\HOMEWORK 0" Last updated Friday 27th December, 2019 at 2:44am. (Not officially to be turned in) Try your best for the following. Using the first half of Shurman's text is a good start. I will possibly add some notes to supplement this. There are also some geometric puzzles below for you to think about. Suggested winter break homework - Learn the proof to Cauchy-Schwarz inequality. - Find out what open sets, closed sets, compact set (closed and bounded in Rn, as well as the definition with open covering cf. Heine-Borel theorem), and connected set means in Rn. - Find out what supremum (least upper bound) and infimum (greatest lower bound) of a subset of real numbers mean. - Find out what it means for a sequence and series to converge. - Find out what does it mean for a function f : Rn ! Rm to be continuous. There are three equivalent characterizations: (1) Limit preserving definition. (2) Inverse image of open sets are open. (3) −δ definition. - Find the statements to extreme value theorem, intermediate value theorem, and mean value theorem. - Find out what is the fundamental theorem of calculus. - Find out how to perform matrix multiplication, computation of matrix inverse, and matrix deter- minant. - Find out what is the geometric meaning of matrix determinantDecember, 2019 . - Find out how to solve a linear system of equation using row reduction. n m - Find out what a linear map from R to R is, and what is its relationth to matrix multiplication by an m × n matrix. How is matrix product related to composition of linear maps? - Find out what it means to be differentiable at a point a for a multivariate function, what is the derivative of such function at a, and how to compute it. The answer involves a linear transformation / matrix. - Find out the multivariate version of chain rule. - Find the statements to inverse function theorem and implicit function theorem. Prepared by Bon-Soon Lin on: Friday 27 Puzzles. Problem 1. Draw a convex polygon P with n sides. What is the sum of the exterior angles of P ? Does it depend on n? What if P is not convex? θ6 θ1 θ5 θ2 P θ4 θ3 Problem 2. Draw a generic five-pointed star like below. What is the sum of the sharp angles? 1 \HOMEWORK 0" Math 32BH θ1 θ2 θ5 θ1 + θ2 + θ3 + θ4 + θ5 =?? θ3 θ4 What about a generic seven-pointed star? Can you make a conjecture? Problem 3. Draw a 3D solid shape where each face is a flat polygon. For example a cube. For each vertex v of this solid, we define its angular defect at vertex v to be X k(v) = 2π − the angles of the faces that meet at v For example, each vertex of the cube is met with three right angles, so the angular defect k(v) for each vertex v of the cube is k(v) = 2π − (π=2 + π=2 + π=2) = π=2. P Calculate the total angular defect for a cube, v a vertex of the cube k(v). Calculate the total angular defect for a generic tetrahedron (four faces, each face a triangle). Calculate the total angular defect of an octahedron (eight faces, each face a triangle, with four triangle sitting above a quadrilateral, and four sitting below) December, 2019 cube tetrahedron octahedron Draw any 3D solid shape with each face a flat polygon, and calculateth its total angular defect. Can you make a conjecture? What is the total angular defect of the following two shapes with a hole drilled through? (Note. the angular defect at a vertex can be negative if the sum of the angles exceed 2π.) Prepared by Bon-Soon Lin on: Friday 27 square prism with a square prism hole drilled out triangular prism with a triangular prism hole drilled out Problem 4. Show in fact: For any 3D solid shape with each face a flat polygon, the total angular defect is always 2π(V − E + F ); where V is the number of vertices, E is the number of edges, and F the number of faces. Verify this with the shapes above. Try to prove this if you can. Note, the quantity χ = V − E + F is the so-called Euler 2 \HOMEWORK 0" Math 32BH χ−2 characteristic of the shape. It is what distinguishes a donut from a sphere! (Roughly speaking 2 is how many \holes" this shape got.) Remark. Somewhat surprisingly, these problems are all related, and are all discrete versions of a famous theorem in differential geometry. I will be mysterious about it for now... December, 2019 th Prepared by Bon-Soon Lin on: Friday 27 3.