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Show that an rotation by an infinitesimal angle \(\Delta \theta\) about an axis \(hat{n}\) is equivalent to successive rotations by infinitesimal angles \(\alpha, \beta, \gamma\) about the three coordinate axes. Keeping first order terms in \(\Delta \theta\), find expressions for the angles \(\alpha, \beta, \gamma\) in terms of components of \(hat{n}\) and \(\Delta\theta\).

I Question 
Let \(S_{ij}\), \(A_{ij}\) and be symmetric, ansymmetric tensors respectively. Then for arbitrary tensor \(T_{ij}\), {\bf prove any one} of the following identities. \[(a)\ \ S_{ij}T_{ij}=\frac{1}{2}S_{ij}\big(T_{ij}+T_{ji}\big); \quad \quad (b)\ \ A_{ij}T_{ij}=\frac{1}{2}A_{ij}\big(T_{ij}-T_{ji}\big).\]
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II Solution 

Show that the trace of rotation matrix for rotation by an angle \(\phi\)is given by \((1+\cos2\phi)\).

Show that \begin{equation*} [\vec{A},\vec{B},\vec{C}]^2 = \begin{vmatrix} \vec{A}.\vec{A} & \vec{A}.\vec{B} & \vec{A}.\vec{C} \\ \vec{B}.\vec{A} & \vec{B}.\vec{B} & \vec{B}.\vec{C} \\ \vec{C}.\vec{A} & \vec{C}.\vec{B} & \vec{C}.\vec{C} \end{vmatrix} \end{equation*} where \([\vec{A},\vec{B},\vec{C}]=\vec{A} . (\vec{B} \times \vec{C} ). \)

Write \(3\times 3\) rotation matrix corresponding to a rotation about axis \(\hat{n}\) by an angle \(\theta\).\hfill \GetQSource using the result \begin{equation}\label{EQ14} \vec{x}{'} = (\vec{x}\cdot\hat{n})\hat{n} - (\hat{n}\times\vec{x})\sin\theta - \hat{n}\times(\hat{n}\times\vec{x})\cos\theta. \end{equation} on the rotations by angle \(theta\) about axis \(hat{n}\) and

Using definition of Levi-Civita symbol, prove the following identities. \begin{eqnarray} \epsilon_{i\,j\,k}\,\epsilon_{i\,j\,k} \ &=&\ 6 \\ \epsilon_{i\,j\,k}\,\epsilon_{l\,j\,k} \ &=&\ 2\,\delta_{il} \\ \epsilon_{i\,j\,k}\,\epsilon_{l\,m\,k}\ &=&\ (\,\delta_{il}\,\delta_{jm}\,-\,\delta_{im}\,\delta_{jl}) \\ \epsilon_{i\,j\,k}\,\epsilon_{l\,m\,n}\ &=&\ \left(\begin{array}{ccc} \delta_{il}&\delta_{im}&\delta_{in}\\ \delta_{jl}&\delta_{jm}&\delta_{jn}\\ \delta_{kl}&\delta_{km}&\delta_{kn} \end{array} \right) \end{eqnarray} Use result on \(\vec{A}\times(\vec{B}\times{\vec{C})}\) to derive identity (3) of Q[1].

Read the following theorem of Rodrigues and Hamilton
taken from Whittaker.
The theorem of Rodrigues and Hamilton.
Any two successive rotations about a fixed point can be compounded intoa single
rotation by means of a theorem, which may be stated as follows:Successive
rotations about three concurrent lines fixed in space, through twicethe angles
of the planes formed by them, restore a body to its original position. For let
the lines be denoted by OP, OQ, OR. Draw. Op, Oq, Orperpendicular to the planes
QOR, ROP, POQ respectively. Then if a body isrotated through two right angles
about Oq, and afterwards through two rightangles about Or, the position of OP is
on the whole unaffected, while Oq ismoved to the position occupied by its image
in the line Or; the effect is therefore the same as that of a rotation round OP
through twice the anglebetween the planes PR and PQ, which we may call the angle
RPQ. It follows that successive rotations round OP, OQ, OR through twice the
angles RPQ, PQR, QRP, respectively, are equivalent to successive rotations
through two right angles about the lines Oq, Or, Or, Op, Op, Oq; but the
latter rotations will clearly on the whole produce no displacement; which
establishes the theorem. 

Now solve the following problem. Following two rotations are preformed in
succession
(i) rotation by angle \(\alpha\) about axis \(\hat{n}\);
(ii) rotation by angle \(\beta\) about axis \(\hat{m}\).
Find the angle and axis of rotation that will produce the same result as the
combined effect of above two rotations.

Let \({\bf A, B,..}\) be objects with components written as \(\vec{A}=(A_1,A_2,A_3), \vec{B}=(B_1,B_2,B_3)\). Introduce \(\vec{A}{'}=(A_1{'},A_2{'},A_3{'}), \vec{B}{'}=(B_1{'},B_2{'},B_3{'})\) etc. by means of equation \begin{equation}\label{EQ01} \begin{pmatrix}A_1{'}\\A_2{'}\\A_3{'}\end{pmatrix} = R \begin{pmatrix}A_1\\A_2\\A_3\end{pmatrix} \end{equation} and with similar equations for other vectors.\\ Using the fact that the matrix \(R\) is orthodgonal matrix show that

• \({A}_i{'}\vec{B}_i{'}=A_i B_i\);
• If \(C_i{'}=\epsilon_{ijk}A_j{'} B_k{'}\), then \(\vec{C}{'}\) is given by an equation similar to \eqRef{EQ01}, where, of course, \(C_i=\epsilon_{ijk}A_j B_k\)
• The value of \( \vec{A}{'}\cdot(\vec{B}{'}\times\vec{C}{'})\) is independent of the matrix \(R\), if \(\det R=1\). What happens if \(\det R=-1\)?