ?
    In Thomas' Calculus, that's in section 12.5. See the box `Vector equation of a line' and make sure to digest the concept, not just memorize a formula you don't understand.
  • OK, you understand how to add vectors in 3D space, but how do you subtract vectors? Pick 2 arbitrary vectors a and b, make sketches showing a+b and a-b. Learn to quickly recognize/reconstruct a+b and a-b. Use vector methods to show that the diagonals of a parallelogram intersect at their midpoint.

• Homework 3, Mon. 1/25/2010 : STUDY SECTIONS 1.3, 1.4 of the vector notes (you *may* skip the definition of dot product and norm in Rⁿ on pages 9 and 10)
  • Watch out! UFO, 2 miles, azimuth 030 deg, inclination 20 deg (= elevation 70 deg), how far East of you is the UFO?
  • Watch out! UFO, 2 miles, azimuth 030 deg, inclination 20 deg, heading 190 deg, constant altitude. How close to you will it pass?
  • What is the equation of the plane perpendicular to direction D, passing through point A? Formulate a vector equation and spell it out in good old cartesian coords. How would you specify explicit data for D and A?
  • What is the vector equation of a plane? What does that equation look like when expressed in terms of cartesian coordinates x, y, z ?
    In Thomas' Calculus, that's in section 12.5. See the box `Equation for a plane', you should understand the basic concepts and how to express them using vectors, then be able to derive the equation in cartesian coordinates. You should also be able to go from cartesian coordinates back to the vector equation.
  • You know the (non zero) lengths a₁, a₂, a₃ of three vectors a₁, a₂, a₃, respectively. You also know the angle α₁ between a₂ and a₃, the angle α₂ between a₃ and a₁, and the angle α₃ between a₁ and a₂. (1) What conditions must the angles satisfy for the vectors to be linearly independent? (2) Assume those conditions are satisfied, and that 2 vectors u and v are given to you in terms of the basis a₁, a₂, a₃, that is u = u₁a₁ + u₂a₂ + u₃a₃ and v = v₁a₁ + v₂a₂ + v₃a₃ with (u₁,u₂,u₃) and (v₁,v₂,v₃) known. Express the dot product u.v in terms of the known data (derive a formula).
  • Exercises: 1.3. 1, 2, 3, 7, 8, 9
  • Given three point A, B, C in 3D space, use vectors to find the intersection of the 3 heights (these are the lines passing through one vertex, perpendicular to the opposite side). Write an algorithm to find the intersection, i.e. a pseudo computer program. You can assume that the points are specified in cartesian coordinates.
  • Section 1.4.: understand and learn formula (14) and (15)
  • Exercises: 1.4. 1, 2, 3, 4, 5
  • Consider 3 points A, B, C lying on a circle. If AB is a diameter of that circle show that the angle ACB is a right angle. (Hint: use vectors and try to not use the next hint!) (Next Hint: let O be the middle of AB, then use vectors OA and OC).
  • Homework 3 extras, Tue. 1/26/2010 :
    • Find an (x,y,z) that satisfies a x + b y + c z = d and minimizes x²+ y² + z². Interpret geometrically.
    • Find an (x,y,z) that satisfies a x + b y + c z = d and minimizes the Euclidean distance from (x,y,z) to (x₀,y₀,z₀). Interpret geometrically.
    • What is the distance between the point with cartesian coords (a,b,c) and the plane perpendicular to the vector with cartesian coords (v₁,v₂,v₃) that passes through the point (x₀,y₀,z₀)?

• Homework 4, Wed. 1/27/2010 : STUDY SECTIONS 1.5 and 1.6.1 of the vector notes
  • Exercises: 1.5. 1, 2, 3, 4, 5, 7, 8
  • Consider an arbitrary tetrahedron in 3D space. Label the vertices A, B, C and D in such a way that the vectors AB, AC and AD are left handed. This can always be done, right? Now define 4 area vectors, one for each face of the tetrahedron, such that the magnitude of an area vector is equal to the triangular area of the face and its direction perpendicular to that face pointing out of the tetrahedron . Prove that the sum of those 4 area vectors is zero, for any tetrahedron.
    What if you glue two tetrahedra together along a common triangular face? you get a polyhedron with 8-2 = 6 triangular faces. What is the sum of the 6 outward area vectors? What if you keep on gluing tetrahedra? Each time you glue, you add 4 triangular faces but lose 2. What if you take any polyhedron and divide each of its faces into triangles?
  • The area of the triangle P₁, P₂, P₃ is given by the magnitude of 1/2 vec(P₁P₂) x vec(P₁P₃) (right?). If O is any point in the same plane, show geometrically and algebraically that the area is also given by the magnitude of 1/2 vec(OP₁) x vec(OP₂) + 1/2 vec(OP₂) x vec(OP₃) + 1/2 vec(OP₃) x vec(OP₁). Is that true also if O is NOT in the same plane?
    If (x₁, y₁), (x₂, y₂), (x₃, y₃), are the cartesian coordinates of three arbitrary points in the xy plane, prove that the area of the triangle that they form is +/- 1/2 of (x₁ y₂-x₂ y₁) + (x₂ y₃-x₃ y₂) + (x₃ y₁-x₁ y₃) . Discuss the meaning of the sign. What if you `glue' triangles together? Could you derive a formula for the area of an arbitrary n-gon?

