Step-by-Step Proof of the Divergence Theorem Explained
You must have read about Quantum Theory or Matrix Theory or Probability Theory in your math class or in some sci-fi movie. But have you ever heard of Divergence Theory? Well, here we are today to learn more about this theory. Divergence Theorem is a theorem that talks about the flux of a vector field through a closed area to the volume enclosed in the divergence of the field. It is a part of vector calculus where the divergence theorem is also called Gauss's divergence theorem or Ostrogradsky's theorem.
State and Prove the Gauss's Divergence Theorem
The divergence theorem is the one in which the surface integral is related to the volume integral. More precisely, the Divergence theorem relates the flux through the closed surface of a vector field to the divergence in the enclosed volume of the field. It states that the outward flux through a closed surface is equal to the integral volume within the surface of the divergence over the area.
The net flow of an area will be received by subtracting the sum of all sources by the sum of every sink. The result describes the flow by a surface of a vector and the behavior of the vector field within it.
To state Gauss's divergence theorem in an easier way, let’s break it into parts. The surface integral of a vector field over a sealed area is known as flux through the surface. Therefore, the flux through the surface is equivalent to the volume integral of the divergence over the area inside the surface. Thus, the total sum of all sources of the field in an area gives the net flux out of the area.
This is an essential result for mathematics in engineering and physics. It is one of the most important theorems and is used to solve tough integral problems in calculus. It is used particularly in the field of electrostatic and fluid dynamics.
Divergence Theorem is generally applied in 3 dimensions, but it can be used in any number of dimensions. When you use it in 2 dimensions, it becomes equivalent to Green’s theorem which states that the line integral around any simple closed curve is equal to a double integral over the plane region. When you use it in 1 dimension, it becomes equivalent to integration by parts.
The Divergence Theorem Proof
Let us consider a surface denoted by S which encloses a volume denoted by V.
Suppose vector A is the vector field in the given region. Suppose this volume is made up of a large number of parallelepipeds (1- 6 parallelepipeds) which represent elementary volumes.
Consider the volume of the jth parallelepiped is \[ \Delta V_{j} \]which is bounded by a surface \[ S_{j} \] of area \[ d \overrightarrow{S_{j}}\] . The surface integral of \[ \overrightarrow{A} \] over the surface \[ S_{j} \] will be-
\[ \oint \oint\limits_{S} \overrightarrow{A} . d \overrightarrow{S_{j}}\]
Now consider if the whole volume is divided into elementary volumes I, II, and III as shown below.
(Image will be uploaded soon)
The elementary volume I outward is the elementary volume II inward and the elementary volume II outward is the elementary III inward and so on.
Thus, the sum of the elementary volumes integrals will cancel each other out and the surface integral arising from the surface S will be left.
\[ \sum \oint \oint \limits _{S_{j}} \overrightarrow {A}. d \overrightarrow {S_{j}} = \oint \oint \limits _{S} \overrightarrow {A}. d \overrightarrow {S} …. (1) \]
Multiplying and dividing the left-hand side of the equation (1) by \[ \Delta V_{i} \], we get
\[ \oint \limits _{S_{j}} \overrightarrow {A} . d \overrightarrow {S} = \sum \frac{1}{\Delta V_{i}} (\oint \oint \limits _{S_{i}} \overrightarrow {A} . d \overrightarrow {S_{i}} ) \Delta V_{i} \]
Now, suppose the volume of surface S is divided into infinite elementary volumes such that \[ \Delta V_{i} \rightarrow 0\].
\[ \oint \oint \limits _{S} \overrightarrow {A} . d \overrightarrow{S} = \lim\limits_{\Delta V_{i} \rightarrow {0}}\sum \frac{1}{\Delta V_{i}} (\oint \oint \limits _{S_{i}} \overrightarrow {A} . d \overrightarrow {S} ) \Delta V_{i} ….. (2) \]
Now,
\[ \lim \limits_ {\Delta V_{i} \rightarrow {0}} (\frac{1}{\Delta V_{i}} (\oint \oint \limits _{S_{i}} \overrightarrow {A} . d \overrightarrow {S})) = (\overrightarrow {V}.\overrightarrow {A} ) \]
Therefore, eq (2) becomes
\[ \oint \oint \overrightarrow {A} . d \overrightarrow {S} = \sum (\overrightarrow{\nabla} . \overrightarrow {A} ) \Delta V_{i} …. (3) \]
We know that \[ \Delta V_{i}\rightarrow{0} \], thus\[\sum \Delta V_{i} \] will become the integral over volume V.
This is the Gauss divergence theorem.
