I have never really understood what divergence does in a vector field although I was able to pass exams because I learned to calculate it. It occurred to me that if I made my own field in 2-dimensions then I could test my own hypothesis about what divergence means. I made the vector field
F above. It is a vector field because there is an arrow at each coordinate showing the size and direction of the function
F. I have set it up so that the function has an x component Fx along the
i direction that is equal to the size of the x coordinate but there is no component in the
j direction. The word divergence in the English language means that things that were travelling parallel to each other split apart and separate due to direction change. That's how I pictured divergence in a vector field. This vector field has been set up because the vectors change size but remain parallel. My idea would mean no divergence. So here comes the calculation.
The upside down triangle is the differential operator nabla or del. Here it is differentiating by the dot product so it is to do with the gradient along the line of the vector; perhaps the rate of change of size along the direction of the vector. The divergence calculates out as 1. So there is divergence of this vector field even though the vectors remain parallel. So the stretching of the vector, the increase in size as it goes along, constitutes divergence.