• Homework 5, Fri. 1/29/2010 : STUDY double cross and mixed products in SECTIONS 1.5 and 1.7 of the vector notes
  • Exercises: 1.7. 1, 2, 3, 7, 8, 9, 10, 11, 12
  • Digest what you can of section 1.6.1
  • Section 1.8: we've covered all the concepts, you should be able to use them to solve all those problems and similar problems. If all the data is given to you in cartesian coordinates, what would those problems look like?

• Homework 6, Mon. 2/1/2010 : INDEX NOTATION: Study section 1.6 of vector notes
  • Study section 1.6, don't just skip it and go straight to the exercises or the TA. Study every line, every expression, every equation. Does this mean you should memorize everything?!? What does it mean to study equation (38) for instance?
  • Write a×b, (a×b)_m, (a×b) ×c, ((a×b) ×c)_m and ((a×b) ×c) .e_i in index notation using Einstein's summation convention.
  • Evaluate ε_ijk ε_ijl
  • True or False: ε_ijk a_i b_j + ε_klm a_m b_l = 0. WHY?

• Homework 7, Wed. 2/3/2010 : INDEX NOTATION: Study section 1.6 of vector notes
  • Study section 1.6, don't just skip it and go straight to the exercises or the TA. Do you know and understand equation (39) for instance? Could you derive/show it `from scratch'? Can you reconstruct it quickly for different vectors and for different components (say, c×r)?
  • Write the position vector r in index notation.
  • Could you derive/show eqns (54), (55) and (56) `from scratch' and can you reconstruct it quickly for different vectors and with different indices? (Warning: the `answer' to this is not just `yes' or `no'...)
  • You are responsible for section 1.8. We covered most of that material. Off to chapter 2 we go.

• Homework 8, Fri. 2/5/2010 : Matrices and orthogonal transformations: Study section 2.1 and 2.2. of the vector notes .
  (Note: section 2 of the notes does not use the summation convention, but I used it in class and in the following exercises and you should be able to use it.)
  1. If (e₁, e₂, e₃ ) is an orthonormal basis and v is an arbitrary vector, (1) express "orthonormal basis" in compact, explicit terms, (2) what does v= v_ie_i mean? (3) What does v_i = v.e_i mean? (4) Why does (2) imply (3)? (5) Why does (3) imply (2)? (6) Assume that (2) is true, then derive (3) from (2). Combining (2) and (3), explain why we can always write v= (v.e_i) e_i = (v.e_j) e_j. (Yes, this is all `review,' of section 1.4 and 1.6, but is it really for you?)
  2. A vector v is expressed in terms of two distinct orthonormal bases, say (e₁, e₂, e₃ ) and (e'₁, e'₂, e'₃ ). Derive explicit formula connecting the components in one basis to those in the other basis. Write your formula compactly using both index notation and matrix notation.
  3. If the primed basis is the same as the unprimed basis except that e'₁=-e₁, what is the matrix Q such that v'_i=Q_ijv_j?
  4. If the primed basis corresponds to a rotation of the unprimed basis about the e₂ direction by angle β what is the matrix Q such that v'_i=Q_ijv_j?
  5. If any two sets of cartesian components are related by v'_i=Q_ijv_j? What is the determinant of Q? Does your answer match the two previous specific examples?
  6. If (v₁,v₂,v₃), (v'₁,v'₂,v'₃) and (v''₁,v''₂,v''₃) are the components of a vector v with respect to three distinct orthonormal bases (i.e. cartesian bases if you wish), and v'_i=A_ijv_j, v''_i=B_ijv'_j, (1) what is A_klA_km=?, (2) if v''_i = C_ijv_j, what is C_ij=? in terms of the A's and B's, (3) what is C_ijC_ik=?, (4) what is C_ijC_kj=? Can you prove/explain/justify your answers?
  7. The unprimed basis (e₁, e₂, e₃ ) corresponds to cartesian coordinates centered at the center of the earth, with the x direction corresponding to longitude 0, latitude 0 and the z axis as the polar axis. The primed basis (e'₁, e'₂, e'₃ ) corresponds to cartesian coordinates centered at Madison, WI, with the z' axis vertical and the x' axis pointing east. Both systems are right handed. If a vector v has components (v₁,v₂,v₃) in the earth basis, what are its components in the Madison basis? If a point has cartesian coordinates (x₁,x₂,x₃) in the earth basis, what are its coordinates in the Madison basis? Madison, WI is at 43° 4' 23" N, 89° 24' 4" W

