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\begin{center}
  {\bf University of Kentucky, Physics 416G \\
    Problem Set \#2 (Rev. D) due Wednesday, 2011-09-14 }
\end{center}

\vspace{.3 in}

1. The {\bf complex plane} is a vector space consisting of the real and imaginary
components $\bs z=(x,y)$ of complex numbers $z=x + iy$.  In this exercise we will
compare complex and vector rotations.

a) Which two complex numbers form the basis of the complex plane in the
above  components?  

b) Show that the dot and cross product of two points $\bs z_1=(x_1,y_1)$ and
$\bs z_2=(x_2,y_2)$ can be represented by the real and imaginary parts of the
complex product $z_1^* z_2 = \bs z_1\cdot \bs z_2 + i (\bs z_1 \times \bs
z_2)_z$, where the $z^*=x-iy$ is called the {\it complex conjugate} of $z$.
Identify the symmetric and antisymmetric parts of this product.  Note 1: this
shows that the vector and complex magnitudes coincide, $|z|^2 = z^*z = \bs z \cdot
\bs z$.  Note 2: for vectors in the $xy$-plane, only the $z$-component of the
cross product is nontrivial, thus the cross-product is not actually a vector
in the original space!

c) Show graphically with coordinates that the operation of multiplying by $i$
rotates a point $z$ by the angle 90$^\circ$ CCW to become the point $iz$.

d) Show graphically that the operation $z \to (1+i\, d\theta) z = z +i z\,
d\theta$ produces a new point with the same magnitude (assuming
$d\theta\,^2=0$), but rotated CCW by the angle $d\theta$.

e) Finite rotations can be obtained by infinite composition of these
infinitessimal rotations, which can be solved as follows: Formally integrate
the equation $dz = i z\, d\theta$ with the initial condition $z(\theta=0)=z_0$
to obtain the rotation formula $z(\theta)=R_\theta z_0 = z_0 e^{i\theta}$.

f) Collect the real and imaginary terms of the Taylor expansion of
$e^{i\theta}$ to show \mbox{$e^{i\theta}=\cos\theta + i \sin\theta$}.

g) Show that complex multiplication by $i$ is equivalent to the vector
operation $\hat {\bs z} \times$.  

h) Determine the matrix representation of the operator of $\hat {\bs z}
\times$, such that $M_z \bs r = \hat {\bs z} \times \bs r$.  Do the same for
$M_x$ and $M_y$.  For any vector $\bs v$, show that $\bs v\times = \bs M \cdot
\bs v = M_x v_x + M_y v_y + M_z v_z$ and derive the matrix representation of
$\bs v \times$.  Note that each of these matrices in antisymmetric.

i) Calculate $M_x^2$, $M_y^2$, $M_z^2$, and $M_xM_y$.  Note that
$(M_x,M_y,M_z)$ are similar to Hamilton's qua\-ter\-nions $(i,j,k)$, which
extend the rotational properties of complex numbers to 3 dimensions.
%%% MxMy-MyMx   or   zhat x + zhat zhat cdot

j) Using identifications above, show that matrix for a vector rotation of
angle $\theta$ CCW in the $xy$-plane can be written $R_\theta = e^{M_z \theta}
= I \cos\theta + M_z \sin\theta = \pmatrix{\cos\theta & -\sin\theta\cr
  \sin\theta & \cos\theta}$, neglecting $z$-components.  In general, you have
shown that the matrix for a rotation by the angle $v=|\bs v|$ in the CCW
direction about the $\hat{\bs v}$-axis can be written as $R_{\bs v} = 
%(e^{\bs M \cdot \bs v} = 
I \cos v + \bs M{\cdot} \hat{\bs v} \sin v + \hat{\bs v} \hat{\bs v}^T (1-\cos
v)$.  The third term deals with the component of the vector that is
out-of-plane of the rotation.

k) Calculate the eigenvalues and eigenvectors of $M_z=\pmatrix{0&-1\cr 1&0}$
to show that $M_z = VWV^\dagger = \frac12 \pmatrix{1&1 \cr i&-i}
\pmatrix{-i&0\cr0&i} \pmatrix{1&-i\cr1&i}$ and $e^{M_z \theta} = V
e^{W\theta}V^\dagger = \frac12 \pmatrix{1&1 \cr i&-i}
\pmatrix{e^{-i\theta}&0\cr0&e^{i\theta}} \pmatrix{1&-i\cr1&i}$.  Multiply this
out to verify part j.  Thus real symmetric matrices have real eigenvalues
while antisymmetric matrices have Tr=0 and imaginary eigenvalues.  The
exponential of a symmetric matrix remains symmetric, while the exponential of
an antisymmetric matrix is an orthogonal rotation with Det=1 and unit length
eigenvalues.  The symmetric/antisymmetric decomposition of matrices is
frighteningly similar to the real/imaginary decomposition of complex numbers.

