Fabian Waleffe -- Math 321 -- U. of Wisconsin

Math 321 homework, lecture pointers and learning objectives: Fall 2010

Fall 2010, Math 321 TA page

Are you looking for exercises on PART 2: VECTOR CALCULUS?

Are you looking for exercises on PART 3: COMPLEX CALCULUS?

PART 1. Vector algebra: geometric concepts and index notation

Numbered exercises refer to the Vector 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 1, Thur. 9/2/10 : We discussed the basics of geometric/physical vectors, magnitude, direction, unit vectors. Discussed arbitrary displacement vector a in 3D space. Gibbs notation. Magnitude and direction. How to express a in cartesian, cylindrical and spherical coordinates, what are different ways of expressing those coordinates (e.g. azimuthal angle; `altitude' or `elevation' or `inclination' or `polar' or `zenith' angle, many different names and slight differences in definition to describe the same concepts). We will follow the ISO 31-11 convention for azimuthal, φ, and polar, θ, angles, not the elementary calculus book convention.
- Review and complete your own lecture notes so they'll make sense later on.
- Wikipedia on spherical coordinates (pretty good today... also contains cylindrical)
- The original book on vector analysis (Gibbs/Wilson)
- Who was Gibbs? When and where did he live? What did he do in science and math?
- What is Gibbs notation? How do we indicate magnitude of a vector? direction of a vector? What is the smallest number of numbers needed to specify an arbitrary vector direction in 3D space?
- What are the units of a unit vector? What is the magnitude of a unit vector? in what units of length?
- Make a simple 3D perspective sketch showing cartesian, cylindrical and spherical representations of vectors in 3D space without just looking them up in your notes or a book
- Write an arbitrary vector v in cartesian, cylindrical and spherical representation. You should be able to derive all the formulas `from scratch'. 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 ... know your basic trig! Pythagoras, cosine, sine, remember those?
- What is the vector equation that expresses the equality between the cartesian, cylindrical and spherical representation of a vector v? (vector v NOT vector a). Deduce all algebraic relations between the various representations (i.e. cartesian and cylindrical, cylindrical and spherical, cartesian and spherical). Write the direction vectors for v and the horizontal radial direction ρ in terms of the cartesian direction vectors.
- What are the cylindrical and spherical coordinates for the points whose cartesian coordinates are (0,0,1), (0,-1,0), (1,1,1)?
- Write your Math 321 TA's name in Chinese.
Homework 2, Tue. 9/7/10 : SECTIONS 1.1, 1.2 in the Vector notes. Vector space, Linear Combinations, Linear independence, Bases and Components
1. What is a scalar? We will use the word in both its math and its physics meaning. Can a scalar be a function of time and/or position, or is it always a constant? Give physical examples of a scalar.
2. Visualize (i.e. draw sketches to illustrate/demonstrate) the 8 properties (1)--(8) of vector addition and multiplication by a real number for displacements in 3D space.
3. What is a linear combination of vectors? What does it mean for vectors to be linearly independent? linearly dependent?
4. What is a vector basis? in 2D space? in 3D space? Do basis vectors have to be unit vectors? Do basis vectors have to be orthogonal (perpendicular) to each other? Is it perpendicular or perpindicular?!?
5. Prove that the medians of a (arbitrary) triangle are concurrent (i.e. go through the same point) using (1) analytic geometry (cartesian coordinates, equations, etc.), (2) vectors as done in class (but without looking at your notes).
6. Use vectors to show that the diagonals of a parallelogram intersect at their midpoint.
7. 3 pts are specified by their cartesian coordinates (1,2,3), (2,3,1), (3,2,1). Provide an explicit expression that delivers any point that is inside the triangle formed by the 3 points. What is the explicit cartesian equation of the line that passes through (2,3,1) and (3,2,1)?
8. If r_A and r_B are the radius vectors of points A and B, respectively, with respect to some origin O, show that (α r_A + β r_B )/(α + β) is (the radius vector of) a point on the AB line for any real α, β with α+β not 0. What about (α r_A + β r_B ), for any real α, β, what would that be?
9. Vector notes: study sections 1.1, 1.2. Exercises 1.2. 1, 5, 6, 7, 8 (yes, some of these exercises are related) [in 1.2.1 (i) both α and β are between 0 and 1]
Homework 3, Thur. 9/9/10 : SECTIONS 1.3, 1.4 in the Vector notes. Dot product, orthonormal basis, sigma notation, indices,...
- Section 1.3.: Digest the concepts in eqn (13) and the accompanying figure. This should all follow naturally from your understanding of the definition (11). You do not want to memorize the formulas, you should understand dot product well enough to be able to rederive these formulas on the fly.
