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

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

3-2 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

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

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

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.

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

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

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

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

(4-2-4) [Prove] ：

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

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

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

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)

[Note] [prove]

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

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

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