If f is a scalar function, then 4-1 gradient of a scalar But grad ; hence 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 、 tensor calculus is the convariant component of

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

We can now write Introduce the notation 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 (4-2-1) (4-2-2)

A direct calculation of is more instructive; with F=,we have Now, whence The covariant derivative of writing as, is defined as the covariant component of ; hence 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 (4-2-3)

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 N th –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 Eq (4-5-1) is easily verified directly, since (4-5-1) 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 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 (4-5-2) (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

The familiar divergence theorem relating integrals over a volume V and its boundary surface S can obvious be written in tensor notation as Where N i 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 t k is the unit tangent vector to C, and the usual handedness rules apply for direction of N i and t i 4-6 Integral Relations

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