Differential geometry and complex analysis: a volume by I. Chavel, H.M. Farkas

By I. Chavel, H.M. Farkas

Chavel I., Farkas H.M. (eds.) Differential geometry and intricate research (Springer, 1985)(ISBN 354013543X)(236s)

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I ¤ j; i; j D 1; 2; : : : ; n/. i ¤ j; i; j D 1; 2; : : : ; n/. n 1/, respectively. 152) that any basis of Skewn complements any basis of Symn to a basis of Linn . 159) called axial vector. 161) Spaces characterized by this property are called orthogonal. Additive decomposition into a spherical and a deviatoric part. 163) n n denote its spherical and deviatoric part, respectively. Thus, every spherical tensor S can be represented by S D ˛I, where ˛ is a scalar number. In turn, every deviatoric tensor D is characterized by the condition trD D 0.

41) Accordingly, the vector and tensor components xi , Aij and x i , Aij are called covariant and contravariant, respectively. The tensor components Aij are referred to as mixed variant. 41) can similarly be written for tensors of higher orders as well. 42) From the very beginning we have supplied coordinates with upper indices which imply the contravariant transformation rule. i D 1; 2; : : : ; n/. 33). 3. 17). 23) by inverting the matrix @x @x i @x i . i D 1; 2; : : : ; n/. Such functions of coordinates are generally referred to as fields, as for example, the scalar field, the vector field or the tensor field.

One can verify that the tangent vectors are linearly independent and form thus a basis of En . 28) be linearly dependent. i D 1; 2; : : : ; n/, not all zero, such that ˛ i g i D 0. i D 1; 2; : : : ; n/ be linear coordinates in En with respect to a basis H D fh1 ; h2 ; : : : ; hn g. Then, 0 D ˛i g i D ˛i @r @r @x j @x j D ˛i j D ˛ i i hj : i i @Â @x @Â @Â 40 2 Vector and Tensor Analysis in Euclidean Space gk Δr θk r(θ k + s) r(θ k) Fig. i D 1; 2; : : : ; n/. 25). 2. Tangent vectors and metric coefficients of cylindrical coordinates in E3 .

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