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math121b:02-10 [2020/02/10 00:29]
pzhou 		math121b:02-10 [2020/02/22 18:03] (current)
pzhou
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 ====== 2020-02-10, Monday ====== ====== 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}
  
  
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 See Example 2 on page 523, for how to consider a general curvilinear coordinate (x1,x2,x3)(x_1, x_2, x_3) on R3\R^3.  See Example 2 on page 523, for how to consider a general curvilinear coordinate (x1,x2,x3)(x_1, x_2, x_3) on R3\R^3. 
  
 +The notation dsd \b s corresponds to 
 +∑i=1n∂∂xi⊗dxi∈(TpRn)⊗(TpRn)∗. \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 TT in V⊗V∗V \otimes V^* can be viewed as a linear operator V→VV \to V, by inserting v∈Vv \in V to the second slot of TT. In this sense dsd \b s is the identity operator on TpRnT_p \R^n. You might have seen in Quantum mechanics the bra-ket notation 1=∑n∣n⟩⊗⟨n∣1 = \sum_n | n \rangle \otimes \langle n | (( ⊗\otimes sometimes omitted as usual in physics.)) It is the same thing, where ∣n⟩∈V| n \rangle \in V forms a basis and ⟨n∣∈V∗ \langle n | \in V^* are the dual basis.  
 +
 +** Orthogonal coordinate system (ortho-curvilinear coordinate) **, the matrix gijg_{ij} is diagonal, with entries hi2h_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 (x1,x2,x3)(x_1, x_2, x_3), and **unit** basis vectors ei\b e_i, we have
 +ai=∂∂xi=hiei. \b a_i = \frac{\d }{\d x_i} = h_i \b e_i.
 +
 +Given a vector field VV, we write its component in the basis of ei\b e_i (warning! this is not our usual notation, we usual write with basis ∂∂xi\frac{\d }{\d x_i}) 
 +V=∑iViei \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 To compute the curl, we note the following rule
 ∇×(fV)=∇(f)×V+f∇×V \b \nabla \times (f \b V) = \b \nabla(f) \times \b V + f \b \nabla \times \b V  ∇×(fV)=∇(f)×V+f∇×V \b \nabla \times (f \b V) = \b \nabla(f) \times \b V + f \b \nabla \times \b V
math121b/02-10.1581323375.txt.gz · Last modified: 2020/02/10 00:29 by pzhou

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