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The difference between differential operators and vectors

May 5, 2016

A vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction.

It can be expressed like

the general form of a vector

based on three perpendicular spatial axes generally designated x, y, and z. Here, i,j,k are the unit vectors in the coordinate system respectively.

While, the differential operator “nabra” is similarly expressed as well as a vector.

nabra

So, this can be thought as a kind of vector.

But, we need to know there is big differences between a general vector and this differential operator, “nabra”.

 

A general vector is defined at a point. But, differential operator “nabra” is defined in the small area, not at a point.

Many scientist neglect this differences. For example, there is an article about Helmhltz Decomposition in Wikipedia like bellow

Part_Longitudinal and Transverse fields

In the upper quoted article, the first(upper) group of equations is OK, because they are expressed at a point. But the second(under) group of equations is not allowed.

We can easily get a sample flow which has divergences by transvers components and rotations by longitudinal components. They are not 0.

For example, let us think a flow like following Figure.

a flow with both components

The sample shows that a flow field is squared off at 50m intervals. And we can get the velocities at each grid point. Their values are shown in the Table.

So, we can get two components at any grid. They are the longitudinal component and transverse component.

transvers and longitudinal components

And, we can calculate the divergence with transverse components, and rotation with longitudinal components.

A sample flow with curl and divergence

 

They both may not be 0.

The mistake in Wiki’s article is caused by the neglect of the differences between the general vector and the difference operator.

 

Someone might say that you could decompose a vector into two components which are the longitudinal component and transvers component at the center point by using Fourier transformation and Invers Fourier transformation.

But, I want to say that the differential is not given at a point even if using Fourier transformation and invers Fourier trans formation. It is given in the very small area, but not at a point.

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