31

the X system.
a
From the diagram, we can write immediately
for &1, a2(using again for
simplicity the 2Dspace):
These relations can be rewritten, realizing that
as
a cos al = a( cos 61 cos 0\1 + sin 61 cos 0\2)
a cos a2 = a(cos 62 cos a 2 + sin 62 co
15
regarded as a scaled difference of two infinitesimally small vectors
+
connecting three infinitesimally close points on the curve.
Hence n
has to lay in the plane given by the three infinitesimally close
.
points, i.e. in the osculating
plane.
_,.
+
Th
1) VECTORS IN RECTANGULAR CARTESIAN COORDINATES
1.1) Basic Definitions
The Cartesian power E3 , where Eisa set of real numbers, is
called the System of Coordinates in threedimensional space (futher
only 3Dspace).
Any element 1EE 3 is said to describe a
8
+
ii) Vector
+
+
+
+
A x B of two vectors A and B is the vector C
given by
fT=
(Cl, C2, C3)
=
It can be easily shown that the following formulae hold:
+
+
(kA) x + = k(A
B
X
+
+
= + x (kB)
A
B)
and particularly
+
+
+
+
Ax B
+
A x (B +
+B X +
A,
c) = + x
27
Expressing a 1 , a 2 in terms of S1 , S2 and w we get:
= ab
= ab
= ab
sin
a2
cos w(sin
s2 cos s1 +sin s1 cos s2 )+sin
w(sin
s1 sin s2sin s1 sin s2 )
sin
cos w  sin
, . The last equation is obviously equivalent to the one we set to prove.
I
Problem:
S
23
2)
VECTORS IN OTHER COORDINATE SYSTEMS
2.1) Vectors in Skew Cartesian Coordinates
In order to see one of the difficulties encountered when dealing
with curvilinear coordinates let us have a look first at the simplest
departure from rectangular Cartesia
90
=R
R
cos 6
This theorem is known as the Ueusnier's
theorem.
"R('X)
(.t.
!. 1; :t4.
.)
4.3
Euler's Equation
Let us now adopt the. point P for the origin of a new coor+
dinate system t;, n, r;, such that the
coincides with n and 1; ; n
+
are oriented
3) TENSORS
3.1)
Definition of a Tensor
As we have said in the last paragraph, there are spaces in which
we cannot define Cartesian coordinate systems.
In these spaces (but
not only these) it is usually difficult to even recognise if a quantity
is a vector
58
(in rectangular Cartesian
det ( gij)
= 1).
Taking
ers t
we assume for simplicity
in another arbitrary coordinate system U,
i t has to satisfy the following transformation law:
( *)
To prove that the above equation is satisfied we have to prove first
th
70
We may note that the covariant and contravariant derivatives of
the four pseudoconstant tensors (mentioned in
3.4.3) are again zero.
This is easily seen when we consider the derivatives in Cartesian
coordinates.
The covariant and contravariant derivat
8'2
4)
SOME APPLICATIONS OF
TENSORS IN DIFFERENTIAL GEOMETRY OF
SURFACES
4.1)
First and Second Fundamental Forms of a Surface
(Relation of
the first fundamental form to the metric tensor.)
The equation for the line element on a surface
( ds )
2
= gaf3
du
85
We shall show later, that this is the case, if and only if
the surface is smooth, i.e., if the curvature changes continuously.
The reason for being asked the question we started with will be also seen
later.
As we know already (see 1. 3. 3) the unit
47
derive the metric tensor of"the U system from the above formulae.
It can be shown that if the U system is again rectangular Cartesian.with the
same scale along all the axes, we end up with
i = j
=/1
gij
\0
i :f j
a tensor equal to the Kronecker
o and
d