• Homework 9, Mon. 2/8/2010 : Matrices and orthogonal transformations continued in succession: Study section 2.1 and 2.2. of the vector notes .
  What was that? oiler?! angles.
  1. What is Q for the transformation of components in one RHOB to another one that is a rotation of the first by α about the y axis? If the vector had components (3,2,1) in the first basis, what are its components in the 2nd basis?
  2. If one basis corresponds to the earth basis described in HWK 8 and the local basis is an East, North, Up basis located at longitude φ, latitude λ,
     1. What is the matrix to go from the earth components to the local components?
     2. What is the matrix to go from the local components to the earth components?
     3. What is the matrix to go from local (φ, λ) to local (φ', λ') basis?
  3. What is the transpose of a matrix? How do we denote that?
  4. How do we multiply matrices? Provide an explicit formula in index notation
  5. If Q is an orthogonal transformation matrix, what is special about its rows and columns? write out explicitly using index notation. What is special about its determinant? Can you explain/justify why?
  6. If Q is an orthogonal transformation matrix, what is the (i,j) element of QQ? of Q^TQ? of QQ^T? Provide explicit formula in index notation.
  7. Can the Euler angles α, β and/or γ be negative?

• Homework 10, Wed. 2/10/2010 : Catch up while you can!
  We discussed HWK 8 #7, and HWK 9 #2. Actually, we derived an explicit formula (in matrix form, so it's a bit cleaner). You can start plugging numbers in that formula for fun, for instance what is the matrix for the Madison basis, or a New York basis, or an Honolulu basis? How about Sidney, Australia? What is φ for Madison? But our objective is of course not just to know what the formula is, or to practice taking sines and cosines of given angles, but to know how to derive it, what it means and how to use it. We reviewed lecture 1 and cylindrical and spherical coordinates in the process. We derived the full matrix `directly' although we did that by expressing direction vectors in 2 steps: cartesian to cylindrical, cylindrical to spherical. We also discussed how to do this using a sequence of simple rotations as in the Euler angles business. That gives us the final matrix as a product of simpler rotation matrices (check it!). These two apparently distinct approaches are in fact essentially identical, just performed and written in a different way. The first approach focuses on connecting direction vectors, the second focuses on connecting components.
  • What is the Q that corresponds to π/2 rotation of a basis about the z-axis?

• Homework 11, Fri. 2/12/2010 : Aie, aie, aie! Rotation of bases before,... now rotation of vectors! I'm spinning out of control!
  Wait! It's just old stuff: parallel and perpendicular components, dot and cross products, definition of cosine and sine,...
  • Write an algorithm to compute the rotation of a vector (V₁,V₂,V₃) about the direction (a₁,a₂,a₃) by an angle α. All components are given in cartesian coordinates. Since it is easy to teach a computer how to calculate dot and cross products in cartesian components, you can simply call those formulas (say as dot(a,b) and cross(a,b)).
  • Exercise: 2.2. 11: problem could be interpreted in two ways:
    1. Find the matrix to go from given components (V₁,V₂,V₃) to components (V'₁,V'₂,V'₃) (both in the same basis (e₁,e₂,e₃ )) of the vector V' that corresponds to the rotation of V about the direction (e₁ + e₂ + e₃ ) by α. [Same basis, rotated vector]
    2. Find the matrix to go from components (v₁,v₂,v₃) of vector V to the components (v'₁,v'₂,v'₃) of the same vector V in the basis (e'₁, e'₂, e'₃ ) that corresponds to the rotation of (e₁, e₂, e₃ ) about a=(e₁ + e₂ + e₃ ) by α. [Same vector, rotated basis]
    (1.) was done in class today using the simplest approach (although scary because one needs to understand dot and cross products in order to understand it).
    (2.) can be done using (1.) 3 times with (V₁,V₂,V₃) = (1,0,0), (0,1,0) and (0,0,1) (to obtain the components of (e'₁, e'₂, e'₃ ) respectively). The three (V'₁,V'₂,V'₃) that result are the rows of our famous Q! In other words, The matrix in (2.) is the transpose of that obtained in (1.)!

