By John B. Conway
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Additional info for A Course in Functional Analysis (Graduate Texts in Mathematics, Volume 96)
U V U ) = v u - vv + u v 2 v . 80) v. 81) Switching u and u gives (UVU) = v u * vu + vv2u. 80, the result is v . ( U V V ) - v . ( V V U ) = u v 2 v - vv2u. 82 over a volume V and apply Gauss’s Theorem, to generate one form of Green’s Theorem: j d B . (uvv - vvu) = d r [ u V 2 v- v V 2 u ] . 83) S In this expression, the closed surface S surrounds the volume V. 80 produces a second form of Green’s Theorem: /dZF. (uvv) = S d r F u * v v+ u V 2 v ] . 85) where the path C encloses the differential surface d Z in the right-hand sense.
13. 86) where the path C I + is~ the closed path surrounding both dal and daz. The line integrals along the common edge of C1 and C2 cancel exactly. Any number of these differential areas can be added up to give an arbitrary surface S and the closed contour C which surrounds S . 87) There is an important consequenceof Stokes’s Theorem for vector fields that have zero curl. Such a field can always be derived from a scalar potential. That is to say, if V X = 0 everywhere, then there exists a scalar function a@) such that A = -v@.
30) Physical Picture of the Gradient The gradient of a scalar field is a vector that describes, at each point, how the field changes with position. 32) 25 DIFFERENTIAL OPERATIONS The RHS of this expression can be recognized as the total differential change of Q, due to a differential change of position dF. The result can be written entirely in vector notation as dQ, = VQ, dF. 33, it is clear that the largest value of dQ, occurs when dT points in On the other hand, a displacement perpendicular to VQ, the same direction as VQ,.
A Course in Functional Analysis (Graduate Texts in Mathematics, Volume 96) by John B. Conway
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