How will you measure divergence in a way other than using the dot product of Nabla (∇)operator? For example, here I have another way of finding the gradient (normal) without using the Nabla (∇) operator.
Objective:To find the gradient (normal) for $x^2 + y^2 -8$ at (2,2)
The usual method is below:-
Grad($x^2 + y^2 -8$)= ∇($x^2 + y^2 -8$)=2xi+2yj -> normal
Therefore, the gradient at (2,2) is obtained by substituting in the above ‘normal’.
Gradient is 4i+4j-> vector1
Here is another method:-
dy/dx=-x/y
dy/dx(2,2)=-1
We know that the product of slope of normal and slope of tangent is -1. So, the slope of normal here is 1.
The vector for line with slope ‘m’ is i+mj
Our required vector is i+(1)j = i+j -> vector2
See, vector1= vector2 (I have obtained normal (gradient) without using nabla operator).
Can anyone provide a different (intuitive) way of measuring curl and divergence? Maybe, something like take a infinitesimally small circle and integrate all the radial components to find the curl.