    We rotated vectors, but how about rotating a point P about an axis parallel to a passing by point A?
    Example: rotate P≡(3,2,1) about axis through A≡(1,2,3) in direction a ≡ (4,5,1). [answer: rotated P ≡ (0.9494, 3.6934, 0.7351)]

• Homework 12, Mon. 2/15/2010 : Onward to Vector calculus, now our vectors are functions, for instance function of time t. Study section 1.9 of the vector notes
  1. Can you prove a product rule? any of them on page 23? Where do these products rules comes from? Santa?
  2. Prove the derivative of a determinant rule using index notation.
  3. What does a . da/dt=0 tell you about a? (i) if this eqn. applies at one specific time, (ii) if it applies for all times.
  4. What does a x da/dt=0 tell you about a? (i) if this eqn. applies at one specific time, (ii) if it applies for all times.
  5. Suppose you know a and da/dt. Derive expressions for d|a|/dt and d/dt (a/|a|) in terms of a and da/dt (recall that (a/|a| is what we call and write a-hat). Do your equations say what you expect them to say? (i.e. make a sketch and interpret your results)
  6. If v is the velocity of a particle at position r such that v=&omega x r, what is the acceleration a= dv/dt of the particle? (express in the simplest, clearest way in terms of &omega and r). What is the acceleration when &omega is a constant? Do you know what those accelerations are called?
  7. Consider the vector differential equation da/dt = &omega x a, where &omega is a constant vector, show that |a| and (&omega . a) are constants. What is that telling you about a?
  8. Write dr/dt = &omega x r in index notation and in matrix notation.
  9. Write dr/dt = &omega x (r-r_A) in cartesian coordinates/matrix notation.
  10. If da/dt = &omega x a, db/dt = &omega x b and dc/dt = &omega x c where &omega is not necessarily constant, i.e. &omega=&omega(t), what is d/dt [det(a, b, c)] = ? Interpret geometrically, what's going on in this problem?
      You should be able to derive and justify the result from your geometric understanding of the concepts without any calculations!
      You should be able to derive the result using algebraic manipulations, if you know your mixed and double cross product identities. [Hint: the Jacobi identity, problem 4, section 1.5 comes in handy. ]

• Wed. 2/17/2010 : Exam 1 ... GULP!

• Homework 13, Fri. 2/19/2010 : Study section 1.10 of vector notes. We solved #7 and #8 from HWK 12, and `processed' #9 into #7. #5 had been solved in class on Monday 2/15. We discussed `Euler's method' for vector differential equations.
  • Solve da/dt = &omega x a where &omega is constant. Write an explicit form for the vector solution and verify that it is indeed a solution of the equation. What data do you need to know to specify a(t) for all t's?

CHAPTER 2: Vector Calculus.

• Homework 14, Mon. 2/22/2010 : Study section 1.1 of the vector calc notes. You may also want to look back at your Math 222 book for conics and parametric curves (e.g. Thomas 11th, Chap 10)
  • Exercises: 1.2. 1, 2, 3 (largely done in class today)
  • What are the implicit and parametric equations of an ellipse? What is the parametric vector equation of an ellipse? (done in class, but you should be able to rederive from scratch if you understand the concepts).
  • What is the meaning of θ in the parametric equation of an ellipse? (done in class, but do you understand it?)
  • A circle is the locus of all points that are equidistant from a given point C called the center of the circle. A similar geometric definition of an ellipse is that it is the locus of points P such that the sum of the distances from P to two given points F₁ and F₂, called the foci, is a constant. This provides a mechanical way to draw ellipses. (see Figure 10.5 in Thomas, 11th edition)
    1. Taking appropriate axes, show that this definition indeed corresponds to the standard algebraic equation for an ellipse. What are the connections between the major and minor radii a, b and the geometric data?
    2. A+: Use the geometric definition of an ellipse to show that the lines F₁P and F₂P make equal angles with the tangent to the ellipse at P. Use vectors to do this in 2 lines, instead of doing it in 2 pages using coordinates. All you need is to understand tangent vector and the meaning of the position vector r=r(t), no need for anything more! Beautiful!
  • Find a simple vector parametric equation (i.e. r=r(t)) for the curve that goes from point P₁ to point P₂ (in 3D space), what is the range of your t?
  • Find a simple vector parametric equation for the curve that goes from points P₁ to P₂ and is parallel to a at P₁. (hint: use products of (1-t) and t as needed with t=0 → 1. Go back to previous problem first).
  • A: Find a simple vector parametric equation for the curve that goes from point P₁ to P₂ and is parallel to a at P₁ and to b at P₂. (hint: use products of (1-t) and t as needed with t=0 → 1). Got to do previous two problems before this one.
  • A+: Find the vector equation for the circle that passes through points P₁, P₂ and P₃. Is that circle unique?

