What's the difference between these differential operators?

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SUMMARY

The discussion clarifies the differences between three differential operators relevant to fluid mechanics: the convective derivative, the variational derivative, and the partial derivative. The convective derivative accounts for changes in a velocity field as a function of both space and time, representing a first-order tensor or vector field. The variational derivative is used in the context of functionals and optimization, while the partial derivative measures the rate of change of a function with respect to one variable, holding others constant. Understanding these distinctions is crucial for fluid mechanics applications.

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I'm learning fluid mechanics, and I am confused about the following differential operators. What's the difference between each?
 

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the first one is the so called convective derivative. This arises from the fact that in the euler picture we have a velocity field, that is a first order tensor field or vector field, describing the velocity as a function of space and time. Hence velocity or any other vector field doesn't change only by time but by place too. So the differential operator is.

The second one is the variational derivative.

The third one is the usual partial derivative.
 

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