Tuỳ thuộc vào các hiện tượng vật lý khác nhau của mỗi chuyên ngành kỹ thuật phải xem xét mà chúng ta sử dụng những công cụ toán học cho phù hợp nhằm mô tả hiện tượng, ước lượng và tối ưu hoa chúng trên ý nghĩa kỹ thuật. | We have arrived at this result for an element of area dx dy in the XV-plane. Is this result general Does it apply to any small area whatever Ils orientation with respect to the coordinate axes It does if it is invariant. We have already seen that the scalar product is invariant. Thus the above line integral is invariant. We have also seen that the operator F and the vector product are invariant. Therefore F X B is invariant. This means that F X fi IS a vector whose value defined by Eq. 1-41 is independent of the particular coordinate axes used as long as they form .1 right-handed Cartesian system. Then Eq. 1-41 is indeed invariant it does apply to any clement of area dsfc and rxB rt lim 4 í B 1-42 - o Jc I bus the component of the curl of a vector normal to a small surface of irca is equal to the line integral of the vector around the periphery c I 1 the surface divided by s4- when this area approaches zero. In general F X B is not normal to B. See Prob. 1-7. rhe curl of a gradient is identically equal to zero Fx F O. 1-43 STOKES S THEOREM surface any finite surface I bounded by the path of integration in question into elements of circle I J. 2 J cl T1 cl so forth J cis in. Fig. 1-8. For any one of these small areas. B dl - Fx H dsA. 1-46 We add the left-hand sides of these equations for all the dsd s and then all the right-hand sides. The sum of the left-hand sides is the line integral around the external boundary since there are always two equal and opposite contributions to the sum along every common side between adjacent djrf s. The sum of the right-hand sides is merely the integral of F X B dsđ over the finite surface. Thus 4 B dl f FxB .dstf 1-47 Jc .J where .5 is the area of any open surface bounded by the curve c. 10 THE LAPLACIAN OPERATOR 72 I he divergence of the gradient of f is the Laplacian of f _ _ . d2f d2f 32f F F r2 3 1-50 J J 3x - ay7 3z2 7 where F is the Laplacian operator. The Laplacian is invariant because it is the result of two successive