\newpage


2. {\bf Affine transformation}.  The affine space (points not vectors) is
distinguished its lack of significance of the origin.  Thus the only two
meaningful operations on points are weighed combinations $w_1x_1 + w_2x_2 +
\ldots$ of points with total weight $w=w_1+w_2+\ldots$ equals to 0 (vector
difference of points) or 1 (weighted average).  This can be formalized by
adding a fourth dimension (weight) which equals 0 for vectors and 1 for
points.
                         
a) Show that matrix transformation $T=\pmatrix{R & \bs t \cr \bs 0 & 1}$
translates a point $\pmatrix{\bs r\cr 1}$ by the displacement $\bs t$ and
rotation $R$ about the origin.  Which operation is done first, in the active
sense?  What operations are done by the same transformation on vectors
$\pmatrix{\bs v \cr 0}$?

b) A robot actuation can be described by the Denavit-Hartenberg parameters
$(d,\theta,r,\alpha)$, which are described in the video clip
\href{http://tekkotsu.no-ip.org/movie/dh-sd.mp4}{http://tekkotsu.no-ip.org/movie/dh-sd.mp4},
as a translation $d$ along the z-axis, a rotation $\theta$ about the $z$-axis,
a translation $r$ along the new $x$-axis, and a rotation $\alpha$ about the
new $x$-axis.  Build a matrix for each of the above transformations and
compose them into a single transformation matrix.  Explain how you order the
transformations, such that the axes is in the proper location in that sequence
of the transformation.

c) Given $n$ measurements $x_i\pm \delta x_i$ of a quantity $x$ with gaussian
(rms) error $\delta x$, the most accurate estimate of $x$ is made by taking a
average of each measurement weighted by $w_i = \delta^{-2}x_i = 1/(\delta
x_i)^2$.  The uncertainty of the average is calculated from the sum of
weights, $w = \sum_i w_i = \delta^{-2}x$.  Show how these can be calculated by
adding 2-d weighted vectors $w_i(x_i,1)$.

d) Likewise show how weighted (4-d) vectors $m_i(\bs r_i,1)$ can be added or
integrated to calculate the center of mass of an extended object.

   
3. {\bf Coordinate basis}

The {\it coordinate basis} $\bs b_i=\partial \bs r/\partial q^i$, {\it
  reciprocal basis} $\bs b^i=\bs\nabla q^i=\partial q^i/\partial \bs r$, and
{\it unit basis} $\hat e_i = \bs b_i / h_i$ (where the {\it scale factor} is
$h_i=|\bs b_i|$), are the natural bases to describe components of a vector
field within a specific coordinate system $(q^1,q^2,q^3)$.  Note that these
basis vectors change from one point to the next.  Calculate each of the
following in both cylindrical and spherical coordinates.

a) Calculate $(\bs b_s, \bs b_\phi, \bs b_z)$ as a functions of $q^i=(s,\phi,z)$
and $(\bs b_r, \bs b_\theta, \bs b_\phi)$ as functions of $(r,\theta,\phi)$.

b) Calculate the resulting scale factors $h_i$ to get unit vectors.  Write out
the basis transformation matrix in terms of these vectors, i.e. $(\hat{\bs s},
\hat{\bs \phi}, \hat{\bs z}) = (\hat{\bs x}, \hat{\bs y}, \hat{\bs z}) R$.

c) Construct the transformation matrices between unit bases, by considering
rotations $R_{\hat{\bs z}}(\phi)$ (rotation about an angle $\phi$ about the
$z$-axis) and $R_{\hat{\bs \phi}}(\theta)$ and compare with part b.
            
d) Calculate the reciprocal vectors $(\bs b^s, \bs b^\phi, \bs b^z)$ and $(\bs
b^r, \bs b^\theta, \bs b^\phi)$. \mbox{Show that $\bs b_i\cdot \bs b^j =
\delta_i{}^j$ in general.}

e) Calculate the metric $g_{ij} = \bs b_i \cdot \bs b_j$ and $g^{ij} = \bs b^i
\cdot \bs b^j$ in terms of $h_i$ and show that they \mbox{are inverses}.

f) Calculate the line elements $\bs{dl} = \bs b_i dq^i$, area elements
$\bs{da} = \frac12 \bs{dl}\times\bs{dl}$, and volume elements $d\tau =
\frac16\bs{dl}\cdot\bs{dl}\times\bs{dl}$.  Note that since differentials
anticommute, the cross products are not zero.

g) Calculate the derivatives of the basis vectors, $\bs\Gamma_{\ij} =
\Gamma_{\ij}{}^k \bs b_k = \partial \bs b_i / \partial q^j = \partial^2 \bs r
/ \partial q^i \,\partial q^j$. These {\it Christoffel symbols} and are
needed to calculate derivatives of vectors in curvilinear coordinates.


\vspace{.166 in}

Also, Griffiths chapter 1, problems 13, 19, 21, 25.

% 1.(p4) distributive \qquad
% %2. associative p4\\
% 4(p7). normal to the plane \qquad
% 5.(p8) BAC-CAB \qquad
% %6. Jacobi identity
% 8.(p12) orthogonal \qquad
% 10.(p12) transformations pseudo \qquad
% 12.(p15) hill gradient \qquad
% 15.(p18) calc divergence \qquad
% 16.(p18) rhat/$r^2$\qquad
% 18.(p20) calc curl\qquad

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