Intuitive understanding of gradient, rotation, and divergence

Reprinted, this is very realistic and straightforward. We suggest you don't look at it when you eat ....

 

If the divergence is zero, it indicates that it is a passive field. If the divergence is not zero, it indicates that it is an active field (with a positive source or a negative source)

If your field is a flow field, the dispersion of the field is the net flow of the fluid flowing out of the unit volume at a certain point in time. if the divergence of a field is not zero at a certain point, it indicates that the field is in the active source. For example, if the electric field is not zero at a certain point, it indicates that the field has a charge. If the velocity field is not zero, A table is generated or disappears in an endless stream at this point (if the divergence is negative ).

A field is located at a certain place and performs ring integral along an infinitely small plane boundary. The plane normal vector is given by the degree vector, and the length of the degree vector is the integral value of the ring per area. basically, the degree of rotation measures whether a vector field turns at a certain point.

 

Gradient: The calculated object is like a pure volume. The calculated result is a vector in a pure volume field. The gradient calculation result is "calculate a vector at each position, and the direction of this vector will be from any point around it (very close to the surrounding, learning calculus should know what is the limit ?), The minimum value points to the maximum value of the surrounding pure value.

The size of this vector will be the minimum and maximum gap mentioned above"

For example, it will be relatively simple. If the current pure field is represented by a mountain,

The higher the pure value, the lower the contrast. After the operation of the gradient operation,

A vector is calculated on each vertex of the mountain, which points to the steep

That direction, and the vector size represents how steep this steep direction is.

 

Divergence: The operator is like a vector, and the result is a pure amount.

The role of divergence is like a vector field,

If we consider any point (or a very small area around it ),

At this point, the degree of divergence of the vector field,

If it is positive, it indicates that these vector fields are exclusive.

If it is negative, it indicates that these vector fields are in the inner set.

For example:

Because the effect of divergence is like a vector field, we cannot use the mountains mentioned above to imagine it,

This time I want to squeeze a lot of people in a big square. If everyone is walking around,

Can we regard everyone's actions as a vector,

If a person put a fart, all the people around him (may include himself) want to get a little farther away,

We will find that all people in this area are moving in the direction outside this small area.

Yes. This is the divergence (you can also say that it is a transient distance ....),

If you scatter more people, the larger the divergence is.

 

Degree of rotation: The operator is a vector, and the result is a vector.

The rotation object is also a vector field. The following example is used:

If all the people who are now scattered are straight to the opposite direction of the fart,

At this time, do you see that the dynamic lines of these people are a standard radiation ??

But in fact, when everyone smells fart, they won't know exactly

Which direction does the fart come from.

However, they may go in the wrong direction. After trying, they will find that something is wrong.

At this time, you can see that the direction of the people is not necessarily a radiation (everyone is moving radial ),

However, there may be some components that are moving to the center (from the perspective of fart points)

The degree of rotation corresponds to the moving of the tangent. Relatively speaking,

The divergence actually corresponds to radial movement.

While a fart, although it may cause some tangent movement like the above,

However, in theory, it does not make the scattered people tend to turn around in a timely manner or reverse clock.

In this case, the time-to-minute conversion can be seen as offset from the reverse clock conversion,

Therefore, in this case, the rotation is still zero.

That is to say, a fart can cause divergence without causing a degree of rotation ....

When is there a spin ??

If the music is played at this time,

Everyone started to dance around the middle fire hand in hand (of course, it's the kind of conversion around the fire)

At this time, there will be no degree of rotation. (except for those who have been farting for the toilet)

 

1.1 of the three above will be remembered.

Whether it is gradient, divergence, or rotation, it is a local volume (pure volume, vector ),

All of the considerations are the case of any point (the surrounding area is very close, very small scope.

The preceding examples are for the second-degree space vectors because they are easy to understand,

And they are big things, but the square is a point, and the fire party is also a point,

Najomi is a Chinese Mustard. You can imagine it on your own.