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By Bichowsky F.R., Urey H.C.

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Extra resources for A Possible Explanation of the Relativity Doublets and Anomalous Zeeman Effect by Means of a Magnetic Electron

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Evaluate the line integral of the gradient of the function f= r sin 0 over each of the contours shown. x Section 1-4 14. Find the divergence of the following vectors: (a) (b) (c) (d) A= xi, + i,+zi, = ri, A= (xy 2)[i. +i, + i] A= rcos Oi,+[(z/r) sin 0)]i, A= r 2 sin 0 cos 4 [i, +ie +ii 15.

1-10 The cross product between the two vectors in Example 1-3. 5 ° THE GRADIENT AND THE DEL OPERATOR The Gradient Often we are concerned with the properties of a scalar field f(x, y, z) around a particular point. +dy i,+dz i, (2) (1) can be written as the dot product: df = a-f i, + -- i, + - i)dl ax ay az = grad f - dl (3) where the spatial derivative terms in brackets are defined as the gradient of f: grad f = Vf=V i + fi,+ fi Ox ay az (4) The symbol V with the gradient term is introduced as a general vector operator, termed the del operator: V = ix-+i,-7 ax ay +is Oz (5) By itself the del operator is meaningless, but when it premultiplies a scalar function, the gradient operation is defined.

A,, and A,) are themselves scalars. The direction of each of the components is given by the unit vectors. We could describe a vector in any of the coordinate systems replacing the subscripts (x, y, z) by (r, 0, z) or (r, 0, 4); however, for conciseness we often use rectangular coordinates for general discussion. 1-2-2 Multiplication of a Vector by a Scalar If a vector is multiplied by a positive scalar, its direction remains unchanged but its magnitude is multiplied by the _AY A = Aý I IAI=A =(A Figure 1-4 directions.

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