Gauss's Divergence Theorem History
Lagrange was the first one to discover the Divergence Theorem in 1762. Later on in 1813, it was rediscovered independently by Carl Friedrich Gauss. He also gave the first proof of the general theorem in 1826. Many other mathematicians like Green, Simeon-Denis Poisson, and Frédéric Sarrus also discovered this theorem.
Conclusion
This article entails detailed information on Divergence Theory and its derivation. You can go through it for a comprehensive understanding. Also, download the PDFs to read at your ease.
FAQs on Mastering Divergence Theory in Mathematics
1. What is the Divergence Theorem in mathematics?
The Divergence Theorem, also known as Gauss's Theorem, is a fundamental concept in vector calculus. It establishes a relationship between the total outward flow (flux) of a vector field through a closed surface and the behaviour of the vector field inside that surface. Specifically, it states that the surface integral of the normal component of a vector field over a closed surface is equal to the volume integral of the divergence of that field over the volume enclosed by the surface.
2. What does the divergence of a vector field represent conceptually?
Conceptually, the divergence of a vector field at a specific point measures the rate at which 'something' is flowing out from that point. Think of it as the field's 'source strength' or 'sink strength'.
- A positive divergence at a point indicates it is a source, where the field originates (like a water faucet).
- A negative divergence indicates it is a sink, where the field terminates (like a drain).
- A zero divergence means the flow into the point equals the flow out, indicating the field is incompressible at that point.
3. What is the main importance of using the Divergence Theorem?
The primary importance of the Divergence Theorem is that it simplifies complex calculations. It allows us to convert a potentially difficult surface integral (calculating flux over a 3D shape's boundary) into a volume integral (calculating divergence throughout the interior of the shape). In many practical problems in physics and engineering, evaluating the volume integral is significantly easier than evaluating the surface integral directly.
4. What are some key real-world applications of the Divergence Theorem?
The Divergence Theorem is crucial in many scientific and engineering fields. Key applications include:
- Electromagnetism: It is the foundation for Gauss's Law, which relates electric charge to the electric field it produces.
- Fluid Dynamics: It is used to derive the continuity equation, which describes the conservation of mass in a fluid flow. It helps determine if a fluid is compressible.
- Heat Transfer: It can be used to describe the flow of heat, where the divergence of the heat flux represents the rate at which heat is being generated or absorbed at a point.
5. How does the Divergence Theorem fundamentally differ from Stokes' Theorem?
The fundamental difference lies in the dimensions they relate. The Divergence Theorem relates a 3D volume to its 2D closed boundary surface (e.g., a sphere's interior to its surface). In contrast, Stokes' Theorem relates a 2D open surface to its 1D closed boundary curve (e.g., a hemisphere's surface to the circle at its base). In simple terms, Divergence Theorem is for volumes and their surfaces, while Stokes' Theorem is for surfaces and their edges.
6. Why is it a requirement for the Divergence Theorem to be applied only on a closed surface?
The Divergence Theorem is only applicable to a closed surface because the theorem is about balancing the net flow 'out' of an enclosed volume. A surface must be closed (like a sphere or a cube) to fully contain a volume. An open surface, like a flat disc or a hemisphere without its base, does not enclose a volume. Therefore, the concept of a 'total outward flux' from an 'enclosed volume' is meaningless for an open surface, making the theorem invalid.
7. What does it signify if the divergence of a vector field is zero everywhere?
If the divergence of a vector field is zero everywhere, the field is called solenoidal or incompressible. This signifies that there are no sources or sinks anywhere in the field. The amount of the field's flux entering any region is exactly equal to the amount leaving it. A prime example from physics is the magnetic field, whose divergence is always zero, which mathematically expresses the fact that there are no magnetic monopoles (isolated north or south poles).
8. What is the mathematical formula for the Divergence Theorem?
The mathematical formula for the Divergence Theorem is expressed as:
∯_S (F ⋅ n̂) dS = ∭_V (∇ ⋅ F) dV
Where:
- The left side is the surface integral of the vector field F over a closed surface S.
- n̂ is the outward pointing unit normal vector to the surface.
- The right side is the volume integral of the divergence of F (∇ ⋅ F) over the volume V enclosed by the surface S.
9. Can the Divergence Theorem be applied to a 2D vector field?
The Divergence Theorem as stated is a 3D concept. However, there is a 2D analogue often called Green's Theorem in its divergence form. It relates a double integral over a 2D region to a line integral around its closed boundary curve. It serves the same purpose of relating the microscopic divergence within a region to the macroscopic flux across its boundary, but it operates in a two-dimensional plane instead of three-dimensional space.
