math121b:02-10
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math121b:02-10 [2020/02/10 07:55] pzhou |
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| ====== 2020-02-10, Monday ====== | ====== 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. | + | |
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| + | 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 $\R^3$). | ||
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| + | Then, we will follow Boas 10.8 and 10.9, to reconcilliate | ||
| - | 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)===== | ===== Differential of a function is a 1-form (covector field)===== | ||
| In Cartesian coordinate, the differential of a function $f$ is | 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 | In general coordinate $(u_1, \cdots, u_n)$, the differential of a function $f$ is | ||
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| Note that $g_{ij}$ and $g^{ij}$ depends on the coordinate system. | Note that $g_{ij}$ and $g^{ij}$ depends on the coordinate system. | ||
| $$ g_{ij} = g(\frac{\d}{\d u_i}, \frac{\d}{\d u_j}), \quad g^{ij} = g^*(d u_i, d u_j). $$ | $$ g_{ij} = g(\frac{\d}{\d u_i}, \frac{\d}{\d u_j}), \quad g^{ij} = g^*(d u_i, d u_j). $$ | ||
| + | Beware that $\nabla u_i \neq \frac{\d}[\d u_i}$. | ||
| ** Notation ** $$ \nabla f = \grad f.$$ | ** Notation ** $$ \nabla f = \grad f.$$ | ||
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| $$ \div (\b V) = \sum_{i=1}^n \frac{1}{\sqrt{|g|}} \frac{\d (\sqrt{|g|} V^i)}{\d u_i} $$ | $$ \div (\b V) = \sum_{i=1}^n \frac{1}{\sqrt{|g|}} \frac{\d (\sqrt{|g|} V^i)}{\d u_i} $$ | ||
| - | 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 |
| - | $$ \int_{\R^n} (\nabla \cdot \b V)(u) f(u) \sqrt{|g|(u)} du_1\cdots d u_n = \int_{\R^n} | + | $$ \int_{\R^n} (\nabla \cdot \b V)\, \varphi\, |
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| + | ====== Back to Boas ====== | ||
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| + | ===== Section 10.8 ===== | ||
| + | For Cartesian coordinate, we have basis vectors $\b i, \b j, \b k$. | ||
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| + | 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: | ||
| + | $$ \b a_r = \frac{\d }{\d r}, \quad \b a_\theta = \frac{\d }{\d \theta}, \cdots $$ | ||
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| + | See Example 2 on page 523, for how to consider a general curvilinear coordinate $(x_1, x_2, x_3)$ on $\R^3$. | ||
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| + | 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. | ||
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| + | ** 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. | ||
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| + | ===== Section 10.9 ===== | ||
| + | Suppose we have orthogonal coordinate system $(x_1, x_2, x_3)$, and **unit** basis vectors $\b e_i$, we have | ||
| + | $$ \b a_i = \frac{\d }{\d x_i} = h_i \b e_i. $$ | ||
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| + | 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 $\frac{\d }{\d x_i}$) | ||
| + | $$ \b V = \sum_i V^i \b e_i $$. | ||
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| + | ==== Divergence. ==== | ||
| + | Try to do problem 1. | ||
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| + | An important property is the " | ||
| + | $$ \b \nabla \cdot (f \b V) = \b \nabla f \cdot \b V + f \b \nabla | ||
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| + | ==== Curl ==== | ||
| + | To compute the curl, we note the following rule | ||
| + | $$ \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 $$. | ||
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| + | 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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