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By Hans J. Weber and George B. Arfken (Auth.)
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Example text
11) for the dot product, we obtain A BxC = Ax (ByC2 - BzCy) + Ay (BZCX - BXCZ) + Az {BxCy = B C x A = C A x B = - A C x B = - C · B x A = - B A x C. 46) The high degree of symmetry present in the component expansion should be noted. Every term contains the factors Ai, Bj, and Ck> If i, j , and k are in cyclic order (χ,ΐ/,ζ), the sign is positive. If the order is ant icy clic, the sign is negative. Further, the dot and the cross may be interchanged, A B x C = AxBC. 47) A convenient representation of the component expansion of Eq.
Axb c = a-bxc' and a - b x c / 0 , show that (a) x ; -y' = 0, if x φ y and xx · y' = 1, if x = y, for (x,y = a , b , c ) , (b) a ' ^ b ' x c ' r ( a - b x c ) " 1 , 6 This is Jacobi's identity for vector products. 36 CHAPTER . (c) w a = 1. VECTOR ANALYSIS b'xc' . 11) by V = (V · a) a' + (V · b) b ' + (V ■ c) c'. 14 An electric charge q\ moving with velocity νχ produces a magnetic in duction B given by μο vi x r B = —c —q\—τ,— 4π' (mks units), where f points from qi to the point at which B is measured (Biot and Savart law).
A curve on the intersection of the surface with the xz-plane going through y. (s + ft,y) -φ(χ,ν) Λ-»Ο /i ,χ lim ^ is the partial derivative of φ with respect to x defined with the understand ing that the other variable y is held fixed. It is useful for drawing tangents and locating a maximum or minimum on the curve where the slope is zero. The partial derivative Θφ/dy is similarly defined holding x fixed, that is, it is the slope of the surface in the ^-direction, and so on for the higher partial derivatives.