Week 3 -- Sections 2.4-3.2

Short-term objectives

  1. Construct the tangent hyperplane to the graph of a function from R^n to R, and interpret it in terms of the approximate change in the value of the function for a small change in its argument. (Sec. 2.4)
  2. Use the gradient vector to determine the directions of fastest increase and of level surfaces. (Sec. 2.4)
  3. Use the chain rule to calculate derivatives of composite functions R -> R^n -> R, including functions that depend on a scalar variable in two or more ways. (Sec. 2.4)
  4. Calculate 2 x 2 and 3 x 3 determinants and cross products. (Sec. 2.5)
  5. Use the algebraic properties of addition and scalar multiplication. (Sec. 3.1)
  6. Recognize familiar classes of mathematical entities on which these vector operations are defined. (Sec. 3.1)
  7. Determine whether a function [with domain R^n, P_n, or C^N(a,b)] is linear, or affine. (Sec. 3.2)
  8. Find the matrix of a linear function from R^n to R^p. (Sec. 3.2)
  9. Recognize or construct the matrices representing simple geometrical transformations in 2- and 3-dimensional space, such as rotations, reflections, and shears.

Long-term objectives

  1. Distinguish the various geometrical representations/interpretations of scalar-valued functions of a vector argument. (Sec. 2.4)
  2. Understand the concepts of vector space, domain, and codomain. (Sec. 3.1)