Divergence Theorem Formula with Proof, Applications & Examples
Download As PDFIMPORTANT LINKS
Divergence theorem states that the surface integral of a vector space field over a closed surface, known as the “flux” through the surface, is equal to the volume integral of the divergence and over region within the surface. It states intuitively that “the total of all sources of the field in a region (with sinks regarded as negative sources) gives the net flux out of the region.” One of the most important theorems in vector calculus is the “Divergence Theorem.” This theorem is applied to a wide range of difficult integral problems. In the divergence theorem, the surface integral is compared to the volume integral. The divergence theorem is a theorem in vector algebra that relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.
What is Divergence Theorem?
Divergence Theorem is a theorem that compares the surface integral to the volume integral. It aids in determining the flux of a vector field through a closed area with the help of the volume encompassed by the vector field‘s divergence. In vector calculus, it is also known as Gauss’ Divergence Theorem.
Divergence Theorem Statement: Divergence theorem relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed by the surface. It says that the outward flux through such a closed surface is equal to the volume integral of the divergence of the field within the surface.
It is one of the most important theorems in calculus and is used to solve difficult integral problems. It is particularly useful in electrostatics and fluid dynamics.
Divergence Theorem Formula
The surface integral of a vector field A over a closed surface equals the volume integral of the divergence of a vector field A over the volume (V) enclosed by the closed surface, according to the Gauss Divergence Theorem.
Divergence Theorem Proof
Take a surface S that surrounds a volume V. Let vector A represent the vector field in the specified region. Let this volume be composed of many elementary volumes in the form of parallelepiped.
Consider the jth parallelopiped, which has a volume
Step 1: Consider the entire volume divided into elementary volumes I, II, and III, as shown in the figure above. The outward direction of elementary volume I is inward direction of elementary volume II, and the outward direction of elementary volume II is inward direction of elementary volume III, and so on.
Step 2: As a result, the sum of integrals of elementary volumes cancels out, and we are left with the surface integral arising from the surface of S.
Step 3: When we multiply and divide the left-hand side of equation (1) by
Step 4: Let us now suppose that the volume of surface S is divided into infinite elementary volumes, resulting in
Step 5: Now,
Step 6: As a result, equation (2) becomes
Step 7: Given that
Difference between Stokes Theorem and Divergence Theorem
Let us learn about the Difference between Stokes Theorem and Divergence Theorem below:
| Stokes theorem | Divergence Theorem |
| The flux passing through a single surface is calculated by Stokes Theorem | The flux entering and exiting a solid through its surface is measured by the Divergence Theorem. |
| According to Stoke’s Theorem, “the surface integral of a curl of a function across a surface limited by a closed surface is equal to the line integral of the specific vector function around that surface. | According to the divergence theorem, the total outward flow from a volume, as determined by the flux through its surface, may be calculated by adding up all the little amounts of outward flow in that volume using a triple integral of divergence. |
| Stokes theorem relates surface integral to line integral. | Divergence theorem relates volume integral to surface integral. |
| Stokes theorem formula: | Divergence theorem formula: |
Applications of Divergence Theorem
Divergence Theorem applications in calculus are given below:
- In vector fields governed by the inverse-square law, such as electrostatics, gravity, and quantum physics.
- In calculus, it is used to calculate the flux of the vector field through a closed area to the volume encircled by the divergence field.
Solved Examples of Divergence Theorem
Example 1: Solve the,
where
S is the box’s surface
Solution: Given the ugliness of the vector field, computing this integral directly would be difficult. The divergence of F, on the other hand, is appealing:
div F = 3+2y +x
The divergence theorem is used to convert the surface integral to a triple integral.
Where B denotes the box
The triple integral of div F = 3+2y +x over the box B is computed:
=36+3 =39
Example 2: Assume there are four stationary point charges in space, each of which has a charge of 0.002 Coulombs (C). The charges are (0,0,1), (1,1,4), (1,0,0), and (2,2,2). Let
Solution: Gauss’ law states that the flux of
Do you want to score well in your Math exams? Then, you are at the right place. The Testbook platform offers weekly test preparation, live classes, and exam series. Prepare a smart and high-ranking strategy for the exam by downloading the Testbook App right now.
| If you are checking Divergence Theorem article, also check related maths articles: | |
| Converse of Pythagoras Theorem | Mean Value Theorem |
| Pythagoras Theorem | Total Probability Theorem |
| Green’s Theorem | Alternate Segment Theorem |
Divergence Theorem FAQs
What is Divergence Theorem?
Divergence Theorem is a theorem that compares the surface integral to the volume integral. It aids in determining the flux of a vector field through a closed area with the help of the volume encompassed by the vector field‘s divergence.
What is divergence theorem statement?
Divergence theorem states that the surface integral of a vector space field over a closed surface, known as the “flux” through the surface, is equal to the volume integral of the divergence and over region within the surface.
What is divergence theorem formula?
Divergence theorem formula is
What is the application of divergence?
Divergence is used to determine whether a field is free of sources. Remember that a source-free field is a vector field with a stream function; alternatively, a source-free field has a flux that is zero along any closed curve.
What is the difference between stokes theorem and divergence theorem?
The flux passing through a single surface is calculated by stokes theorem while the flux entering and exiting a solid through its surface is measured by the divergence theorem.
- 3 Live Test
- 163 Class XI Chapter Tests
- 157 Class XII Chapter Tests