# Divergence and Stokes’ Theorems

JoVE Core
Physik
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JoVE Core Physik
Divergence and Stokes’ Theorems
##### Vorheriges Video2.14: Line, Surface, and Volume Integrals

The divergence theorem states that the integral of the divergence of a vector field within a volume equals the flux of the vector field through the surface enclosing the volume.

To understand this, consider the velocity field of a fluid flowing smoothly in a pipe. If a hole is poked through its surface, the fluid flows through it. The velocity field then has a positive divergence near the hole.

The total amount of fluid flowing out from the pipe per unit time is equivalent to the net divergence of the velocity in the entire pipe volume.

Stokes' theorem states that the surface integral of the curl of a vector field over a closed surface equals the line integral of the vector field around that surface.

The curl of a vector represents the circulation of that vector along a closed loop. On the surface, the adjacent loops have opposite circulation and cancel each other out. So, the net circulation is only around the edge of the loops.

## Divergence and Stokes’ Theorems

The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write numerous physical laws in both integral form and differential form. Each theorem has an important implication in fluid dynamics and electromagnetism. Through the divergence theorem, a difficult surface integral can be transformed easily into a volume integral, and vice versa. The rate of flow or discharge of any material across a solid surface in a vector field, like electric flow, wind flow, etc., can be determined using the divergence theorem. Similarly, Stokes' theorem can be used to transform a difficult surface integral into an easier line integral, and vice versa. The line integral in itself can be evaluated using a simple surface with a boundary.

## Suggested Reading

1. Griffiths, D.J. (2013). Fourth Edition. Introduction to Electrodynamics. San Francisco, CA: Pearson. Pp. 31-35
2. Ida, N. (2015). Fourth Edition. Engineering Electromagnetics. Switzerland: Springer International Publishing. Pp. 93-106