Week 9 -- Sections 5.5, 6.1

Short-term objectives

1. Find the differential (or Jacobian matrix) of a nonlinear function from that of its inverse. (Sec. 5.5)
2. Find normal vectors to the coordinate surfaces (or lines) of a curvilinear coordinate system. (Sec. 5.5)
3. Find the differential (or partial derivatives) of a multivariable function defined implicitly. (Sec. 5.5)
4. Test whether a function of two vector variables qualifies as an inner product. (Sec. 6.1)
5. Use the Schwartz, triangle, and norm-difference inequalities. (Sec. 6.1)

Long-term objectives

1. Appreciate the implicit and inverse function theorems in matrix-algebra terms. (Sec. 5.5)
2. Become alert to the algebraic and geometrical symptoms of breakdown of existence, uniqueness, or smoothness of inverse and implicit functions. (Sec. 5.5)
3. Understand the concepts of inner product, norm, and metric (on spaces more general than R^n) and how each is a specialization of the next. (Sec. 6.1)