東京大学 新領域創成科学研究科 複雑理工学専攻 2024年8月実施 専門基礎科目

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• Tokyo-University

Problem No. 1 (Calculus)

Answer the following questions. e is the base of the natural logarithm and i is the imaginary unit.

Omit the derivations and write only the answers.

(Q.1) Define the real function f(x) as follows:

$f(x) = \int_0^x \mathrm{d}t \,\mathrm{e}^{-t^2}.$

• (1) Obtain the derivative of f(x).
• (2) Obtain the Taylor expansion of f(x) around x = 0 up to the third order.
• (Q.2) Two real functions x(t) and y(t) satisfy the following coupled differential equations:

$\frac{\mathrm{d}x}{\mathrm{d}t} = -\nu x - y + A\cos at,$ $\frac{\mathrm{d}y}{\mathrm{d}t} = -\nu y + x + A\sin at,$

where A, a, and $\nu$ ( $\nu > 0$ ) are real constants.

• (1) Let z(t) = x(t) + iy(t). Obtain the differential equation for z(t).

• (2) Let A = 0. Obtain the general solution x(t) and y(t) of the coupled differential equations.

• (3) Let A > 0. x(t) asymptotically approaches $x_1(t) = B\cos(bt + \phi)$ as $t \to \infty$ regardless of the initial condition. Here, B (B > 0), b, and $\phi$ are real constants. Obtain B, b, and $\phi$ .

• (Q.3) Define the real function f(x) as follows:

$f(x) = xe^{-x}.$

• (1) Obtain all the extrema of f(x) and the corresponding values of x.
• (2) In an xy Cartesian coordinate system, obtain the area of the region defined by $0 \le y \le f(x)$ and x > 0.
• (3) In an xyz Cartesian coordinate system, obtain the volume of the solid formed by rotating the xy-plane region defined in (2) around the x-axis.
• (4) The surface of the solid defined in (3) can be expressed in terms of the real function

$g(x, y, z) = y^2 + z^2 - x^2 e^{-2x},$

as g(x, y, z) = 0. Obtain the gradient vector of g(x, y, z).

(5) Obtain the coordinates (x, y, z) where h(x, y, z) = xyz is maximized on the surface of the solid defined in (3) in the x > 0 region. In addition, obtain the maximum.

Problem No. 2.1 (Linear algebra)

The transpose of a matrix or a vector is denoted by the superscript $\top$ . The exponential of a square matrix J is defined as

$e^J = \sum_{k=0}^{\infty} \frac{1}{k!} J^k.$

Answer the following questions.

(Q.1) Consider the following real matrix A and real vectors $\boldsymbol{b}$ and $\boldsymbol{c}$ :

$A = \begin{pmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{pmatrix}, \quad \boldsymbol{b} = \begin{pmatrix} 1 \\ 1 \\ \beta \end{pmatrix}, \quad \boldsymbol{c} = \begin{pmatrix} 3 \\ 2 \\ 1 \end{pmatrix},$

where $\beta$ is a constant.

Omit the derivations and write only the answers.

• (1) Obtain the eigenvalues $\lambda_1 \leq \lambda_2 \leq \lambda_3$ of A.
• (2) Obtain the matrix P that satisfies

$P^{\top}AP = \begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & \lambda_3 \end{pmatrix}.$

• (3) Suppose that the equation $A\mathbf{x} = \mathbf{b}$ for the variable $\mathbf{x} \in \mathbb{R}^3$ has a solution. Obtain the value of $\beta$ . In addition, obtain all the solutions of the equation.

• (4) Obtain $\lim_{t\to\infty} e^{-At} \boldsymbol{c}$ , where t is a real number.

• (Q.2) Let B be an $n \times n$ real asymmetric matrix.

  • (1) Show that the eigenvalues of B and $B^{\top}$ are the same. You may use the fact that the determinants of a square matrix and its transposed matrix are the same.
  • (2) Let $\lambda_i$ (i = 1, ..., n) be the eigenvalues of B. Let $\boldsymbol{u}_i$ and $\boldsymbol{v}_i$ be the eigenvectors of B and $B^{\top}$ corresponding to the eigenvalue $\lambda_i$ , respectively. Namely,

$B\boldsymbol{u}_i = \lambda_i \boldsymbol{u}_i,$ $B^{\top} \boldsymbol{v}_i = \lambda_i \boldsymbol{v}_i$

hold true. Show that $\boldsymbol{v}_i^{\top} \boldsymbol{u}_i = 0$ for $\lambda_i \neq \lambda_j$ .

Problem No. 2.2 (Probability and Statistics)

Answer the following questions. e is the base of the natural logarithm, and i is the imaginary unit.

