\documentclass{jarticle}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{bm}

\title{�����s��̓��o}

\begin{document}
\maketitle

\section{��`}

�����s��i�����e���\���j�̒�`�͉��L�̒ʂ�D
\begin{align}
 I &= 
\begin{bmatrix}
 I_{x}	& -I_{xy}	& -I_{xz} \\
-I_{xy}	&  I_{y}	& -I_{yz} \\
-I_{xz} & -I_{yz}	&  I_{z}
\end{bmatrix}
\end{align}
\begin{align*}
I_x &= \rho \int_V (y^2 + z^2) dV, \\
I_y &= \rho \int_V (x^2 + z^2) dV, \\
I_z &= \rho \int_V (x^2 + y^2) dV, \\
I_{xy} &= \rho \int_V xy \, dV, \\
I_{yz} &= \rho \int_V yz \, dV, \\
I_{xz} &= \rho \int_V xz \, dV.
\end{align*}

�x�N�g���ŏ�����
\begin{align*}
I
&= - \rho \int_V (r^\times)^2 dV \\
&= \rho \int_V (\|r\|^2 \bm{1} - r\,r^\mathsf{T}) dV \\
&= \rho \int_V (\mathrm{tr}(r\,r^\mathsf{T}) \bm{1} - r\,r^\mathsf{T}) dV \\
&= \rho ( \mathrm{tr}(\int_V r\,r^\mathsf{T} dV) \bm{1} - (\int_V r\,r^\mathsf{T} dV) )
\end{align*}
�Ƃ�������D
�����
\begin{align}
C = \int_V r\,r^\mathsf{T} dV
\end{align}
�Ƃ�����
\begin{align}
I = \rho (\mathrm{tr}(C)\bm{1} - C)
\end{align}
�ł���D

����`��̊e�_$r$��$r' = r + b$�ƕ��s�ړ����ē�����`��̊����s������߂�D
���̌`��̊����s���$I$�C�ϊ����ꂽ�`��̊����s���$I'$�Ƃ���D
\begin{align*}
I'
&= -\rho \int_V ((r + b)^\times)^2 dV \\
&= -\rho \int_V ((r^\times)^2 + 2 r^\times b^\times + (b^\times)^2) dV \\
&= I - 2m (\bar{r})^\times b^\times - m(b^\times)^2
\end{align*}
������$\bar{r}$�͊�_����݂����`��̏d�S�D
��_���d�S�ɂ���ꍇ��$\bar{r} = 0$�ƂȂ�̂�
\begin{align}
I' = I - m (b^\times)^2
\end{align}
�ƂȂ�C�悭�m��ꂽ���s���藝�𓾂�D

���ɁC����`��̊e�_$r$��$r' = Ar$�Ɛ��`�ϊ����ē�����`��̊����s����l����D
���̌`��̊����s���$I$�C�ϊ����ꂽ�`��̊����s���$I'$�Ƃ���D
�܂���q��$C$�����߂��
\begin{align*}
C'
&= (\det A) \int_V (Ar)(Ar)^\mathsf{T} dV \\
&= (\det A) A C A^\mathsf{T}
\end{align*}
������$\det A$��������̂͐��`�ϊ��ɂ���Ĕ����̐ς�$dV$����$(\det A)dV$�ƂȂ邽�߁D
������
\begin{align}
I' = \rho (\mathrm{tr}(C')\bm{1} - C')
\end{align}

\section{������}

���_�𒆐S�Ƃ��钼���́D
�e�ӂ̒�����$l_x$�C$l_y$�C$l_z$�Ƃ���D

�̐ς�
\begin{align}
V = l_x l_y l_z
\end{align}

�������[�����g��
\begin{align*}
I_x
&= \rho \int_x \int_y \int_z y^2 + z^2 dz dy dx \\
&= \rho \int_x \int_y l_z y^2 + \frac{{l_z}^3}{12} dy dx \\
&= \rho \int_x \frac{l_y l_z (l_y^2 + l_z^2)}{12} dx \\
&= \frac{\rho V (l_y^2 + l_z^2)}{12}
\end{align*}

����Ċ����e���\����
\begin{align}
I =
\frac{\rho V}{12}
\begin{bmatrix}
I_y^2 + I_z^2	& 0				& 0 			\\
0				& I_x^2 + I_z^2	& 0				\\
0				& 0				& I_x^2 + I_y^2
\end{bmatrix}
\end{align}

