By Derek F. Lawden
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Extra info for A Course in Applied Mathematics, Vol.1, 2
150) here. 152) can be similarly obtained. Ascalar ﬁeld φ is deﬁned in V and we divide the body into an inﬁnite number of parallel inﬁnitesimal cylinders. 6 with ends Aand B. Ends Aand B are parts of the surface B area S of the body and has unit outward normal nA i or ni , respectively. We now consider two neighboring points P(xi ) and P (xi +dxi ) in the inﬁnitesimal cylinder. 6 and is a constant. 153) where ds is the length of dxi . 155) Cartesian Tensors 39 Note that ni ui is negative at A and positive at B.
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. By writing in the component form, verify the following identities: (a) ∇(φ + ψ) = ∇φ + ∇ψ (b) Div(v + w) = Div v + Div w (c) Div(φw) = (∇φ) · v + φ(Div v) (28) Consider the integral L e3ij ni vj dL, where L is the closed boundary of a plane surface S with normal (0, 0, 1); vi is a vector; ni is the outward normal of the contour L. Show that this integral is the circulation of vector vi along L denoted by L v · dL.
14) In the coordinate transformation from xi to xi , the transformation rule for the components of a second-rank tensor is Tpq = Qip Qjq Tij , where Qij is the transformation matrix. Derive the reversed expression of this equation, that is, express Tij in terms of Tij and Qij . (15) Investigate the inﬂuence of the coordinate transformation x1 = −x1 , x2 = x2 , x3 = x3 on the components of a tensor T. (16) Prove that ei ⊗ ei = I where I is the identity tensor. (17) Show that det(ei ⊗ ei ) = 1 and det(e1 ⊗ e1 ) = 0.
A Course in Applied Mathematics, Vol.1, 2 by Derek F. Lawden