# Understanding the divergence theorem

• polaris90
In summary: Divergence theorem is a mathematical statement that tells you how much flux is diverging from a given volume (or surface).
polaris90
I'm having some trouble understanding what divergence of a vector field is in my "Fields and Waves" course. Divergence is defined as $divE=∇E = (∂Ex/∂x) + (∂Ey/∂y) + (∂Ez/∂z)$. As far as I understand this gives the strength of vector E at the point(x,y,z).
Divergence theorem is defined as $∫∇Eds$, where ds represents the area or volume of the vector field. In other words, I understand it as the overall strength of the vector field at a group of points composing a volume defined by the integral.
Could someone verify this for me?

At least pointing out what's wrong would help.

Divergence is not the strength of the field at a particular point. Divergence is the total amount of flux going into or leaving a volume at that point.

Thanks, so would it be correct to say that divergence theorem refers to the amount of flux passing through a surface or volume?

Almost. Not through the volume, but rather being sourced or swallowed by the volume. For example, you cited ∇E which is often seen as part of one of Maxwell's equations. In this case, ∇E = q, which says the flux leaving a volume (the divergence) is equal to the amount of charge q contained in the volume. The charge q is sourcing e-field flux in this example.

thank you all, that was very helpful

polaris90 said:
Thanks, so would it be correct to say that divergence theorem refers to the amount of flux passing through a surface or volume?

Divergence theorem just makes it very clear.
What is flux?
Product of normal component of vector field and the surface element.

What is divergence of vector field?
Net flux diverging per unit volume.

What is Divergence?
Positive or negative divergence of a vector field at a point indicates whether the lines of force are diverging or converging at that given point.

By divergence theorem... The flux diverging from a given volume will be equal to the flux passing through the closed surface enclosing the volume.

For your question, the flux as explained above will be 'passing' through the closed surface enclosing the volume and the volume will be acting as the source of flux.

Consider a charge kept at the center of the sphere. The charge enclosed by this volume will be equal to electric flux lines passing normal to the closed surface. So the volume here is the source of flux whereas the flux is passing through the surface of the sphere.

Simple :]

Correct me if i am wrong.

Last edited:

## What is the divergence theorem?

The divergence theorem, also known as Gauss's theorem, is a mathematical concept in vector calculus that relates the surface integral of a vector field to the volume integral of the divergence of that field.

## What is the significance of the divergence theorem?

The divergence theorem is significant because it allows us to evaluate a difficult surface integral by converting it into a simpler volume integral. This is particularly useful in solving problems in electromagnetism, fluid mechanics, and heat transfer.

## How is the divergence theorem derived?

The divergence theorem is derived from the fundamental theorem of calculus and Green's theorem. It can also be derived using the concept of flux, which is the amount of a vector field passing through a surface.

## What are the assumptions of the divergence theorem?

The divergence theorem assumes that the vector field is continuous and differentiable, and that the surface or volume being evaluated is simply connected (i.e. has no holes).

## Can the divergence theorem be applied in three-dimensional space?

Yes, the divergence theorem can be applied in three-dimensional space. It is a generalization of Green's theorem, which is used in two-dimensional space.

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