Differences

This shows you the differences between two versions of the page.

--- math121b:02-10 [2020/02/10 08:29]
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 72: / Line 72: @@
 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
+the case we will be considering mainly.
+===== Section 10.9 =====
+Suppose we have orthogonal coordinate system $(x_1, x_2, x_3)$, and **unit** basis vectors $\b e_i$, we have
+$$ \b a_i = \frac{\d }{\d x_i} = h_i \b e_i. $$
+Given a vector field $V$, we write its component in the basis of $\b e_i$ (warning! this is not our usual notation, we usual write with basis $\frac{\d }{\d x_i}$)
+$$ \b V = \sum_i V^i \b e_i $$.
+==== Divergence. ====
+Try to do problem 1.
+An important property is the "Leibniz rule"
+$$ \b \nabla \cdot (f \b V) = \b \nabla f \cdot \b V + f  \b \nabla  \cdot \b V.$$
+==== Curl ====
 To compute the curl, we note the following rule
 $$ \b \nabla \times (f \b V) = \b \nabla(f) \times \b V + f \b \nabla \times \b V $$

Lecture Notes

User Tools

Site Tools

Differences

Page Tools