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Problem: Find the curvature and the torsion of the curve: \[ x=\cos ^{2} t, \quad y=\sin ^{3} t, \quad z=\cos 2 t . \]

7.4 Differential geometry

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Problem: \( A=\{a, b, c\}, B=\{1,2,3,4\}, P_{1} \subseteq A \cdot B, P_{2} \subseteq B^{2} \). Represent \( P_{1}, P_{2} \) graphically. Find the matrix \( \left(P_{1} \circ P_{2}\right)^{-1} \). Using the matrix, check whether the relation \( P_{2} \) is reflexive, symmetric, antisymmetric, transitive? What class of relations does it belong to? \[ \begin{array}{l} P_{1}=\{(a, 1),(a, 2),(a, 3),(a, 4),(b, 3),(c, 2)\} \\ P_{2}=\{(1,1),(1,4),(2,2),(2,3),(3,3),(3,2),(4,1),(4,4)\} . \end{array} \]

6.2.1 Binary relations

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Problem: Find the domain of definition, the domain of values of the relation \( P \). Is the relation \( P \) reflexive, antireflexive, symmetrical, antisymmetric, nonsymmetrical, transitive? What class of relations does it belong to? Justify your answer. \[ P \subseteq R^{2},(x, y) \in P \Leftrightarrow x \cdot y>1 . \]

6.2.2 Binary relations

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Problem: Present the given relationship in the form of a graph and a matrix. Determine the properties of the relation (symmetry, reflexivity, transitivity). \[ R=\left\{\begin{array}{l} a b \\ a c \\ c b \\ a a \end{array}\right\} . \]

6.2.3 Binary relations

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Problem: Show that \( R \circ R \subseteq R \) for any transitive binary relation \( R \). Can this inclusion be strict?

6.2.4 Binary relations

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Problem: Let's consider on the set \( \mathbb{Z}^{2} \) the binary relation \( (k, l) \sim(m, n) \), which means that \( m+n-k-l \) is divisible by 3 . Is it an equivalence? Draw on checkered paper all such points \( (m, n), 0 \leq \) \( m, n \leq 10 \) that \( (m, n) \sim(0,0) \). Find the maximum number of pairwise incomparable points.

6.2.5 Binary relations

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Problem: Show that the relation \( x \equiv y(\bmod \mathbb{Z}) \), meaning that \( x-y \in \mathbb{Z} \), gives an equivalence on the set of real numbers \( R \), and construct an explicit bijection between the quotient set \( R / \mathbb{Z} \) and the unit circle \( S^{1} \stackrel{\text { def }}{=}\left\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2}=1\right\} \) in \( \mathbb{R}^{2} \). Does the standard order in \( \mathbb{R} \) induce any order in \( \mathbb{R} / \mathbb{Z} \) ? Construct an explicit bijection between the functions \( S^{1} \rightarrow \mathbb{R} \) and the periodic functions \( \mathbb{R} \rightarrow \mathbb{R} \) of period 1 .

6.2.6 Binary relations

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Problem: Prove the identity, using the laws of Boolean algebra. Represent one of the expressions in the basis of elementary functions: \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|} \hline\( y_{1} \) & \( y_{2} \) & \( y_{3} \) & \( y_{4} \) & \( y_{5} \) & \( y_{6} \) & \( y_{7} \) & \( y_{8} \) & \( y_{9} \) & \( y_{10} \) & \( y_{11} \) \\ \hline 0 & \begin{tabular}{l} \( a \) \\ \( \wedge b \) \end{tabular} & \( a \) & \( a \oplus b \) & \begin{tabular}{l} \( a \) \\ \( \vee \vee b \) \end{tabular} & \begin{tabular}{l} \( a \) \\ \( \downarrow b \) \end{tabular} & \begin{tabular}{l} \( a \) \\ \( \Leftrightarrow b \) \end{tabular} & \( \bar{a} \) & \begin{tabular}{l} \( a \) \\ \( \rightarrow b \) \end{tabular} & \( a \mid b \) & 1 \\ \hline \end{tabular} The set of basic function numbers should include the numbers of your option. For example, for option 1 \( y_{1}, y_{11}, y_{10} \) can be taken, the missing functions are selected based on Post's completeness theorem. \[ \begin{array}{l} ((a \wedge \bar{c}) \downarrow(b \wedge \bar{c})) \wedge((a \mid d)(\overrightarrow{b \wedge d}))= \\ =((a \mid b) \mid(a \oplus \bar{b})) \rightarrow((c \oplus d) \wedge(d \rightarrow c)), \text { option } 2 . \end{array} \]

6.3.1 Boolean algebra

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Problem: Find the image of domain \( D=\{z \in \mathbb{C}:|z+1-2 i|<3\} \) for the linear - fractional transformation \[ f(z)=\frac{z+1+i}{z-1-i} \text {. } \]

10.5.1 Conformal mappings

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Problem: Find the image of domain \[ D=\{|z|>1\} \cup\left\{|z| \leq 1, I_{m} z>0\right\} \backslash\{z=i t,-4 \leq t<-1\} \] when mapped by the Joukowsky transform: \[ f(z)=\frac{1}{2}\left(z+\frac{1}{z}\right) \text {. } \]

10.5.2 Conformal mappings

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Problem: Find the image of domain \[ D=\{\operatorname{Re} z<0\} \cup\{0<\operatorname{Im} z<2 \pi\} \backslash\{z \neq t+2 \pi,-\infty

10.5.3 Conformal mappings

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Problem: Find the image of domain \[ D=\{\operatorname{Re} z<0\} \cup\{0<\operatorname{Im} z<2\} \backslash\{z \neq t+i,-\infty

10.5.4 Conformal mappings

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Problem: Find the image of domain \( D=\{0<\operatorname{Re} z<1\} \cup\{\operatorname{Im} z>0\} \) for the function mapping \( f(z)=\tanh i \pi z \).

10.5.5 Conformal mappings

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Problem: Find the tangent planes of the surface \( z=x^{4}-2 x y^{3} \), orthogonal to the vector \( \vec{a}(-2,6,1) \).

7.6 Differential geometry

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Problem: Find the angle between the lines \( u+v=0 \), \( u=\tan v \) at their common point on the surface \[ x=u \cos v, y=u \sin v, z=u+v . \]

7.7 Differential geometry

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