operates on vector field producing a scalar field giving quantity of the gector field’s source at each points.
represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
definition
the divergence of a vector field F(x) at point x0 is defined as the limit of the ratio of the surface integral of F out of the closed surface of a volume V enclosing x0 to the volume of V, as V shrinks to zero
divFx0=V→0lim∣V∣1∬S(V)F⋅n^dS
where ∣V∣ is the volume of V, S(V) is the boundary of V and n^ is the outward unit normal to that surface.
Cartesian coordinates
for a continuously differentiable vector field F=Fxi+Fyj+Fzk, divergence is defined as the scalar-valued function:
Suppose a function f:Rn→Rm is a function such that each of its first-order partial derivatives exists on Rn, then the Jacobian matrix of f is defined as follows:
When m=n, the Jacobian matrix is a square, so its determinant is a well-defined function of x1
When m=1, or f:Rn→R is a scalar-valued function, then Jacobian matrix reduced to the row vector ∇Tf, and this row vector of all first-order partial derivatives of f is the transpose of the gradient of f, or Jf=∇Tf
gradient
a vector field ∇f whose value at a point p gives the direction and the rate of fastest increase.
In coordinate-free term, the gradient of a function f(r) maybe defined by:
df=∇f⋅dr
where df is the infinitesimal change in f for an infinitesimal displacement dr, and is seen to be maximal when dr is in the direction of the gradient ∇f
definition
the gradient of f (∇f) is defined as the unique vector field whose dot product with any vector v at each point x is the directional derivative of f along v, such that: 2