On representation of expansion in fluid flow

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Helmholtz's thesis discusses the changes in an infinitesimal volume of water, highlighting three motions, including expansion along three axes. Modern interpretations use the negative gradient of a scalar function to represent this expansion. The question arises as to how a gradient, which suggests a single direction, can encapsulate the three axes of dilatation. It is noted that dilatation is typically expressed through divergence, which lacks directionality. The discussion seeks clarity on the relationship between gradient and divergence in the context of fluid dynamics.
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In Helmholtz original thesis On integrals of the hydrodynamical equations, which express vortex-motion, he mentioned in the first section that the change undergone by an arbitrary infinitesimal volume of water under the time dt is composed of three different motions. One of them is an expansion or contraction of the particle parallel to three main axes of dilatation. However in modern language, this part is represented by the negative gradient of a scalar function. My question is why the gradient which I think contains one direction can represent three main directions of dilatation?
 
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Dilatation is usually expressed with a divergence, not a gradient, and the divergence has no direction.
 

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