\section{�~��}

���_�𒆐S�Ƃ��C������$d$�C���a��$r$�Cx�������Ƃ���~���D

�̐ς�
\begin{align}
V = \pi r^2 d
\end{align}

�������̊������[�����g��
\begin{align*}
I_x
&= \rho d \int_0^r s^2 (2 \pi s) ds = \frac{\rho \pi r^4 d}{2} = \rho V \frac{r^2}{2}
\end{align*}
���ɒ�����������̊������[�����g�́C
\begin{align*}
I_y 
&= \rho \int_{0}^{\pi} \frac{(2r\sin\theta)^2 + d^2}{12} \cdot 2r\sin\theta \cdot d \cdot r\sin\theta d\theta \\
&= \rho \int_{0}^{\pi} \frac{2r^2d\sin^2\theta(4r^2\sin^2\theta + d^2)}{12}d\theta \\
&= \rho \int_{0}^{\pi} \frac{r^2d (1 - \cos2\theta)(2r^2(1 - \cos2\theta) + d^2)}{12}d\theta \\
&= \rho \int_{0}^{\pi} \frac{2r^4d(1-\cos2\theta)^2 + r^2d^3(1-\cos2\theta)}{12} d\theta \\
&= \rho \int_{0}^{\pi} \frac{(2r^4d + r^2d^3) - (4r^4d + r^2d^3) \cos2\theta + 2r^4d \cos^2 2\theta}{12} d\theta \\
&= \rho \int_{0}^{\pi} \frac{(3r^4d + r^2d^3) - (4r^4d + r^2d^3) \cos2\theta + r^4d \cos4\theta}{12} d\theta \\
&= \pi \rho \frac{3r^4d + r^2d^3}{12} \\
&= \rho V \frac{3r^2 + d^2}{12}
\end{align*}
�������ϕ��̒��Œ����̂̊������[�����g�𗘗p�D

����Ċ����e���\����
\begin{align}
I = \rho V
\begin{bmatrix}
\frac{r^2}{2}	& 0				        & 0 			        \\
0				& \frac{3r^2 + d^2}{12}	& 0				        \\
0				& 0				        & \frac{3r^2 + d^2}{12}
\end{bmatrix}
\end{align}

\section{��}

���_�𒆐S�Ƃ������a$r$�̋��D

�̐ς�
\begin{align}
V = \frac{4}{3} \pi r^3
\end{align}

�������[�����g��
\begin{align*}
I_x
&= \rho \int_{-r}^{r} \frac{\pi}{2}(r^2 - x^2)^2 dx \\
&= \rho \int_{-r}^{r} \frac{\pi}{2}(r^4 - 2r^2x^2 + x^4) dx \\
&= \rho \frac{\pi}{2} (2r^5 - \frac{4}{3}r^5 + \frac{2}{5} r^5) \\
&= \frac{8}{15} \rho \pi r^5 \\
&= \rho V \frac{2}{5} r^2
\end{align*}

����Ċ����e���\����
\begin{align}
I =
\rho V \frac{2}{5}r^2 \bm{1}
\end{align}

\section{����}

���_�𒆐S�Ƃ��C$x > 0$���݂̂̔��a$r$�̔����D

�̐ς�
\begin{align}
V = \frac{2}{3} \pi r^3
\end{align}

�d�S��
\begin{align*}
&\int_0^r \pi (r^2 - x^2) x dx = \frac{\pi}{4}r^4,\\
& (\frac{\pi}{4}r^4) / (\frac{2}{3} \pi r^3) = \frac{3}{8}r
\end{align*}
���
\begin{align}
c = \begin{bmatrix}\frac{3}{8}r \\ 0 \\ 0\end{bmatrix}
\end{align}

���_�Ɋւ��銵�����[�����g��
\begin{align*}
I_x
&= \rho \int_0^r \frac{\pi}{2} (r^2 - x^2)^2 dx = \frac{4}{15} \rho \pi r^5
\end{align*}
\begin{align*}
I_y
&= \rho \int_0^r (\frac{\pi}{4} (r^2 - x^2)^2 + \pi (r^2 - x^2) x^2) dx \\
&= \frac{4}{15} \rho \pi r^5
\end{align*}
�������ϕ��̒��ŉ~���̊������[�����g�𗘗p�D
����Ċ����s���
\begin{align}
I = \rho V \frac{2}{5} r^2 \bm{1}
\end{align}
�ƌ`�͋��ƕς�炸�D�Ώ̂ɑ̐ς������ɂȂ��Ă���̂œ��R�̌��ʁD
�������d�S�Ɋւ��銵���s��𓾂�ɂ͕��s���藝�ŕϊ����K�v�Ȃ̂Œ��ӁD

\section{�~��}

���a$r$�C����$d$�̉~���D
��������ʂ�yz�ʂɂ���C����x���C���_��$x > 0$���Ƃ���D

�̐ς�
\begin{align}
V = \frac{1}{3} \pi r^2 d
\end{align}

�d�S��
\begin{align*}
&\int_0^d \pi (\frac{r}{d}(d-x))^2 x dx = \frac{1}{12} \pi r^2 d^2, \\
&(\frac{1}{12} \pi r^2 d^2) / (\frac{1}{3} \pi r^2 d) = \frac{1}{4}d
\end{align*}
���
\begin{align}
c = \begin{bmatrix}\frac{1}{4}d \\ 0 \\ 0\end{bmatrix}
\end{align}