Omit the derivations and write only the answers.

• (Q.1) Let X and Y be random variables with the means $\overline{X} = 1$ , $\overline{Y} = 2$ , the variances $s_X = 1$ , $s_Y = 2$ , respectively, and the covariance $s_{XY} = -1$ . Random variables U and V are defined by U = X + 3 and V = X + Y.
  • (1) Obtain the means $\overline{U}$ , $\overline{V}$ of U, V.
  • (2) Obtain the variances $s_U$ , $s_V$ of U, V.
  • (3) Obtain the covariance $s_{UV}$ between U and V.
• (Q.2) Let X be a random variable obeying the standard Cauchy distribution:

$p(x) = \frac{1}{\pi (1 + x^2)}.$

Answer the following questions.

• (1) Obtain the cumulative distribution function F(x) of the standard Cauchy distribution.

• (2) Let U be a random variable obeying the uniform distribution on the interval [0,1]. Obtain the probability $\Pr(U \leq F(x))$ , where F(x) is the cumulative distribution function in (1).

• (3) For U defined in (2), obtain the real function g(U) that satisfies X = g(U).

• (4) Obtain the characteristic function of the standard Cauchy distribution: $\psi(t) = \int_{-\infty}^{\infty} e^{itx} p(x) dx$ , where t is a real number.

• (5) Let $X_1, X_2, ..., X_n$ be n random variables that are independent and identically distributed and obey the standard Cauchy distribution. A random variable Z is defined by $Z = X_1 + X_2 + ... + X_n$ . Obtain the probability density function of Z.

• (Q.3) Let $(x_1, y_1), (x_2, y_2), \ldots, (x_N, y_N)$ be N sample pairs of random variables X and Y. Define the following statistics on the sample pairs:

$\overline{x} = \frac{1}{N} \sum_{i=1}^{N} x_i, \quad \overline{y} = \frac{1}{N} \sum_{i=1}^{N} y_i,$

$\sigma_{xx} = \frac{1}{N} \sum_{i=1}^{N} x_i x_i, \quad \sigma_{xy} = \frac{1}{N} \sum_{i=1}^{N} x_i y_i.$

The regression line $Y = A_0 + A_1 X$ is obtained by minimizing the following function l:

$l(A_0, A_1) = \sum_{i=1}^{N} (y_i - A_0 - A_1 x_i)^2,$

which is assumed to be minimum at $A_0 = a_0$ and $A_1 = a_1$ . Express $a_0$ and $a_1$ in terms of $\overline{x}$ , $\overline{y}$ , $\sigma_{xx}$ , and $\sigma_{xy}$ . In addition, show the condition under which both $a_0$ and $a_1$ are uniquely determined.

Problem No. 2.3 (Mechanics)

Answer the following questions. Let the gravitational acceleration be q > 0, constant). Write only the answers.

• (Q.1) A liner molecule ABA is composed of Atoms A and B. Atom B is located between two Atoms A. Consider Molecule ABA as a system composed of three point masses connected by two massless springs of spring constant k. The masses of Atoms A and B are m and M, respectively. The atoms move along a common line. Ignore the gravity.
  • (1) Write the equations of motion for the three atoms. The displacements of Atoms A, B, A from the natural lengths are defined as x, y, and z, respectively, with the positive direction from one Atom A to the other Atom A.
  • (2) Introduce $Q_1 = x + z$ and $Q_2 = x z$ . Answer the angular frequencies $\omega_1$ and $\omega_2$ of their oscillation. Let the center of the gravity of Molecule ABA do not move.
• (Q.2) Consider a point mass thrown at an elevation angle $\theta$ with initial velocity $v \neq 0$ . x axis is taken in the positive direction of the horizontal component of the initial velocity. y axis is taken in the

vertical upward direction. The origin is the initial position of the point mass.

• (1) Answer the vertical position y of the point mass at x = X (> 0), using X, g, $\theta$ , and v.
• (2) Answer the range of the vertical position y where the point mass cannot reach at x = X (> 0) for any angles $\theta$ , using X, g, and v.
• (Q.3) A uniform-density rigid sphere of mass m and radius r is at rest on a frictionless horizonal plate. We want to roll the sphere without slipping by hitting horizontally with a stick as shown in the figure below. Answer the vertical distance h from the center of the sphere to hit using m, r, and the moment of inertia I of the rigid sphere.

(Q.4) Suppose the Earth is a rigid sphere that rotates eastward around the axis passing through the south pole and the north pole at a constant angular velocity $\omega$ . Let (x, y, z) be the coordinate system fixed to the Earth's surface with the origin at Point P on the northern hemisphere at latitude $\theta$ ( $0 < \theta < \frac{\pi}{2}$ ). Positive direction of z axis is defined in the direction from the Earth's center to Point P, and positive directions of x axis and y axis are defined towards the south and east, respectively, on the tangential plane at Point P. A point mass of mass m is thrown from Point P to the positive direction of z axis at initial velocity v.

