### Math 321 homework : Spring 2004

Read the Notes on vectors FIRST and try out as many of the implicit and explicit exercises given there as possible.

• 2005/09/06 Lecture 1: sections 1.1-1.5 with all exercises. Yes, you are responsible for 1.3 and 1.5.
1. Consider the course grades example (i.e. the vector space of student grades). Which of the following are good/bad norms for those vectors: the average (i.e. mean), the maximum grade, the minimum grade, Joe's grade? Justify. Recall that x = (x1,...,xn) is the list of grades for students 1 to n, for exam X. We discussed how such grade lists must be added (i.e. we defined addition of such lists) and how they should be rescaled (i.e. we defined how to multiply such lists by a scalar). Since these fundamental operations are defined and have the correct properties (1) to (8) in the notes, these ordered lists of n real numbers are vectors.
2. What is the geometric picture associated with the vector equation a + b + c =0?
3. Show that the lines dropped from the vertices of a triangle perpendicularly (=orthogonally) to the opposite side all intersect at the same point.
4. For any triangle, show that the lines connecting each vertex to the middle of the opposite side intersect at the same point which is 2/3 of the way down from the vertex. Do this in 2 ways: (1) using the methods of plane Euclidean geometry, (2) using vectors (but NOT using cartesian coordinates).
5. while we're having fun: show (geometrically of course, but no need for vectors) that the sum of the inner angles of a triangle is always Pi.
6. If A, B and C are three arbitrary points on a circle of center O, show (yes, geometrically) that the angle BOC is always twice the angle BAC. (OK there are two possible angles BOC, the one we want is that which `subtends' the same circular arc BC as the angle BAC). Figure out what all this means. I have not tried to figure this one out using vectors, but if you can, let us know.
• 2005/09/08 Lecture 2: Solved 2, 3 above. Discussed dot product, orthonormal bases, Kronecker symbol, Einstein's summation convention. Dot product exercises: See page 6 of the notes.
• 2005/09/13 Lecture 3: section 1.6, cross (or vector or area) product and permutation symbol.
• 2005/09/15 Lecture 4: Sect. 1.6 Double vector product, Sect. 1.7 Mixed (or Box) product, determinants.
1. Explain/show geometrically why - |a|,|b|,|c| =< det(a,b,c) =< |a|,|b|,|c|. (Note =< means less-or-equal)
2. If the angle between a and b is PHI and the angle between a X b and c is THETA, express det(a,b,c) in terms of the magnitudes of the vectors and the angles PHI and THETA.
• 2005/09/20 Lecture 5: Sect. 1.7 continued, 3 fundamental properties of determinants with geometric interpretation. Reciprocal basis. Sect 1.8: points, lines and planes.
• 2005/09/22 Lecture 6: Exercises, Reciprocal basis. Sect 1.8: points, lines and planes.
• 2005/09/27 Lecture 7: Exercises,sects 1.9 and 1.10: vector functions of a scalar variable, e.g. position vector as a function of time. Applications to Newtonian Mechanics.
• 2005/09/29 Lecture 8: 1.10 continued: Motion due to a central force, Kepler's law, angular momentum, Kinetic and potential energy. Sect 1.11 Motion of a system of particles.
• 2005/10/4 Lecture 9: Motion of a rigid body. Poisson vector.
• 2005/10/6 Lecture 10: Cartesian coordinates, change of coordinates, orthogonal transformations.
• 2005/10/11: EXAM 1
• 2005/10/13: solutions to exam 1 by Chris Chrobak, Paul Ellison, Grant Kudert, Michael Line, Ben Payne, Ian Saunders, Johanna Wendlandt and Mikhail Wolfson (and FW).
• 2005/10/18: Matrices, Matrix-vector and Matrix-matrix products. Transpose.
• 2005/10/20: Orthogonal matrices, Euler angles. Gram-Schmidt. (See section 2.1 Exercises) Linear Systems Ax=b, geometric interpretations (sections 2.2.1 and 2.2.2 only)
• 2005/10/25: BEGIN QUICK REVIEW OF VECTOR CALCULUS
Parametric representations of curves and surfaces. Curves and tangent vector, line element dr. Surfaces: tangent vectors and normal vector, surface element dS. Line integrals, surface integrals as limits of a sum and their parametric representations as iterated integrals.
1. What kind of a curve is defined by r(t)= e1 a cos t +e2 a sin t + e3 c t, where a and c are constants and e1 e2 e3 is a right-handed orthonormal basis (e.g. Cartesian unit vectors)? What is a tangent vector to the curve at r(t)? What is the length of the curve from t=0 to t=2 Pi? If A and B are two points on that curve, what is the integral over the curve from A to B of dr? of r. dr? (dot product of r and dr)
2. Consider the curve r(t)= e1 a cos t +e2 b sin t, where a and b are constants. What kind of curve is this? What is the geometrical meaning of the integral over the curve of r X dr? (cross product of r with dr) Express the integral over the curve of f(x,y) dx in terms of t. Write that last integral in vector form.
3. Calculate the surface element dS for the Cartesian parametrization of a surface, i.e. z=f(x,y).
4. Calculate the surface element for a sphere using the spherical coordinates discussed in class. Calculate the area of the arctic `circle' (i.e. area of a spherical cap).
• 2005/10/27: Gradient, div, curl. The "del" operator (D, on this web page). Divergence and curl. Vector identities: D . (f v), D X (f v),etc.
1. Calculate the divergence and the curl of r/r^3 (use vector identities for a slick calculation).
• 2005/11/03: Further vector identities. Fundamental theorem(s) of vector calculus: Gauss, Green and Stokes theorems.
• See notes on Vector Calculus and Complex Calculus for theory, example and exercises.