Page 191 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes  X i1...ir X 1 i ...i r (t h,x ,x ,x ) X 1 i ...i r (t,x ,x ,x )
1
2
3
3
2
1


3. Prove theorem j1...js  lim 1 j ...j s 1 j ...j s . For this purpose
t  h 0 h
consider another Cartesian coordinate system x , x , x  related to x , x , x via
1
2
3
3
2
1


2
1
1
1
1
1
3
1
1
2
1
3
1
1
1
1
 x  S x  S x  S x  a ,  x  T x  T x  T x   a ,



3
1
2
3
1
2
 2 2 2 2 2 3 2  2 2 1 2 2 2 3 a , . Then in the new
2





 x  S x1 S x  S x  a , and  x  T x  T x  T x  
3
2
2
1
1
3
 3 3 1 3 2 3 3 a .  x  T x  T x  T x   a .
3
3
3
1
3
3
2
3
3




 x  S x  S x  S x   1 2 3
2
3
1
coordinate system consider the partial derivatives

 X 1 i ...i r
Y qj 1 i ...i r s   x 1 j ...j s

q
1 ...j
 X  1 i ...i r  X 1 i ...i r
and derive relationships binding Y  qj 1 i ...i r s   x 1 j ...j s and Y qj 1 i ...i r s   x 1 j ...j s .
1 ...j
q
1 ...j
q
3 3 3
k
ri
4. Formula (rot X)r   g   j X can be generalized for the case when X is an arbitrary
ijk
i 1 j 1 k 1



3
tensor field with at least one upper index. By analogy with (div X) ...............   s X ........s........ .
...............

..................
s 1

suggest your version of such a generalization.
3 3 3 3 3
k
ri
ij
Note that formulas    g   and (rot X)r   g   j X for the Laplace
j
ijk
i
i 1 j 1 i 1 j 1 k 1





operator and for the rotor are different from those that are commonly used. Here are
standard formulas:
2 2 2
        
   1    2    3  ,
  x    x    x 
e 1 e 2 e 3
rot X  det    .
x  1 x  2 x  3
X 1 X 2 X 3
3 3 3 3 3
ij
ri
k
The truth is that formulas    g   and (rot X)r   g   j X are written
j
ijk
i
i 1 j 1 i 1 j 1 k 1





for a general skew-angular coordinate system with a SAB as a basis. The standard formulas
e 1 e 2 e 3
rot X  det    are valid only for orthonormal coordinates with ONB as a
x  1 x  2 x  3
X 1 X 2 X 3
basis.
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