Divergence (divergence) and curl (curl)

original link divergence (divergence)

The discussion of divergence should start with vectors and vector fields. Vectors are the basic concepts for studying multidimensional computing in mathematics. For example, the speed can be decomposed into independent components, then the speed is a multidimensional vector. If each position in the space has a vector attribute, this space is called the vector field. For example, the speed of water in a swimming pool is a vector field.

Divergence is the operator that acts on the vector field. It maps the vector field to a scalar field. The scalar of one of these points represents the "inflow" or "outflow" of the vector of the point.

For example, consider a closed cube area in the swimming pool, where there is either liquid outflow or liquid inflow in the six surfaces of the area. The outflow is positive, the inflow is negative, the flow of six faces is added, if positive, the region has a positive divergence. The inverse is the negative divergence. This is the concept of "flux" in the definition of divergence.

Now if an arbitrary closed surface is taken, its flux is the integral of the component of the vector field on the surface normal vector. If the volume of the closed surface is infinitely small, then the limit is a certain point in the surface, the limit flux is the divergence. From this point of view, the above "inflow" (divergence of less than 0) represents the annihilation of the flux, while the "outflow" (divergence greater than 0) indicates that the region has a new flux generation.

As mentioned in the previous fluid, as the flow does not produce or disappear from thin air, the total dispersion of incompressible fluid must be zero. In fluid mechanics, divergence refers to the rate of change in unit volume when the fluid is moving.

Mathematical representation of divergence

In Cartesian coordinate system oxyz, if the vector field is (P (x, Y, z), Q (x, y, x, y, z)
Then the above three components of x, Y, Z for partial derivative, and then add three results.

In general, use a div or a downward triangle to add a bit to indicate divergence.

If there is a divergence of zero, it is said that the place divergence free.

The concept of divergence is derived from the study of electrostatics in physics. At the end of 19th century, British physicist Hevy (Heaviside) suggested that since electric charges in electrostatic fields were the cause of electric field forces, then the electric field force "aggregation" of the point would be equal to the sum of all the electric field forces emitted by this tiny volume. He also referred to this "aggregation" as "divergence". And he gives a formula for the divergence calculation. As can be seen from the above, the divergence is considered to be the object of a certain point and its close to a very small space.

However, the dispersion of light does not fully describe the properties of the electric field. In 1873, British physicist Maxwell used the four-element method of Hamilton in a paper.

Rotation (Curl)

Remember the four elements you mentioned before?
http://blog.csdn.net/wangxiaojun911/article/details/4644243

For four Elements q = i * A + J * B + K * C + D for partial derivative. Defining the differential operator (D/DX) i + (D/dy) j + (D/dz) k, the differentiation of the first three items in the original four element is very special. Its differential results are divided into two parts, the first part is a, B, C respectively on the x, Y, Z of the partial derivative, if the (a,b,c) as a vector, the result is negative divergence. (Note i*i=-1, J*j=-1, K*k=-1)

The second part is called the "curl" part. for (Dc/dy-db/dz) i+ (DA/DZ-DC/DX) j+ (Db/dx-da/dy) K. Maxwell used the name of curl, which he thought was the attribute of the vector field rotation tendency.

Unlike divergence, which is a scalar on a small bit, the result of the spin operation is still a vector, representing the tendency of rotation on a local micro-element. For example, consider the Arctic Circle as a very small local object discussion, a vector field along the latitude direction will cause the curl and divergence to zero, the vector field along the longitude will cause divergence and curl to zero degrees.

The mathematical representation of rotation degree

If the coordinate system and the vector field are defined at the very beginning (p,q,r), and P () is the partial derivative, the divergence is defined as:
(P (R)/P (Y)-P (Q)/P (z),
P (p)/P (z)-P (R)/p (x)
P (Q)/p (x)-P (p)/P (y))

Divergence (divergence) and curl (curl)