Basic knowledge advanced application.

The form nabla below allows the depiction in vectors, and hence vector operators such as dot, multiplication…

Note not only this nabla(div, grad, curl) is a linear operator, Matrix in general is a linear operator(transformation linearly)

To be more accurate in definition, we define “gradient” as follows – take nabla on a function composed of scalar values(x, y z), the function is the contour.

Divergence, physical meaning is to measure how much the particle field you are studying is expanding(positive scalar of this divergence value), contracting or maintain the same.

for example, compare the following three vector field(equations) represents in math language and plotted:

In sum, it’s the dot product between delta and f .

Third, curl. it’s a determinant as output, it’s the math form of cross product between delta and f.

The physical meaning of curl is that it’s a measure of rotation in the particle field.

supplement on Nov 2021 per Dr. Trefor Bazett, grasp the concept of circulation on line integral and then flux.

He played a trick introducing the k vector pointing up in z direction and then use the cross product of unit tangential vector T and k to get the normal vector n:

Curl is the Circulation Density of a Vector Field.