### Week 6 -- Sections 4.2-4.4

#### Short-term objectives

- Calculate the basis of tangent vectors to the coordinate curves at a
point, in a curvilinear coordinate system. (Sec. 4.2)
NOTE: A better method for finding the normal vectors to the
coordinate surfaces will emerge in Sec. 5.5.
- Use dimension counts to decide certain independence and spanning
questions. (Sec. 4.3 ("summary theorem"))
- Find the expansion of a vector in terms of a basis (in R^n, P_n, and
other familiar spaces). (Sec. 4.4)
- Find the transformation matrix from one basis to another; distinguish
the roles of a matrix and its inverse, transpose, and contragredient.
(Sec. 4.4)
- Find the matrix representing a linear function with respect to given
bases for the domain and codomain. (Sec. 4.4)

#### Long-term objectives

- Understand the distinction between the tangent-vector basis and the
normal-vector basis (and the related orthonormal basis, in the case
of an orthogonal coordinate system). (Sec. 4.2)
- Gain facility in reasoning about independence, spanning, and
dimension. (Sec. 4.3)
- Understand the connection between linear coordinate transformations and
the multivariable chain rule. (Sec. 4.4)