• (1) Answer the x, y, and z components of the Earth's angular velocity vector $\boldsymbol{\omega}$ in the coordinate system (x, y, z).
• (2) The equation of motion for the point mass in the coordinate system (x, y, z) is expressed as follows using position vector $\mathbf{r}$ , its first derivative with respect to time $\dot{\mathbf{r}}$ , and its second derivative $\ddot{\mathbf{r}}$ :

$m\ddot{\boldsymbol{r}} = \boldsymbol{F} + 2m\dot{\boldsymbol{r}} \times \boldsymbol{\omega}$

$\mathbf{F} = (0, 0, -mg)$

Write the acceleration for the point mass in the y direction at the time t elapsed after throwing the point mass, using, g, t, v, $\omega$ , $\theta$ . Ignore the terms of order $\omega^2$ or higher.

Problem No. 2.4 (Electromagnetism)

Answer the following questions, assuming vacuum environment. Except for Q.5(2), write only the answers.

• (Q.1) In the below, $\varepsilon_0$ and $\mu_0$ are dielectric constant and permeability, respectively, in vacuum.
  • (1) Write the light speed, using $\varepsilon_0$ and $\mu_0$ .
  • (2) Two conducting wires are placed in parallel at a distance r. The length of the wires is infinite. Current I is flowing in the same direction in each wire. Write the magnitude and direction of the force acting on the wire per unit length.
• (Q.2) Answer the questions on the electric circuit shown in Figure 1. The $R_1$ , $R_2$ , and $R_3$ represent electric resistors, whose resistance values are $r_1$ , $r_2$ , and $r_3$ , respectively.
  • (1) Write the combined resistance value between A and B, using $r_1, r_2, r_3$ .
  • (2) A constant voltage V is applied between A and B. Write the current flowing in $R_3$ , and write the electric power consumed in $R_2$ . The answers should use V, $r_1$ , $r_2$ , $r_3$ .
• (Q.3) Answer the questions on the electric circuit shown in Figure 2. An AC voltage is applied with an amplitude $\tilde{V}$ and an angular frequency

• $\omega$ . Here, L is the inductance of the coil, C is the capacitance of the capacitor, R is the resistance value of the resistor. Use j as the imaginary unit.
• (1) Write the combined impedance between A and B, using $\omega$ , L, C, R.
• (2) Write the ratio $\tilde{I}_C/\tilde{I}_R$ , where $\tilde{I}_C$ is the amplitude of the current flowing in the capacitor and $\tilde{I}_R$ is the amplitude of the current flowing in the resistor.
• (3) We remove the capacitor from the circuit, namely we handle a circuit without the capacitor. We express the voltage variation between A and B as $V = \tilde{V}e^{j\omega t}$ . Write the current flowing in the coil, using $\omega$ , L, R, V. Write also the effective electric power consumed in the resistor, using $\omega$ , L, R, $\tilde{V}$ . (The effective electric power is the time-averaged electric power.)
• (Q.4) There is a closed circuit shown in Figure 3. The circuit is formed by one turn loop of a conducting wire and its surface area is S.

Using an equation

$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$

from Maxwell's equations, derive the expression on a voltage V in the circuit induced by the magnetic field crossing the circuit (Note: write only the expression on V). Here, $\boldsymbol{B}$ is a magnetic field externally given, and $\boldsymbol{E}$ is an electric field induced by the magnetic field. The magnetic field is uniform in space. The circuit is placed on a flat surface whose normal unit vector is expressed by $\boldsymbol{n}$ . Ignore the thickness of the conducting wire.

Figure 3

(Q.5) There is a square, whose side length is L, on an xy plane in a Cartesian coordinate system as depicted by broken lines in Figure 4(a). We made a closed circuit by winding a conducting wire twice in the same direction along the sides of this square. This wire has a resistance value R per the length L. Here, a magnetic field is applied uniformly in space in the z direction. The magnitude of the magnetic field B varies with time t as shown in Figure 4(b). The value of B is $B_s$ for $t \leq t_s$ , $B_e$ for $t \geq t_e$ and varies at a constant rate for $t_s < t < t_e$ . Answer the following questions for $t_s < t < t_e$ .

Ignore the self-inductance of the circuit and the thickness of the conducting wire.

• (1) Write the electric voltage V induced in the circuit by using the variables used in the above.
• (2) Describe the force on the wire, using V, R, L in about 5 lines.

Figure 4