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math121b:02-10 [2020/02/09 23:34] pzhou |
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====== 2020-02-10, Monday ====== | ====== 2020-02-10, Monday ====== | ||
- | 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. | + | $$\gdef\div{\text{div}} \gdef\vol{\text{Vol}} \gdef\b{\mathbf} \gdef\d{\partial}$$ |
- | 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 ===== | + | 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 the math notation and physics notations. | ||
+ | |||
+ | |||
+ | |||
+ | ===== Differential of a function | ||
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 | ||
- | $$ df = \sum_i \frac{\df }{\d u_i} d u_i. $$ | + | $$ df = \sum_i \frac{\d f }{\d u_i} d u_i. $$ |
- | You can specify the differential of a function directly: $df$ at a point $p \in \R^n$, is a linear function on $T_p \R^n$: it sends an element | + | 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 |
+ | $$ df(p) : \b v_p \mapsto | ||
+ | where $\b v_p(f)$ is the directional derivative of $f$ along $\b v_p$. | ||
===== Gradient of a function (is a vector field) ===== | ===== Gradient of a function (is a vector field) ===== | ||
In Cartesian coordinate, the gradient of a function is | In Cartesian coordinate, the gradient of a function is | ||
- | $$ \gdef\grad{\text{ grad } } \grad f = \sum_i $$ | + | $$ \gdef\grad{\text{ grad } } \grad f = \sum_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) ===== | ===== Divergence of a Vector field (is a function) ===== | ||
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The divergence of $\b V$ is a function on $\R^n$, | The divergence of $\b V$ is a function on $\R^n$, | ||
- | $$ div(\b V) = \sum_{i=1}^n \d_i( V^i ) $$ | + | $$ \div(\b V) = \sum_{i=1}^n \d_i( V^i ) $$ |
recall that $V^i$ is a function on $\R^n$, and $\d_i$ is taking the partial derivative with respect to $x_i$. | 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, | **What does divergence mean?** Geometrically, | ||
Then, we have the geometrical interpretation as | Then, we have the geometrical interpretation as | ||
- | $$ \gdef\div{\text{div}} \gdef\vol{\text{Vol}} | + | $$ \div(\b V) = \lim_{\epsilon \to 0} \frac{1}{\vol(C)} \frac{d \vol(\Phi^t(C))}{dt} \vert_{t=0}. $$ |
That is why, if $S \subset \R^n$ is an open domain, we can compute the change-rate of the volume of $S$ by | That is why, if $S \subset \R^n$ is an open domain, we can compute the change-rate of the volume of $S$ by | ||
$$ \frac{d \vol(\Phi^t(S)) } {dt}\vert_{t=0} = \int_{S} \div(\b V)(\b x) \, d \vol(\b x). $$ | $$ \frac{d \vol(\Phi^t(S)) } {dt}\vert_{t=0} = \int_{S} \div(\b V)(\b x) \, d \vol(\b x). $$ | ||
- | If we are given a curvilinear | + | ** In curvilinear coordinate. ** |
+ | The formula for computing the divergence is the following, suppose $\b V = \sum_i V^i \frac{\d }{\d u_i}$, then | ||
+ | $$ \div (\b V) = \sum_{i=1}^n \frac{1}{\sqrt{|g|}} \frac{\d (\sqrt{|g|} V^i)}{\d u_i} $$ | ||
+ | |||
+ | The reason | ||
+ | $$ \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 $$ | ||
+ | |||
+ | ====== Back to Boas ====== | ||
+ | |||
+ | ===== Section 10.8 ===== | ||
+ | For Cartesian | ||
+ | |||
+ | For spherical coordinate, we have **unit** basis vectors $\b e_r, \b e_\theta, \b e_\phi$, and corresponding coordinate basis vectors $\b a_r, \b a_\theta, \b a_\phi$ (not unit length). These $\b a_n$ corresponds to our coordinate basis tangent vectors: | ||
+ | $$ \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 $$. | ||
+ | |||
+ | ==== Divergence. ==== | ||
+ | Try to do problem 1. | ||
+ | |||
+ | An important property is the " | ||
+ | $$ \b \nabla \cdot (f \b V) = \b \nabla f \cdot \b V + f \b \nabla | ||
+ | |||
+ | ==== 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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