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Since e231 = 1 and e321 = −1, the terms of the last equality cancel out in pairs. 143) The last equality is again obtained by observing that the inner product of an antisymmetric tensor with a symmetric tensor is zero. 147) (e) There are two forms of the divergence of tensor T. 149) The two divergences are in general different, but they are the same if the tensor T is symmetric. 65). 3 The Theorem of Gauss In continuum mechanics, an important application of the theorem of Gauss is to change an area integral into a volume integral and vice versa.
Show that, at an arbitrary point of this surface, Curl v is tangential to the surface or vanishes, while Div v is given by the rate of change of the normal component of v in the direction of the normal of the surface. (26) A tensor M transforms every vector into its mirror image with respect to the plane whose normal is √ 2 n= (e + e2 ) 2 1 (a) Find the matrix of M. (b) Use this linear transformation to ﬁnd the mirror image of a vector a = e1 + 2e2 . (Note that this is not a case of rotation. ) (27) Let φ(x, y, z) and ψ(x, y, z) be scalar functions of positions, and let v(x, y, z) and w(x, y, z) be vector functions of position.
2 Differential operators on vectors There are two differential operators on vector vi which are well known. They are the divergence and the curl. 140) in which vj,k is the velocity gradient if vi is the velocity. The velocity gradient is an important kinematic quantity and is discussed in Chapter 3. 140), we have w1 = v3,2 − v2,3 . Vector wm is the dual vector of tensor vj,k except for a constant multiple factor. Dual vectors are discussed in Chapter 3. 142) We note that ejk1 is antisymmetric and φ,kj is symmetric such that their inner product is zero.