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--- math121b:02-10 [2020/02/10 08:38]
pzhou
+++ math121b:02-10 [2026/02/21 14:41] (current)
@@ 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