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--- math121b:02-10 [2020/02/09 23:55]
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\div{\text{div}} \gdef\vol{\text{Vol}} \gdef\b{\mathbf} \gdef\d{\partial}
-We first finish up the derivation of Laplacian last time. See the lecture note [[02-07]]. Then, we introduce two vector operations, divergence and curl. Finally, we give a summary of the mathematical treatment of tensor analysis.
+We first finish up the derivation of Laplacian last time. See the lecture note [[02-07]]. Then, we review three concepts $df, \nabla f, \nabla \cdot \b V $($ \nabla \times \b V $is a bit special for$ \R^3$).
+Then, we will follow Boas 10.8 and 10.9, to reconcilliate the math notation and physics notations.
-Then, we will follow Boas 10.9 and 10.10, to introduce the physic-engineer notations: the nabla operator  $\gdef\b{\mathbf} \b \nabla$ .
 ===== Differential of a function is a 1-form (covector field)=====
 In Cartesian coordinate, the differential of a function  $f$  is
- df = \sum_i \frac{\df }{\d x_i} dx_i.
+ df = \sum_i \frac{\d f }{\d x_i} dx_i.
 In general coordinate  $(u_1, \cdots, u_n)$ , the differential of a function  $f$  is
@@ Line 26: / Line 31: @@
 Note that  $g_{ij}$  and  $g^{ij}$  depends on the coordinate system.
  $g_{ij} = g(\frac{\d}{\d u_i}, \frac{\d}{\d u_j}), \quad g^{ij} = g^*(d u_i, d u_j).$
+Beware that $\nabla u_i \neq \frac{\d}[\d u_i}$.
 ** Notation ** $$ \nabla f = \grad f.$$
@@ Line 53: / Line 59: @@
  $\div (\b V) = \sum_{i=1}^n \frac{1}{\sqrt{|g|}} \frac{\d (\sqrt{|g|} V^i)}{\d u_i}$
-Exercise: prove that for any compactly supported function  $\varphi$  ((a compactly supported function on  $\R^n$  is a function that vanishes outside a sufficently large ball. )), we have
+The reason we have the above formula is that , for any compactly supported function  $\varphi$  ((a compactly supported function on  $\R^n$  is a function that vanishes outside a sufficently large ball. )), we have
- \int_{\R^n} (\nabla \cdot \b V)(u)  f(u) \sqrt{|g|(u)} du_1\cdots d u_n = \int_{\R^n} g(\b V, \nabla f(u)) \sqrt{|g|(u)} du_1\cdots d u_n
+ \int_{\R^n} (\nabla \cdot \b V)\,  \varphi\, \sqrt{|g|} du_1\cdots d u_n = \int_{\R^n} \b V \cdot (\nabla \varphi)\, \sqrt{|g|} du_1\cdots d u_n
+====== Back to Boas ======
+===== Section 10.8 =====
+For Cartesian coordinate, we have basis vectors  $\b i, \b j, \b k$ .
+For spherical coordinate, we have **unit** basis vectors  $\b e_r, \b e_\theta, \b e_\phi$ , and corresponding coordinate basis vectors  $\b a_r, \b a_\theta, \b a_\phi$  (not unit length). These  $\b a_n$  corresponds to our coordinate basis tangent vectors:
+ $\b a_r = \frac{\d }{\d r}, \quad \b a_\theta = \frac{\d }{\d \theta}, \cdots$
+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
+ $\b \nabla \times (f \b V) = \b \nabla(f) \times \b V + f \b \nabla \times \b V$
+and
+ $\b \nabla \times \nabla f = 0$ .
+In the ortho-curvilinear coordinate, we can use the above rule to get a formula for the curl. I will not test on the curl operator in the orthocurvilinear case.

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