Divergence Theorem

Divergence Theorem is one of the important theorems in Calculus. The divergence theorem relates the surface integral of the vector function to its divergence volume integral over a closed surface.

In this article, we will dive into the depth of the Divergence theorem including the divergence theorem statement, divergence theorem formula, Gauss Divergence theorem statement, Gauss Divergence theorem formula, and Gauss Divergence Theorem proof.

We will also go through some points on Gauss’s Divergence theorem vs Green’s theorem, solve some examples, and answer some FAQs related to the divergence theorem.

What is Divergence Theorem?

Divergence Theorem states that,

“Surface integral of the normal of the vector point function P over a closed surface is equal to the volume integral of the divergence of P under the closed surface”.

What is Divergence?

Divergence of a vector field is defined as the vector operation which results in the scalar field calculating the rate of change of flux. The divergence is denoted by the symbol ∇ or div(vector).

For a vector field P:

Divergence of P(x, y) in 2-D, P = P₁i + P₂j is given by:

∇P(x, y) =

Divergence of P(x, y) in 3-D, P = P₁i + P₂j + P₃k is given by:

∇P(x, y, z) =

Learn more about, Divergence and Curl

Divergence Theorem Formula

The formula for the Divergence Theorem is given by:

Gauss Divergence Theorem

Gauss Divergence theorem gives us the relation between the surface integral of the vector to the volume of the vector in a closed surface. Below we will learn about the Gauss Divergence Theorem in detail.

Statement of Gauss Divergence Theorem

The Gauss Divergence Theorem states that:

“Surface integral of the vector field P over the closed surface is equal to the volume integral of the divergence of the vector over the closed surface”.

It can be mathematically represented as:

Proof of Gauss Divergence Theorem

To prove Gauss Divergence theorem, consider a surface S with volume V. It has a vector field P in it. Let the total volume of solid consists of different small volumes of parallelepipeds.

Since the solid is divided into small elementary volumes, let’s take one of them V_j bounded by the surface S_j of area then the surface integral of the vector V over S_j is represented as

The total volume is divided into parts volume I, II, III …. The outward volume of S_i is the inward volume of S_i+1. So, these elementary volumes cancel each other, and we get the surface integral derived by surface S.

——(1)

Now, we will multiply and divide equation (1) by ΔVi,

Again, we consider that the volume of surface S is divided into infinite parts i.e., ΔVi → 0

——–(2)

Since,

Putting above value in equation 2

As ΔV_i→0, ∑ (ΔV_i ) results in the volume integral V

Gauss’s Divergence Theorem vs. Green’s Theorem

Gauss Divergence Theorem and Green’s Theorem both the theorems have their specific advantages.

Gauss’s Divergence Theorem Vs Green’s Theorem
Gauss Divergence Theorem	Green’s Theorem
Gauss Divergence Theorem deals with 3-D solid bounded by closed curve	Green’s Theorem deals with the 2-D figures bounded by simple closed curve
Gauss divergence theorem uses the volume integralin its results.	Green’s theorem uses the surface integral in its results.

Divergence Theorem – Conclusion

In the above article we have discussed about the Divergence Theorem and Gauss Divergence theorem. After discussing these theorems, we can conclude that the surface integral of the vector field under a closed surface is equal to the volume integral of the divergence of the vector field under the closed region.

Read More,

• Scalars and Vectors
• Vector Calculus
• Vector Algebra

Divergence Theorem Example

Example 1: Compute where, F = (4x + y, y² – cos x² z, xz +ye^3x) and 0 ≤ x ≤ 1, 0 ≤ y ≤ 3 and 0 ≤ z ≤ 2.

Solution:

According to Divergence Theorem

The intervals given are: 0 ≤ x ≤ 1, 0 ≤ y ≤ 3 and 0 ≤ z ≤ 2

First, we find div(F)

div(F) = (4 + 2y + x)

Now, putting the values of div(F) and intervals in the formula

(4 + 2y + x) dz dy dx

Integrate the above expression w.r.t x, y and z respectively.