�������[�����g��
\begin{align*}
I_x
&= \rho \int_0^d \frac{\pi}{2} (\frac{r}{d}(d-x))^4 dx \\
&= \frac{1}{10} \rho \pi r^4 d
\end{align*}
\begin{align*}
I_y
&= \rho \int_0^d (\frac{\pi}{4}(\frac{r}{d}(d-x))^4 + \pi (\frac{r}{d}(d-x)^2)x^2) dx \\
&= \rho \pi (\frac{1}{20} r^4d + \frac{1}{30} r^2d^3)
\end{align*}

\section{�J�v�Z��}

���a$r$�C����$d$�Cx�������Ƃ���J�v�Z���D

�̐ς�
\begin{align}
V = \frac{4}{3}\pi r^3 + \pi r^2 d
\end{align}

�������̊������[�����g��
\begin{align*}
I_x
&= \rho (\frac{4}{3} \pi r^3)(\frac{2}{5} r^2) + \rho (\pi r^2 d)(\frac{1}{2}r^2) \\
&= \rho \pi (\frac{8}{15} r + \frac{1}{2}d)r^4
\end{align*}
���ɒ�����������̊������[�����g�́Cx��������$d/2$���ꂽ����������Ɖ~�����琬��̂�
\begin{align*}
I_y
&= \rho (\frac{4}{3}\pi r^3) (\frac{2}{5}r^2 + \frac{1}{4}d^2) + \rho (\pi r^2 d) (\frac{3r^2 + d^2}{12}) \\
&= \rho \pi (\frac{8}{15}r^5 + \frac{1}{3}r^3d^2 + \frac{1}{4}r^4d + \frac{1}{12}r^2d^3)
\end{align*}

\section{�����l�ʑ�}

\begin{align}
\begin{bmatrix}0 \\ 0 \\ 0\end{bmatrix}, \,
\begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}, \,
\begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix}, \,
\begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix}
\end{align}
��4�_����Ȃ�l�ʑ́D

\begin{align*}
I_x
&= \int_0^1 \int_0^{1-x} \int_0^{1-x-y} y^2 + z^2 dz dy dx \\
&= \int_0^1 \int_0^{1-x} y^2(1-x-y) + \frac{1}{3} (1-x-y)^3 dy dx \\
&= \int_0^1 \int_0^{1-x} (1-x)y^2 - y^3 + \frac{1}{3}(1-x-y)^3 dy dx \\
&= \int_0^1 \frac{1}{3}(1-x)^4 - \frac{1}{4}(1-x)^4 + \frac{1}{12}(1-x)^4 dx \\
&= \int_0^1 \frac{1}{6}(1-x)^4 dx = \frac{1}{30}
\end{align*}
\begin{align*}
I_{xy}
&= \int_0^1 \int_0^{1-x} \int_0^{1-x-y} xy dz dy dx \\
&= \int_0^1 \int_0^{1-x} xy(1-x-y) dy dx \\
&= \int_0^1 \int_0^{1-x} (1-x)xy - xy^2 dy dx \\
&= \int_0^1 \frac{1}{2} x(1-x)^3 - \frac{1}{3} x(1-x)^3 dx \\
&= \int_0^1 \frac{1}{6} x(1-x)^3 dx \\
&= \frac{1}{6} \int_0^1 (1-x)^3 - (1-x)^4 dx \\
&= \frac{1}{120}
\end{align*}
�����
\begin{align}
I =
\begin{bmatrix}
 \frac{1}{30}  & -\frac{1}{120} & -\frac{1}{120} \\
-\frac{1}{120} &  \frac{1}{30}  & -\frac{1}{120} \\
-\frac{1}{120} &  \frac{1}{120} &  \frac{1}{30}
\end{bmatrix}
\end{align}

�܂��C$C$�s���
\begin{align}
I =
\begin{bmatrix}
\frac{1}{60}  & \frac{1}{120} & \frac{1}{120} \\
\frac{1}{120} & \frac{1}{60}  & \frac{1}{120} \\
\frac{1}{120} & \frac{1}{120} & \frac{1}{60}
\end{bmatrix}
\end{align}

\section{�l�ʑ�}

���_$O$�C�_$a$�C$b$�C$c$����Ȃ�l�ʑ́D

�̐ς�
\begin{align}
V = \frac{1}{6} |a \cdot (b \times c)|
\end{align}

�d�S��
\begin{align}
g = \frac{1}{4}(a + b + c)
\end{align}

�����s��͐����l�ʑ̂̊����s��$I$����`�ϊ����ē���D
$[1;\,0;\,0]$�C$[0;\,1;\,0]$�C$[0;\,0;\,1]$�����ꂼ��$a$�C$b$�C$c$�ֈڂ��ϊ���
\begin{align*}
A = [a\, b\, c]
\end{align*}
�ł���̂ŁC�����l�ʑ̂�$C$����`�ϊ�����$C'$�C$I'$�����߂�΂悢�D

\end{document}