• Homework 15, Wed. 2/24/2010 : Study sections 1.1 and 1.2 of the vector calc notes. You may also want to look back at your Math 222/234 book for vector functions, curves and line integrals (e.g. Thomas 11th, Chap 13 + Chap 16 for `flow integral, work and circulation')
  • Concepts of curves, explicit and conceptual r=r(t) (e.g. circles and ellipses, and `connect the dots', splines ).
  • Concept of line element dr, derivative and line integral, study section 1.2
  • Exercises: 1.2. 4, 5, 6 (#5 done in class today)

• Homework 16, Fri. 2/26/2010 : Study sections 1.3 and 1.4 of the vector calc notes. You may also want to look back at your Math 234 book for surfaces, surface area and surface integrals (e.g. Thomas 11th, Chap 16.5, 16.6)
  • Concepts of surface, surface parametrization/coordinates, coordinate curves, surface element
  • Exercises: 1.3. 1, 2, 3, 5
    #2 is especially important, you are expected to be able to derive all the formulas e.g. (vector) surface element dS but you are also expected to know all the relevant formulas e.g. dS = R² sinθ dθ dφ as well as surface area of a sphere S = 4 π R² and how to compute it `from scratch'.
    #5: `Outer' and `inner' radius could be interpreted in various ways. Don't run to mama (or papa, or Boyd) right away. Digest the given formula to figure out the meaning of R and a, this should be fairly elementary my dear Watson! Can you explain to our dear Watson how you deduced the meaning of R and a?
  • Exercises: 1.4. 1, 2, 3, 4

• Homework 17, Mon. 3/1/2010 : CATCH UP!
  • Quickly solved most of problem 1.3.2 in class today. You should be able to rederive everything from scratch: relations between cylindrical, spherical and cartesian coords, 3D view, meridional and `zonal' views, connections between the various unit vectors, derivation of ∂r/∂θ and ∂r/∂φ in (i) `hybrid' formulation, (ii) quick geometric derivation using `infinitesimal' circular arcs, (iii) almost as quick derivation using meridional and zonal (or latitudinal) views. Derivation of dS.
  • Flux of a 1/r<sup>2</sup> vector field, q=constant times r/r<sup>3</sup>, through a sphere S of radius R (e.g. electric flux, gravitational flux, mass flux, neutron flux,...).
    Sphere centered at origin was trivial (wasn't it?) (problem 1.4.4, done in class on 2/26/2010).
    What about for a sphere centered at point C ≠ O? We derived the integral in class, can you derive it from scratch? Hide your notes and try to rederive the integral. Can you calculate it?
  • Can you give a convincing geometric argument why ∫ q . dS = 0 for q=r/r<sup>3</sup> when the integral is over a sphere of radius R centered at a distance c from the origin with c > R? How about when the surface is any reasonably nice closed surface? (like a box or an ellipsoid, or any potato shaped surface).
• Homework 18, Wed. 3/3/2010 : Study sections 1.5 and 1.6 of the vector calc notes. You may also want to look back at your Math 234 book for cylindrical and spherical coordinates (e.g. Thomas 11th, Chap 15.6) and substitution in multiple integrals (Chap 15.7).
  (EXCEPT that Thomas uses the `american mathematician' definition of spherical coords NOT the ISO 31-11 standard notation used in physics and engineering. See the `Conventions' paragraph on the wikipedia spherical coords page).
  • Know how to derive the volume elements in cylindrical, spherical and general curvilinear coords.
  • Know what it means for generalized (curvilinear) coords to be orthogonal and how that simplifies computing the volume element.
  • Introduce 3D toroidal coordinates and compute the volume of a general torus. (Exercise 1.5.4)
  • Exercises: 1.6. 2, 3 (don't just look up formula, derive them)
  • A surface is specified in spherical coords as r = a sin θ, where a is an arbitrary positive constant, and θ is our usual polar angle. Try to visualize it. Explain why this is a closed surface. Compute the volume inside this closed surface.