- Exercises 1.3. 1--9.
- Section 1.4.: Do you understand equations (14), (15) and (16)?
  (You may skip the part of section 1.4 on pages 9 and 10. This is Math 340 material, some of you know it already, the others should learn it soon.)
- Exercises 1.4. 1--5.
Homework 4, Tue. 9/14/10 : SECTION 1.5 (cross & double cross product, Right Handed Orthonormal Bases)
- 1.5. eqn (20) and accompanying figures, huh?! color and symbols on a page, or meaningful to you?
- 1.5. From the geometric definition of the cross product, verify in detail that (α a)xb = α (axb) for any real number α and 3D vectors a and b.
- 1.5. From the geometric definition of the cross product, verify in detail that (a+b)xc = axc + bxc
- 1.5. Can you reconstruct (24) easily and quickly, as done in class with the 1+2=3 trick instead of the (fake) determinant trick?
- 1.5. What is the right way to remember and reconstruct eqn (25)? Is (25) a definition or a derivation?
- Exercises 1.5. 1--5
- A+ level exercise for fun: Can you prove (25) from the geometric definition of the cross product? (Hint: it's done in the notes, but could you do it from scratch, not just regurgitate it?)
Homework 5, Thur. 9/16/10 : Section 1.6: Levi-Civita symbol, Einstein's summation convention
1. What is the permutation symbol? (give a specific definition, don't write a short novel about it!)
2. Section 1.6: Definition (27), digest it. Equation (28) and (29), verify them. Study section 1.6.1, this is really a `reminder' about sigma notation which you used in Math 222 (for series).
3. indices: the dummies and the free
4. Make your own mental summary of concepts: Geometric meaning and cartesian index expressions for a.b, axb, (axb).c .
5. Summation convention: what is it? Eqns (36)--(39)... what do they all mean? can you rederive them on your own?
6. True or False? ε_ijkε_ijm is zero except when k=m. Why?
7. True or False? ε_ijkε_ijm = 3 if k=m. Why?
8. True or False? The k component of axb is ε_ijka_ib_j
9. True or False? The k component of axb is ε_kija_ib_j
10. True or False? The i component of axb is ε_ijka_jb_k
11. Exercises 1.6. 1--4.
12. Let U = (axb)xc and W=(a.c)b - (b.c)a. Write U and W in index notation in terms of the components of a, b and c. Use index manipulations to deduce formula (44), which you should know.
Homework 6, Tue. 9/21/10 : Sections 1.6 + 1.7: Double cross in index notation, Mixed/Box product, determinants
- 1.6. Write (axb)xc and ax(bxc) in index notation (assuming an orthonormal, a.k.a. cartesian, basis).
- 1.6. Derive equation (44) from eqn (25) as done in class and in the notes. But you should be able to do it from scratch without looking at notes! can you?
- Learn to reconstruct (44).
- Evaluate the following expression (which use the summation convention): (1) ε_ijkε_ijk; (2) ε_ijkε_ijl; (3) ε_ijkδ_ij; (4) ε_ijkδ_kl;
- Exercises 1.6. All.
- Exercises 1.7. 1, 2, 3, 7, 8, 9, 10, 11, 12. (Are some of these problems related to one another?)
Homework 7, Thur. 9/23/10 : Section 1.8 + some of 1.9
1. Exercises 1.8. All.
2. What are the geometric properties (e.g. "it is a sphere of radius 5 centered at the point (2,3,9)") of the set of points whose cartesian coordinates (x,y,z) satisfy
  1. A (x-x₁) = B (y-y₁) = C (z-z₁)?
  2. A (x-x₁) + B (y-y₁) = C (z-z₁)?
  3. (x-x₁)² + (y-y₁)² + (z-z₁)² = A
3. Answer this poor guy's question, and this one too at `experts-exchange.com'...
4. What is the distance between the point (x₁,y₁,z₁) and the plane a x + b y + c z + d = 0? Formulate and solve the problem using vectors, specify how to obtain the necessary data from the given data and what actual arithmetic operations must be done to get the actual number. (NOTE: in other words, the question is not about plugging into a formula, it's about deriving the formula)
5. What is the distance between the point (4,2,9) and the plane 3x + 2y + z + 3 = 0? (See NOTE above)
6. How far is the plane a x + b y + c z + d = 0 from the origin? (See NOTE above)
Homework 8, Tue. 9/28/20 : Solved and discussed some problems. (1) Two geometric viewpoints for systems of equations, and geometric solutions: sections 2.4.1 and 2.4.2 in the notes, but look back at your notes for today's lecture first. (2) Problem 1.7.7 using vector operations. Should also be able to do it using index notation. (3) Rotation of a vector about a direction by an angle: vector solution + MATLAB code.
1. Rotate the vector with cartesian components (4,2,6) about the direction (3,1,5) by angle π/6. What are the components of the rotated vector? (NOTE: formulate using vectors, obtain a general solution in compact vector form)
2. Point P has cartesian coordinates (3,2,1). What are its new coordinates after we rotate it by angle π/3 about the axis passing through point A ≡ (1,2,3) with direction D ≡ (4,5,1)? Write the procedure as an "algorithm" (i.e. a `recipe' to follow to obtain the answer from the given data). Write a code (Matlab, C, Fortran,...) to do this for any P, A, D, angle. [ANSWER for this data: new P has coords (0.9494, 3.6934, 0.7351)]. Compare our vector solution procedure to the common matrix solution. Do you feel lucky?
