Advanced calculus; . thout any changes except those implied by the fact that axfi =?= an illustration consider the application of the definition of differen-tiation to the vector product Uxv of two vectors which are supposedto be functions of a numerical variable, say x. Then A(UxV) = (U + Au)x(v + AV) — UxV= UxAv + AUxv + AUxAv, A(Uxv) Av Au , AUxAv-^ L = Ux— + —xV H , Ax Ax Ax Ax f/(uxv) ... A(uxv) dv ———- = Inn — = ux—- -+- — dx Ax dx du dx Here the ordinary rule for a product is seen to hold, except thatthe order of the factors must not be interchanged. The interpretation of the de
Advanced calculus; . thout any changes except those implied by the fact that axfi =?= an illustration consider the application of the definition of differen-tiation to the vector product Uxv of two vectors which are supposedto be functions of a numerical variable, say x. Then A(UxV) = (U + Au)x(v + AV) — UxV= UxAv + AUxv + AUxAv, A(Uxv) Av Au , AUxAv-^ L = Ux— + —xV H , Ax Ax Ax Ax f/(uxv) ... A(uxv) dv ———- = Inn — = ux—- -+- — dx Ax dx du dx Here the ordinary rule for a product is seen to hold, except thatthe order of the factors must not be interchanged. The interpretation of the derivative is important. Let the variablevector r be regarded as a function of some variable, say x, and supposer is laid off from an assumed origin so that, as x varies,the terminal point of r describes a curve. The incre-ment Ar of r corresponding is a vector quantityand in fact is the chord of the curve as derivative dx ,. Ar— = Inn —: dx Ax dx Ar — = Inn — = tas As (42). is therefore a, vector tangent to the curve; in particular ifthe variable x were the arc s, the derivative would havethe magnitude unity and would be a unit vector tangent to the derivative or differential of a vector of constant length is per-pendicular to the vector. This follows from the fact that the vector COMPLEX NUMBERS AND VECTORS 171 then describes a circle concentric with the origin. It may also be seenanalytically from the equation d() = + iv/r = 2 iv/r = d const. = 0. (43) If the vector of constant length is of length unity, the increment Ar isthe chord in a unit circle and, apart from infinitesimals of higherorder, it is equal in magnitude to the angle subtended at the then the derivative of the unit tangent t to a curve withrespect to the arc s. The magnitude of dt is the angle the tangent turnsthrough and the direction of dt is normal to t and hence to the vector quantity, ^ 7^ curvature C = -7- = -7-5 >
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