• Homework 19, Fri. 3/5/2010 : Study sections 1.5, 1.6 and 1.7 of the vector calc notes. Mappings, Change of variables, Jacobian matrix, Jacobian determinant. Inverse mappings.
  • Exercises: 1.7. 1,2, 4,5,6

• Homework 20, Mon. 3/8/2010 : Study sections 2.1, 2.2 of the vector calc notes. What is a gradient? a directional derivative? (e.g. Thomas 11th, Chap 14.5)
  1. Compute the gradient of the scalar field f=f(r) for (i) f=f(|r|) , (ii) f=f(|r-r₁|), (iii)f=f(a.r + c) where r₁, a and c are constants. Do each of these in TWO ways: (1) using your geometric understanding of the gradient, (2) using cartesian coordinates.
  2. What is the general expression of the gradient of (i) f=f(x,y,z) in cartesian coords? (ii) f=f(ρ,&phi,z) in cylindrical coords? f=f(r,θ,φ) in spherical coords? (iv) f=f(q₁,q₂,q₃) in generalized orthogonal coords? (v) A++: f=f(q₁,q₂,q₃) in generalized not necessarily orthogonal coords?
  3. From the geometric definition of an ellipse as the set of points P such that the sum of the distance from P to two fixed points F₁ and F₂ is a constant, |r-r₁|+ |r-r₂| = 2 a, (i) Find the direction of the normal to the ellipse at P, (ii) show that the lines (F₁,P) and (F₂,P) make equal angles with the normal to the ellipse at P.
  4. Is |dr| = d|r|? Discuss and sketch.

• Homework 21, Wed. 3/10/2010 : Study sections 2.1, 2.2 and 2.3 of the vector calc notes. Divergence, Curl.
  • What is ∇ in cartesian coords? in cylindrical coords? in spherical coords? (done in class and in these Extra notes on gradient .)
  • What is ∇ in index notation?
  • What does ∂_i mean?
  • What is ∇.v in index notation? Is ∇.v= v .∇? What is the geometric meaning of v .∇? Compute (v .∇) r, for a general v, where r is the usual position vector, do that in 2 ways: (1) `geometrically' from your understanding of v .∇ and (2) using cartesian coords.
  • What is ∇ × v in index notation?
  • What is ∇ . (∇ × v) in index notation?
  • What is v . (∇ × v) in index notation?
  • Compute v . (∇ × v) for v = α e_z + ω e_z × r, where α and ω are scalar constants and e_z is the unit vector in the fixed z direction.
  • What is ∇.v in spherical coords?

• Homework 22, Fri. 3/12/2010 : Study sections 2.1--2.4 of the vector calc notes. Vector identities.
  • What kind of vector field is v = ω × r, where r is the position vector and ω is a constant vector? Sketch, where does it come up in physics/engineering? Calculate ∇ . v and ∇ × v.
  • Exercises: 2.4. All, 1--9, especially 6, 7, 8, 9.

• Homework 23, Mon. 3/15/2010 : Study sections 3.1, 3.2, 3.3, 3.4 of the vector calc notes. Fundamental Theorems of Calculus.
  • Derived ∇ . v in spherical coords for Noah. Everyone was thrilled!
  • Concepts of Flux through a closed surface and circulation around a closed loop (for example circulation of a velocity field in fluid dynamics and Ampère's law in E&M (in applications, the line element dr is often written dl).
  • Connection between flux through a closed surface and divergence.
  • Connection between circulation around a closed loop and curl.

• Homework 24, Mon. 3/15/2010 : Study Chapter 3 of the vector calc notes. Fundamental Theorems of Calculus, in particular, divergence theorem and Stokes theorem. Warning: lectures covered this material in a muck quicker and intuitive way, focusing on understanding meaning of divergence and curl. You are not responsible for all that is covered in the online notes, but you should thoroughly understand the meaning of formula (86), (87), (111) and (112).
  • What are the geometric, coordinate-free interpretations of the divergence and the curl?
  • Calculate the divergence and curl of (1) v = r, (2) v = r/r³, (3) v = ω × r, (4) v = (ω × r)/|ω × r|², where r is the position vector and ω is an arbitrary constant vector. Visualize the vector fields and explain/motivate the results using your understanding of div and curl (by taking suitable little boxes or loops around points).
  • Exercises: 3.7. 1, 2, 3, 4, 8, 9