Divergence Theorem Example
Example 1: Compute where, F = (4x + y, y2 – cos x2 z, xz +ye3x) 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
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.
Table of Content
- What is Divergence Theorem?
- Divergence Theorem Formula
- Gauss Divergence Theorem
- Proof of Gauss Divergence Theorem
- Gauss’s Divergence Theorem vs. Green’s Theorem
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