Week 10 -- Sections 6.2-6.4

Short-term objectives

  1. Find the expansion of a vector in terms of an orthonormal or orthogonal basis. (Sec. 6.2)
  2. Find the matrix representing a linear function with respect to an orthonormal basis. (Sec. 6.2)
  3. Construct an orthonormal set in R^n, using the Gram-Schmidt process. (Sec. 6.2)
  4. Construct the first few elements of an orthonormal set of polynomials, with respect to an integral inner product of functions. (Sec. 6.2)
  5. Differentiate (with respect to a parameter) the inverse of a matrix. (Sec. 6.3)
  6. Calculate the basic vector differential operations (grad, div, curl). (Sec. 6.4)
  7. Use, and if necessary rederive, identities involving the vector differential operations. (Sec. 6.4)

Long-term objectives

  1. Understand concepts of arc length, unit tangent vector, and line integral (for later comparison with analogous concepts for surfaces). (Sec. 6.3)
  2. Become familiar with normal vector, binormal vector, curvature, and torsion of a curve (their geometrical meaning and their calculational construction). (Sec. 6.3) ._. NOTE: The short-term (testable skills) list could very well include "Calculate the normal vector, ... of a curve." However, our syllabus is too crowded to insist on this.
  3. Understand the origin of the vector differential operations, and the identities satisfied by them, in elementary algebraic and calculus operations and facts. (Sec. 6.4)