1. 3、general curvilinear coordinates in Euclidean 3-D
3-1 coordinate system and general in Euclidean 3-D
Suppose that general coordinates are( ); this means that the position vectors x of a point is a known function of , and ,
then the choice that is usually made for the base vectors is
For consistency with the right-handedness of the εi , the coordinates
must be numbered in such a way that

2. As an example , consider the cylindrical coordinate
With in terms of the Cartesian coordinates and the Cartesian base vectors , where , we have
And so

3. 3-2 metric tensor and jacobian
We have already seen that is a tensor ; it will now be shown why it is called the metric tensor
The definition , together with ,give
Note that
So that an element of arc length satisfies

4. Note that is the same as
The jacobian of the transformation relating Cartesian coordinates and curvilinear coordinates is determinant of the array and the element of volume having the vectors
As edges is

5. 3-3 Transformation rule for change of coordinates
Suppose a new set general coordinates is introduced, with the understanding that the relations between and are known, at least in principle. The rule for changing to new tensor components is

6. 4、tensor calculus 4-1 gradient of a scalar
If f is a scalar function, then
But grad ; hence
is the convariant component of
An alternative way to conclude that is a vector is to note
that is a scalar for all recall that is a vector ,and
invoke the appropriate quotient law.

7. 4-2 Derivative of a vector ; christoffel` symbol; covariant derivative
Consider the partial derivative of a vector F. with F = ,
we have
write
christoffel system of the second kind
the contravriant component of the derivative with respect to of the base vector. Note that

8. We can now write
(4-2-1)
Introduce the notation
(4-2-2)
This means that called the convariant derivative of is definded as the contravariant component of the vector comparing (4-2-1) and (4-2-2) then gives us the formula
Although is not necessarily a tensor, is one , for

9. The covariant derivative of writing as , is defined as the covariant component of ; hence
(4-2-3)
A direct calculation of is more instructive; with F= ,we have
Now , whence
And therefore
consequently
And while this be the same as (4-2-3) it shows the explicit addition to needed to provide the covariant derivative of

10. Other notations are common for convariant derivatiives; they are, in approximate order of popularity
Although is not a third-order tensor, the superscript can nevertheless be lowered by means of the operation
and the resultant quantity , denoted by , is the Christoffel symbol of the first kind. The following relations are easily verified

11. (4-2-4)
[Prove] ：

12. 4-3 covariant derivatives of Nth –order Tensors
Let us work out the formula for the covariant derivatives of write
By definition
This leads directly to the formula

13. 4-4 divergence of a vector
A useful formula for
will be developed for general coordinate systems. We have
But, by determinant theory
Hence
And therefore

14. 4-5 Riemann-Christoffel Tensor
Since the order of differentiation in repeated partial differentiation of Cartesian tensor is irrelevant, it follows that the indices in repeated covariant differentiation of general tensors in Euclidean 3-D may also be interchanged at will. Thus, identities like
and
(4-5-1)
Eq (4-5-1) is easily verified directly, since
However, the assertion of (4-5-1) in Euclidean 3-D leads to some nontrivial information. By direct calculation it can be shown that

15. With help of (4-2-4) it can be shown that , the Riemann-christoffel tensor, is given by
But since the left-hand side of vanishes for all vectors , it follows that
(4-5-2)
Although (4-5-2) represents 81equations, most of them are either identities or redundant, since Only six distinct nontrivial conditions are specified by (4-5-2), and they may be written as
(4-5-3)

16. [Note]
[prove]

17. Since is antisymmetrical in i and j as well as in p and k, no information is lost if (4-5-2) is multiplied by Consequently , a set of six equations equivalent to (4-5-3) is given neatly by
Where is the symmetrical, second-order tensor
The tensor is related simply to the Ricci tensor
So that (4-5-3) is also equivalent to the assertion

18. 4-6 Integral Relations
The familiar divergence theorem relating integrals over a volume V and its boundary surface S can obvious be written in tensor notation as
Where Ni is the unit outward normal vector to S . Similar stokes，theorem for integrals over a surface S and its boundary line C is just
Where tk is the unit tangent vector to C , and the usual handedness rules apply for direction of Ni and ti

19. 廣義相對論