Week 6 -- Sections 4.2-4.4

Short-term objectives

1. 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.
2. Use dimension counts to decide certain independence and spanning questions. (Sec. 4.3 ("summary theorem"))
3. Find the expansion of a vector in terms of a basis (in R^n, P_n, and other familiar spaces). (Sec. 4.4)
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)
5. Find the matrix representing a linear function with respect to given bases for the domain and codomain. (Sec. 4.4)

Long-term objectives

1. 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)
2. Gain facility in reasoning about independence, spanning, and dimension. (Sec. 4.3)
3. Understand the connection between linear coordinate transformations and the multivariable chain rule. (Sec. 4.4)