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--- math121b:02-10 [2020/02/09 22:44]
pzhou created
+++ math121b:02-10 [2020/02/22 18:03] (current)
pzhou
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 ====== 2020-02-10, Monday ======
-We will first finish up the 'mathy' treatment of tensor analysis:
+ $\gdef\div{\text{div}} \gdef\vol{\text{Vol}} \gdef\b{\mathbf} \gdef\d{\partial}$
-  * cotangent vectors and differential 1-form
-  * finish the derivation of Laplacian
-  *
+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.
+===== Differential of a function is a 1-form (covector field)=====
+In Cartesian coordinate, the differential of a function  $f$  is
+ $df = \sum_i \frac{\d f }{\d x_i} dx_i.$
+In general coordinate  $(u_1, \cdots, u_n)$ , the differential of a function  $f$  is
+ $df =  \sum_i \frac{\d f }{\d u_i} d u_i.$
+You can specify the differential of a function directly:  $df$  at a point  $p \in \R^n$  is a linear function on  $T_p \R^n$ ,  $df(p) \in (T_p \R^n)^*$ . It does the following
+ df(p) : \b v_p \mapsto  \b v_p(f), \quad \b v_p \in T_p \R^n
+where  $\b v_p(f)$  is the directional derivative of  $f$  along  $\b v_p$ .
+===== Gradient of a function (is a vector field) =====
+In Cartesian coordinate, the gradient of a function is
+ $\gdef\grad{\text{ grad } }  \grad f = \sum_i \frac{\d f}{\d x_i} \frac{\d }{\d x_i}.$
+In general coordinate, the gradient of a function is more complicated
+ $\grad f = \sum_{i,j} g^{ij} \frac{\d f}{\d u_i} \frac{\d }{\d u_j},$
+where  $g^{ij}$  is the entry of the inverse matrix of the matrix  $[g_{kl}]$ . And it just happens that, for Cartesian coordinate,  $g^{ij} = \delta_{ij}$ .
+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 **  $\nabla f = \grad f.$
+===== Divergence of a Vector field (is a function) =====
+Let  $\R^n$  be the flat space, with standard coordinates  $(x_1, \cdots, x_n)$ .
+Let  $\b V$  be a vector field on  $\R^n$ , that is, for each point  $p \in \R^n$ , we specify a tangent vector
+ $\b V(p) = \sum_{i=1}^n V^i(p) \d_i \in T_p \R^n.$
+We require that  $V^i(p)$  varies smoothly with respect to  $p$ .
+The divergence of  $\b V$  is a function on  $\R^n$ ,
+ $\div(\b V) = \sum_{i=1}^n \d_i( V^i )$
+recall that  $V^i$  is a function on  $\R^n$ , and  $\d_i$  is taking the partial derivative with respect to  $x_i$ .
+** Notation **  $\nabla \cdot \b V = \div (\b V).$
+**What does divergence mean?** Geometrically, it measure the relative change-rate of the volume of an infinitesimal cube situated at point  $p$ . Suppose  $\Phi^t: \R^n \to \R^n$  is the flow generated by  $\b V$  (every point moves as dictated by  $\b V$ ). And let  $C= C(p, \epsilon)$  be a cube of side-length  $\epsilon$ , center at  $p$ .
+Then, we have the geometrical interpretation as
+ \div(\b V) = \lim_{\epsilon \to 0} \frac{1}{\vol(C)} \frac{d \vol(\Phi^t(C))}{dt} \vert_{t=0}.
+That is why, if  $S \subset \R^n$  is an open domain, we can compute the change-rate of the volume of  $S$  by
+ $\frac{d \vol(\Phi^t(S)) } {dt}\vert_{t=0} = \int_{S} \div(\b V)(\b x) \, d \vol(\b x).$
+** In curvilinear coordinate. **
+The formula for computing the divergence is the following, suppose  $\b V = \sum_i V^i \frac{\d }{\d u_i}$ , then
+ $\div (\b V) = \sum_{i=1}^n \frac{1}{\sqrt{|g|}} \frac{\d (\sqrt{|g|} V^i)}{\d u_i}$
+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)\,  \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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