. Carnegie Institution of Washington publication. . Fig. 98. Fig. 99. the suffix 2 denoting that the operation curl is performed only in two dimensions. We shall see presently that the curl of the three-dimensional vector is a vector. But, precisely as in the case of the vector-product, the vector-nature of the curl does not appear if we confine ourselves to the consideration of two-dimensional fields. We can now write every term in the sum appearing as second member of equation (b) in the form curh A da-. This sum then takes the form of an integral extended to the area formed by all the eleme
. Carnegie Institution of Washington publication. . Fig. 98. Fig. 99. the suffix 2 denoting that the operation curl is performed only in two dimensions. We shall see presently that the curl of the three-dimensional vector is a vector. But, precisely as in the case of the vector-product, the vector-nature of the curl does not appear if we confine ourselves to the consideration of two-dimensional fields. We can now write every term in the sum appearing as second member of equation (b) in the form curh A da-. This sum then takes the form of an integral extended to the area formed by all the elements da. Thus we get the formula (e) jAsds = |curl2 Ada that is, the line-integral of the tangential component of a two-dimensional vector taken around a closed curve is equal to the integral of the curl of the vector taken over the area bounded by the closed curve. As the expression 1 cdn dn ?s represented the divergence of the vector-lines, section ds 170 (/), i. e., the curvature of their positive normal curves, the expression — ds Sn will represent the divergence of the positive normal curves, i. e., the negative curvature (— 7) of the vector-lines which are the negative normal curves to the curves n (section 168). That is, we can write the expression of curl2 A (/) curl2A = — ^—--M7 Sn. Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original Carnegie Institution of Washington. Washington, Carnegie Institution of Washington
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