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2 Gradient, Divergence, Curl We consider differentiation and integration in the study of vector calculus. Differentiation is discussed in this section and integration in a later section. There are three kinds of differentiation: the gradient, the divergence, and the curl. 131). 1 Geometric signiﬁcance of the gradient The value of the scalar ﬁeld φ(xk ) deﬁned over the region R varies in general with xi . 5 A constant φ surface. is constant. 5. 135) But, xi has a small value and we may take it as small as possible in the sense of calculus.

We now take the trace of the ﬁrst equation of (a) and get tr B = tr{A − 13 (tr A)I} = tr A − tr 13 (tr A)I = tr A − 13 (tr A)(3) = 0 (c) Therefore, B is traceless, and from (b) A is the sum of a spherical tensor and a traceless tensor. To prove the uniqueness, we ﬁrst assume that this decomposition is not unique. Then, there exists a spherical tensor D and a traceless tensor E such that A=D+E (d) We take the trace of (d) and (b) to ﬁnd tr A = tr D + tr E = tr D = tr C (e) because E and B are traceless.

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. For the same reason the other two terms involving m = 2 and m = 3 are also zero. The above result can also be seen easily by expanding out each term as in the last equality.