Divergence Theory – Proof of the Theorem

# Divergence Theory – Proof of the Theorem

## What Do We Understand By Gauss’s Divergence Theorem?

The Gauss divergence theorem states that the divergence of a vector field is the line integral of the divergence of the vector field over a closed curve.

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## State and Prove the Gauss’s Divergence Theorem

The Gauss divergence theorem states that the divergence of a vector field is the line integral of the field’s curl over a curve.

The proof of the theorem is as follows.

Let be a vector field, a curve, and a point in the plane.

We first show that the divergence of is zero.

Since is a vector field, we can compute its divergence at any point in the plane by taking the divergence of its component vectors.

For any point in the plane,

We now show that the line integral of the curl of over is zero.

We first note that the curl of is the derivative of with respect to :

For any point in the plane,

We now compute the line integral of over :

We can rewrite this line integral as a double integral over :

We now use the fact that is a vector field to split the integral into two parts:

We now compute the two integrals:

Since both integrals are zero, the line integral of the curl of over is also zero.

We have thus shown that the divergence of is zero and the line integral of the curl of over is zero.

Hence, the Gauss divergence theorem is proven.

## The Divergence Theorem Proof

The divergence theorem states that the net divergence of a vector field is zero. This theorem can be proven using the following steps:

1. Let F be a vector field.

2. Let D be the divergence of F.

3. Show that D is a vector field.

4. Show that the net divergence of D is zero.

5. Conclude that the divergence theorem is true.

To show that D is a vector field, we need to show that it has both a magnitude and a direction. We can do this by showing that it satisfies the following properties:

1. The magnitude of D is the sum of the magnitudes of the individual components of D.

2. The direction of D is the direction of the vector sum of the individual components of D.

We can show that D satisfies both of these properties by taking the dot product of D with itself. The dot product of two vectors is the sum of the products of the corresponding components of the vectors.

Since the magnitude of D is the sum of the magnitudes of the individual components of D, and the direction of D is the direction of the vector sum of the individual components of D, it follows that D is a vector field.

Now we need to show that the net divergence of D is zero. To do this, we need to show that the divergence of the sum of the individual components of D is zero

## Gauss’s Divergence Theorem History

The divergence theorem was first formulated by Carl Friedrich Gauss in 1813.

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