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单词 GradientTheorem
释义

gradient theorem


If  u=u(x,y,z)  is continuously differentiable function in a simply connected domain (http://planetmath.org/Domain2) D of 3 and P=(x0,y0,z0)  and  Q=(x1,y1,z1)  lie in this domain, then

PQuds=u(x1,y1,z1)-u(x0,y0,z0)(1)

where the line integralMathworldPlanetmath of the left hand side is taken along an arbitrary path in D.

The equation (1) is illustrated by the fact that

uds=uxdx+uydy+uzdz

is the total differentialMathworldPlanetmath of u, and thus

PQuds=PQ𝑑u.
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