Question

In what ways are the Fundamental Theorem for Line Integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem similar?

Short answer

The Divergence theorem is stated.

Step-by-step solution

Definition of Concept

Integrals: An integral is a mathematical concept that assigns numbers to functions in order to describe displacement, area, volume, and other concepts that arise from combining infinitesimal data. The process of determining integrals is known as integration.

State the divergence Theorem.

Consider a positively oriented solid region E which is simple and enclosed by the surface S as boundary and the vector field F with components that have continuous partial derivatives. Then the surface integral of F over the surface S according to Divergence theorem is as follows,

\(\iint_{S}{F}\cdot dS=\iiint_{E}{\operatorname{div}}\mathrm{F}dV\)

The Divergence theorem is stated as follows:

- The Divergence theorem is a extension of a Green's theorem to vector fields on \({\mathbb{R}^3}\).

- The Divergence theorem is similar to Green's theorem and Stokes' theorem,

- The Divergence theorem demonstrates the relation between the integral of derivative of a field function over a region and the integral of field function over the boundary of a region.

Therefore, the Divergence theorem is stated.