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Problems to Chapter 7

1. Problem 1
2. Problem 2
3. Problem 3
4. Problem 4

Problem 1.

For $\Phi (x,y,t)= t^4 + yt^2 +xt$ find

1. $\Theta: = \{(x,y,t)\colon \Phi_t(x,y, t)=0\}$;
2. $\Theta_{\text{sing}}: = \{(x,y,t)\colon \Phi_{tt} (x,y,t)=\Phi _{tx}(x,y,t) = \Phi _{ty}(x,y,t)=0\}$;
3. Lagrangian manifold $\Lambda := \{(x,y,\Phi_x,\Phi_y) \colon (x,y,t)\in \Theta \setminus \Theta_{\text{sing}}\}$;
4. $\pi_X \Lambda :=\{(x,y)\colon \exists t\ \Phi _t (x,y,t)=0\}$;
5. Caustic set $\Lambda_0:= (x,y)\colon \exists t\ \Phi _t (x,y,t)=\Phi _{tt} (x,y,t)=0 \}$.

Problem 2.

For $\Phi (x,y,z,t)= t^5 + zt^3+ yt^2 +xt$ find

1. $\Theta: = \{(x,y,z,t)\colon \Phi_t(x,y,z, t)=0\}$;
2. $\Theta_{\text{sing}}: = \{(x,y,z,t)\colon \Phi_{tt} (x,y,z,t)=\Phi _{tx}(x,y,z,t) = \Phi _{ty}(x,y,z,t)=\Phi _{tz}(x,y,z,t)=0\}$;
3. Lagrangian manifold $\Lambda := \{(x,y,z,\Phi_x,\Phi_y,\Phi_z) \colon (x,y,z,t)\in \Theta \setminus \Theta_{\text{sing}}\}$;
4. $\pi_X \Lambda :=\{(x,y,z)\colon \exists t\ \Phi _t (x,y,z,t)=0\}$;
5. Caustic set $\Lambda_0:= (x,y,z)\colon \exists t\ \Phi _t (x,y,z,t)=\Phi _{tt} (x,y,z,t)=0 \}$.

Problem 3. Calculate asymptotics as $k\to \infty$ \begin{gather*} \iint_{-\infty}^\infty e^{i k (s^2t^2 -sx -ty)}\,ds dt. \end{gather*}

Problem 4. Calculate asymptotics as $k\to \infty$ \begin{gather*} \iint_{-\infty}^\infty e^{i k (s^3t^3 -sx -ty)}\,ds dt. \end{gather*}

$\Leftarrow$  $\Uparrow$  $\Rightarrow$