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math121b:sample-m1 [2020/02/19 09:04] pzhou |
math121b:sample-m1 [2020/02/23 16:56] (current) pzhou |
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* Find the metric tensor $g$ in terms of $u,v$ (or equivalently, | * Find the metric tensor $g$ in terms of $u,v$ (or equivalently, | ||
* The volume element. | * The volume element. | ||
- | * The expansion of $\d_u$ and $\d_v$ using $\d_x$ and $\d_y$ | + | * The expansion of $\d_u$ and $\d_v$ using $\d_x$ and $\d_y$. |
* (bonus) If a function on $\R^2$ is given by $f(u,v) = 2u + 3v$, find its gradient expressed using basis vectors $\d_u, \d_v$. | * (bonus) If a function on $\R^2$ is given by $f(u,v) = 2u + 3v$, find its gradient expressed using basis vectors $\d_u, \d_v$. | ||
* (bonus) Find the divergence of the vector field $V = \d_u$. | * (bonus) Find the divergence of the vector field $V = \d_u$. | ||
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$$ div(V^i \d_i) = \frac{1}{\sqrt{g}} \d_i(\sqrt{g} V^i) $$ | $$ div(V^i \d_i) = \frac{1}{\sqrt{g}} \d_i(\sqrt{g} V^i) $$ | ||
where we used Einstein summation convention, and $\d_i = \frac{\d}{\d u_i}$, $u_1 = u, u_2 = v$. | where we used Einstein summation convention, and $\d_i = \frac{\d}{\d u_i}$, $u_1 = u, u_2 = v$. | ||
+ | |||
+ | ====== Partial Solution ====== | ||
+ | 1. Draw the picture to see. You should draw a parallogram with $v$ along the diagonal, and two sides parallel to $e_1, | ||
+ | |||
+ | 2. Since $V$ is 3 dim, suffice to show that $f_i$ are linearly independent. That is, if we have | ||
+ | $$ 0 = c_1 f_1(t) + c_2 f_2(t) + c_3 f_3(t) $$ | ||
+ | for all $t$, then we need to show $c_i=0$. Plug in $t=2,3,5$ to conclude this is indeed so. | ||
+ | |||
+ | Same thing for the next problem, suppose we try to find the expansion | ||
+ | $$ f(t) = c_1 f_1(t) + c_2 f_2(t) + c_3 f_3(t) $$ | ||
+ | then we can get $c_i$ by plug-in $t=2, | ||
+ | |||
+ | You may worry about: what about $t$ equal to other values? Well, since we have an equation about degree two polynomials, | ||
+ | |||
+ | 3. This is good place to introduce another trick, the relation about 2-dim orthogonal coordinate and holomorphic functions. | ||
+ | |||
+ | Suppose $z = f(w)$ is a holomorphic function, if we write $z = x+ i y$, $w = u + iv$, we get Cauchy-Riemmann equation | ||
+ | $$ \begin{cases} | ||
+ | \frac{\d x}{\d u} = \frac{\d y}{\d v} \cr | ||
+ | \frac{\d x}{\d v} = - \frac{\d y}{\d u} | ||
+ | \end{cases} | ||
+ | $$ | ||
+ | Hence, we have | ||
+ | $$ g(\frac{\d }{\d u} ,\frac{\d }{\d u}) = \left( \frac{\d x}{\d u} \right)^2 + \left( \frac{\d y}{\d u} \right)^2 = \left( \frac{\d y}{\d v} \right)^2 + \left( - \frac{\d x}{\d v} \right)^2= g(\frac{\d }{\d v} ,\frac{\d }{\d v})$$ | ||
+ | and | ||
+ | $$ g(\frac{\d }{\d u} ,\frac{\d }{\d v}) = \frac{\d x}{\d u} \frac{\d x}{\d v} + \frac{\d y}{\d u} \frac{\d y}{\d v} = - \frac{\d x}{\d u} \frac{\d y}{\d u} + \frac{\d y}{\d u} \frac{\d x}{\d u} = 0 $$ | ||
+ | Hence, we have orthogonal coordinate, and furthermore, | ||
+ | $g_{uu} = g_{vv}$. | ||
+ | |||
+ | In this problem, we have | ||
+ | $$ x + i y = \cosh(u + iv) $$ | ||
+ | so the above method applies. Another examples is | ||
+ | $$ y = uv , x = (u^2 - v^2)/2 $$ | ||
+ | this is from | ||
+ | $$ x + i y = (u + iv)^2/ | ||
+ | |||
+ | Of course, one can do the problem without using the above trick. One then do | ||
+ | $$dx = ... du + ... dv, \quad dy = ... du + ... dv$$ | ||
+ | then plug into | ||
+ | $$ ds^2 = dx^2 + dy^2 $$ | ||
+ | to express $ds^2$ using $du$ and $dv$. You should get in the end | ||
+ | $$ ds^2 = (\cosh^2(u)- \cos^2(v)) (du^2 + dv^2) $$ | ||
+ | Or, the factor can be written in different ways, since | ||
+ | $$ \cosh^2(u)- \cos^2(v) = \sinh^2(u) + \sin^2(v) .$$ | ||
+ | |||
+ | Let $H^2 = \sinh^2(u) + \sin^2(v)$, then we have | ||
+ | $$ g = [g_{ij}] = \begin{pmatrix} H^2 & 0 \cr 0 & H^2 \end{pmatrix} $$ | ||
+ | So we have $g_{ij} = \delta_{ij} H^2, \quad g^{ij} = H^{-2} \delta_{ij}$. | ||
+ | |||
+ | The volume element is | ||
+ | $$ \sqrt{g} du dv = (\sinh^2(u) + \sin^2(v)) du dv$ | ||
+ | |||
+ | The expansion is | ||
+ | $$ \d_u = \d_u(x) \d_x + \d_u(y) \d_y = ... $$ | ||
+ | |||
+ | The gradient is | ||
+ | $$ grad(f) = H^{-2} \d_u(f) \d_u + H^{-2} \d_v(f) \d_v = H^{-2} (2 \d_u + 3 \d_v) $$ | ||
+ | |||
+ | The divergence of $V$ is | ||
+ | $$ div(V) = H^{-2} \d_u (H^2) = \frac{2 \cosh(u)\sinh(u)}{ \sinh^2(u) + \sin^2(v) }$$ | ||
+ | More details: $V = 1 \d_u + 0 \d_v$ so $V^1 = 1$, $V^2=0$, and $\sqrt{g} = \sqrt{H^4} = H^2$. Plug in the divergence formula will yield the answer. | ||
+ | |||
+ | In Boas' notation, we have $e_1 = \d_u / \| \d_u \| = \d_u / H$, so | ||
+ | $$V = \d_u = H e_1$$ | ||
+ | and $V^1_{Boas} = H$, also $h_1 = h_2 = H$. Using that Boas formula, we have | ||
+ | $$div V = \frac{1}{h_1h_2} \d_u( h_2 V^1_{Boas}) = H^{-2} \d_u (H^2) $$ | ||
+ | yield the same answer. | ||
+ | |||
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