Question

check the divergence theorem for the unit cube suited at origin.

Answer 1

Explanation:

Before examining the divergence theorem, it is helpful to begin with an overview of the versions of the Fundamental Theorem of Calculus we have discussed:

The Fundamental Theorem of Calculus:

∫baf′(x)dx=f(b)−f(a).

This theorem relates the integral of derivative f′ over line segment [a,b] along the x-axis to a difference of f evaluated on the boundary.

The Fundamental Theorem for Line Integrals:

∫C∇f⋅dr=f(P1)−f(P0),

where P0 is the initial point of C and P1 is the terminal point of C. The Fundamental Theorem for Line Integrals allows path C to be a path in a plane or in space, not just a line segment on the x-axis. If we think of the gradient as a derivative, then this theorem relates an integral of derivative ∇f over path C to a difference of f evaluated on the boundary of C.