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I am not sure if this is the answer you are looking for, since the question is listed in both Physics and Abstract Algebra, so I will try to give you some examples from physics.
One of the indicators of a divergence of a vector field is the presence of a source.
For example the electric field can be represented by a vector field, with each vector pointing along the field and has a length proportional to the strength of the electric field at that position. A point source then causes an electric field with a divergence at the location of the point source, with the vectors all pointing away from it (positive charge) or towards it (negative charge).
Another example would be some point mass and the Newtonian gravitational field.
One of Maxwell's equations states that the magnetic field cannot have any divergences meaning that there are no magnetic monopoles.
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Definition of Divergence of a vector field?
hedivergence of a vector fieldF= (F(x,y),G(x,y)) with continuous partial derivatives is defined by:
What is physical significance of divergence?
Divergence is a vector operator that measures the magnitude of a vector fields source or sink at a given point.
How does vector calculus apply in fluid mechanics?
The velocity at each point in the fluid is a vector. If the fluid is compressible, the divergence of the velocity vector is nonzero in general. In a vortex the curl is nonzero.
What physical quantity use both magnitude and direction?
Any vector quantity does. Examples of vector quantities include but are not limited to . . . - Displacement - Velocity - Acceleration - Torque - Force - Electric field - Momentum - Poynting vector
Is magnetic field line scalar or vector quantity?
Vector.
