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?
We may reconstruct vector A from scalar Psi using the vector operation - In this case gradient:
Ax = d/dx Psi
Ay = d/dy Psi
Az = d/dz Psi
Thus, we may always switch to scalar, instead of a vector variable, if nabla \times A = 0. This is widely used method not only in fluid mechanics. Similar are concepts of stream function, scalar potential, etc.
Hi. It is not the negative gradient, but the divergence, which is a scalar.
This problem is the vector notation: From grad U = nablaU (without the dot) for div U = nabla . U (with the dot) is only a small problem at the moment of print, on record in the classroom, or failure on the Xerox machine that confusion is made.
To avoid these mistakes simply adopt the following standard. The gradient always increases the order of the tensor, the divergence decreases in the same order and curv retains the same order.
When looking at an equation, if there are doubts check the homogeneity of the same: Soma, always tensioners same order.
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