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--- math121b:02-10 [2020/02/10 00:38]
pzhou
+++ math121b:02-10 [2020/02/22 18:03] (current)
pzhou
@@ Line 1: / Line 1: @@
 ====== 2020-02-10, Monday ======
- $\gdef\div{\text{div}} \gdef\vol{\text{Vol}} \gdef\b{\mathbf}$
+\gdef\div{\text{div}} \gdef\vol{\text{Vol}} \gdef\b{\mathbf} \gdef\d{\partial}
@@ Line 71: / Line 71: @@
 See Example 2 on page 523, for how to consider a general curvilinear coordinate  $(x_1, x_2, x_3)$  on  $\R^3$ .
+The notation  $d \b s$  corresponds to
+ $\sum_{i=1}^n \frac{\d }{\d x_i} \otimes d x_i \in (T_p \R^n) \otimes (T_p \R^n)^*.$
+An element  $T$  in  $V \otimes V^*$  can be viewed as a linear operator  $V \to V$ , by inserting  $v \in V$  to the second slot of  $T$ . In this sense  $d \b s$  is the identity operator on  $T_p \R^n$ . You might have seen in Quantum mechanics the bra-ket notation  $1 = \sum_n | n \rangle \otimes \langle n |$  ((  $\otimes$  sometimes omitted as usual in physics.)) It is the same thing, where  $| n \rangle \in V$  forms a basis and  $\langle n | \in V^*$  are the dual basis.
 ** Orthogonal coordinate system (ortho-curvilinear coordinate) **, the matrix  $g_{ij}$  is diagonal, with entries  $h_i^2$  (not to be confused with our notation for dual basis). This is

Lecture Notes