3. What are the differences between rotating a vector and rotating a point?
Homework 9, Thur. 09/30/10 : Sections 2.1 and 2.2, orthogonal transformations/change of orthonormal basis. Notes do not use summation convention, however lecture and exercises below DO use the summation convention!
1. If (v₁,v₂,v₃) and (v'₁,v'₂,v'₃) are the components of a vector v with respect to two distinct orthonormal bases (i.e. cartesian bases if you wish), how are those components connected? DERIVE the relationships between these two sets of components, both ways v → v' and v' → v. You are not asked to spit back formulas here, but instead to derive and explain what the formulas are and why. You should be able to do this without looking at any notes/book if you understand the concepts.
2. 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?
3. 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?
4. What is Q_ikQ_jk = ? Why? Can you give an algebraic reason why? A geometric reason?
5. What is Q_klQ_km = ? Why? Can you give an algebraic reason why? A geometric reason?
6. 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 (2 and 3 above) and the example done in class?
7. 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?
8. The unprimed basis 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 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 10, Tue. 10/05/10 : Sections 2.1 and 2.2, orthogonal transformations/change of orthonormal basis. Notes do not use summation convention, however lecture and exercises below DO use the summation convention!
- Which Homework 9 problems did we solve/discuss in class today?
- Why are the Q_ij called `direction cosines'? Each of these directions cosines correspond to the cosine of an angle. What angle corresponds to Q₂₃? How many angles are there altogether? Are these angles all independent? Why? What is the minimun number of angles that one needs to specify to determine a Q matrix in 3D, in general?
- A+ level (do you have other problems to worry about before this one?): We discussed rotation of a vector about a direction in lecture and homework 8 on Tuesday Sept 28. We solved the following problem: Rotate the vector with cartesian components (b₁,b₂,b₃) about the direction (a₁,a₂,a₃) by angle φ (right hand convention). We did this in a nifty way using a vector formulation, even coded into Matlab. That rotation gives new components (b₁',b₂',b₃') which are the components of a new vector b' in the same basis (not the components of the same vector in a new basis as we've been talking about in Lectures 9 and 10. OK, you may need to take a deep breath and read that again). Can you write that (lecture 8) transformation (b₁,b₂,b₃) → (b₁',b₂',b₃') as a matrix transformation b'=Ab for some matrix A? Is that A orthogonal just like the Q's we've encountered when changing orthonormal bases?
Honeymoon is over: EXAM 1 is THURSDAY OCT 7, 11:00-12:15. Rooms: 121 Psychology if your last name starts with "H" or above, and 1257 Comp Sci otherwise. Material is all the concepts covered in the homework assignments 1--9 on this page. Remember, this course is not about plug-and-chug into boxed formulas. So far, it's been about understanding basic geometric concepts and vectors in 3D space, how to translate/express them into cartesian form, or cylindrical or spherical, how to think using vectors, how to do grungy cartesian calculations using the powerful index notation with Einstein's summation convention.
Exam 1, Thur. 10/07/10 : When and where did we cover that material?!? Does the homework prepare you for the exams?!
1. (a) Homework 5, Problems 1 and 2. Eqn (27) in the vector notes (NOT eqn (28), (28) is an application of (27)).
  (b) Homework 5. Eqns (38) and (39) in the vector notes and homework 5 and 6.
  (c) Homework 5, #12. Boxed formula (44) derived in the vector notes and in lecture.
  (d) Homework 6. This is exercise 1.7.7. The vector derivation was done in discussion and in class. The index derivation was `left for the exam'.
2. (a) Eqn (25) in notes and homework 4, 5 and 6.
  (b) Eqns (46), (48) and (56) in the notes. Exercises 1.7.2 and 1.7.3, homework 6.
  (c) Exercise 1.5.2, howework 4.
3. Problem 3, homework 7. Lines and planes, section 1.8, homework 7. Approach similar to rotation problem solved and matlab coded in class (homework 8).
4. Homework 8. Section 2.1. in the notes. In this exam question, index notation was avoided and focus was on concept of change of cartesian basis. This is really about understanding sections 1.4 and 2.1 on orthonormal bases and the meanings of eqns (15), (83) and (84) in the notes.
5. Related to change of bases and problem 8 on homework 9, but in fact this is lecture and homework 1: how do you specify a direction in 3D space?
Students protest UW standards but keep enjoying Beer and circus!

ONWARDS TO PART 2: VECTOR CALCULUS !