(8 + 4y + 2x) dy dx

Putting all the limits and integrating

42 +3

45

Example 2: Compute where, F = (x² + 4y, 4y – tan z, z +y) and 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and 0 ≤ z ≤ 1.

Solution:

According to Divergence Theorem

The intervals given are: 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and 0 ≤ z ≤ 1

First, we find div(F)

div(F) = (2x + 4 + 1) = 2x + 5

Now, putting the values of div(F) and intervals in the formula

(2x + 5) dz dy dx

Integrate the above expression w.r.t x, y and z respectively.

(2x + 5) dy dx

Putting all the limits and integrating

1² + 5(1)

6

Example 3: Compute where, F = (x + y + z, y², x³ + z³) and 0 ≤ x ≤ 1, 0 ≤ y ≤ 2 and 0 ≤ z ≤ 2.

Solution:

According to Divergence Theorem

The intervals given are: 0 ≤ x ≤ 1, 0 ≤ y ≤ 2 and 0 ≤ z ≤ 2

First, we find div(F)

div(F) = (1 + 2y + 3z²)

Now, putting the values of div(F) and intervals in the formula

(1 + 2y + 3z²) dz dy dx

Integrate the above expression w.r.t x, y and z respectively.

(2 + 4y + 8) dy dx

Putting all the limits and integrating

28(1)

28

Example 4: Compute where, F = (2x + 3y + 4z, 2y², 5x³ + z) and 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and 0 ≤ z ≤ 3.

Solution:

According to Divergence Theorem

The intervals given are: 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and 0 ≤ z ≤ 3

First, we find div(F)

div(F) = (2 + 4y + 1) = 3 + 4y

Now, putting the values of div(F) and intervals in the formula

(3 + 4y) dz dy dx

Integrate the above expression w.r.t x, y and z respectively.

(9 + 12y) dy dx

Putting all the limits and integrating

15(1)

15

Practice Problems on Gauss Divergence Theorem

P1. By using the Divergence theorem, evaluate where, = sin (πx) i + zy³ j + (z² + 4x) k and S is the surface of the box with -1 ≤ x ≤ 2, 0 ≤ y ≤1 and 1 ≤ z ≤ 4. All the six sides are included in S.

P2. By using the Divergence theorem, evaluate where, = yx² i + (x y – 3x⁵) j + (x – 6y) k and S is the surface of the sphere with radius of sphere is 5, y ≤ 0 and z ≤ 0. All the three surfaces of solid are included in S.

P3. By using Gauss Divergence Theorem, compute where, F = (x⁴ + 4y, 5y² – cot x² , xe^3z) and 0 ≤ x ≤ 2, 1 ≤ y ≤ 5 and -1 ≤ z ≤ 1.

P4. By using Gauss Divergence Theorem, compute where, F = (x² + y² + z², 5y³ – z⁴ , xz⁵e^3y) and -1 ≤ x ≤ 3, 1 ≤ y ≤ 3 and -2 ≤ z ≤ 2.

Divergence Theorem – FAQs

1. What is Divergence Theorem Definition?

Divergence theorem states that, “Surface integral of the normal of vector function P is equal to the volume integral of the divergence of the vector function P over the closed surface.”

2. What is Gauss Divergence Theorem Formula?

Gauss Divergence Theorem Formula is given by,

3. What is Statement of Gauss Divergence Theorem?

Gauss Divergence Theorem statement is, “Surface integral of the vector field P over the closed surface is equal to the volume integral of the divergence of the vector over the closed surface”.

4. What is Formula for Divergence Theorem?

Formula for divergence theorem is,

5. What is Divergence in Vector Calculus?

Divergence in the vector calculus is the operation on vectors that gives us the rate of change in the flux of the vector field in the form of scalar field.