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) = \frac{\partial P_1}{\partial x} + \frac{\partial P_2}{\partial y}
Divergence of P(x, y) in 3-D, P = P1i + P2j + P3k is given by:
∇P(x, y, z) = \frac{\partial P_1}{\partial x} + \frac{\partial P_2}{\partial y}+\frac{\partial P_3}{\partial z}
Learn more about, Divergence and Curl
The formula for the Divergence Theorem is given by:
\oiint\limits_S\overrightarrow{\rm P}.\overrightarrow{\rm n} .d\overrightarrow{\rm S}= \oiiint\limits_V(\overrightarrow{\rm \nabla}.\overrightarrow{\rm P})dv
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:
\oiint\limits_S\overrightarrow{\rm P}.d\overrightarrow{\rm S}= \oiiint\limits_V(\overrightarrow{\rm \nabla}.\overrightarrow{\rm P})dV
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 Vj bounded by the surface Sj of area d\overrightarrow{\rm S}
then the surface integral of the vector V over Sj is represented as \oiint\limits_S\overrightarrow{\rm P}.d\overrightarrow{\rm S_j}
The total volume is divided into parts volume I, II, III .... The outward volume of Si is the inward volume of Si+1. So, these elementary volumes cancel each other, and we get the surface integral derived by surface S.
\sum\oiint\limits_{Sj}\overrightarrow{\rm P}.d\overrightarrow{\rm S_j}=\oiint\limits_{S}\overrightarrow{\rm P}.d\overrightarrow{\rm S}
------(1)
Now, we will multiply and divide equation (1) by ΔVi,
\oiint\limits_S\overrightarrow{\rm P}.d\overrightarrow{\rm S}=\sum\frac{1}{\Delta V_i}(\oiint\limits_{Si}\overrightarrow{\rm P}.d\overrightarrow{\rm S})\Delta V_i
Again, we consider that the volume of surface S is divided into infinite parts i.e., ΔVi → 0
\oiint\limits_S\overrightarrow{\rm P}.d\overrightarrow{\rm S}=lim_{\Delta V_i\rightarrow0}\sum\frac{1}{\Delta V_i}(\oiint\limits_{Si}\overrightarrow{\rm P}.d\overrightarrow{\rm S})\Delta V_i
--------(2)
Since,
lim_{\Delta V_i\rightarrow0}\big[\frac{1}{\Delta V_i}(\oiint\limits_{Si}\overrightarrow{\rm P}.d\overrightarrow{\rm S})\bigg] = (\overrightarrow{\rm V}.\overrightarrow{\rm P})
Putting above value in equation 2
\oiint\limits_S\overrightarrow{\rm P}.d\overrightarrow{\rm S}=\sum(\overrightarrow{\rm \nabla}.\overrightarrow{\rm P})\Delta V_i
As ΔVi→0, ∑ (ΔVi ) results in the volume integral V
\oiint\limits_S\overrightarrow{\rm P}.d\overrightarrow{\rm S}= \oiiint\limits_V(\overrightarrow{\rm \nabla}.\overrightarrow{\rm P})dV
Gauss's Divergence Theorem vs. Green's Theorem
Gauss Divergence Theorem and Green's Theorem both the theorems have their specific advantages.
|
|
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.
|
\oiint\limits_S\overrightarrow{\rm P}.\overrightarrow{\rm n} .d\overrightarrow{\rm S}= \oiiint\limits_V(\overrightarrow{\rm \nabla}.\overrightarrow{\rm P})dv
| \oint\limits_C(Ldx~+~Mdy)~=~\iint\limits_D(\delta M/\delta x~-~\delta L/\delta y)dxdy
|
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,
Divergence Theorem Example
Example 1: Compute \oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}
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
\oiint\limits_S\overrightarrow{\rm P}.\overrightarrow{\rm n} .d\overrightarrow{\rm S}= \oiiint\limits_V(\overrightarrow{\rm \nabla}.\overrightarrow{\rm P})dv
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
\oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}= \iiint\limits_{000}^{132}
(4 + 2y + x) dz dy dx
Integrate the above expression w.r.t x, y and z respectively.
\oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}= \iint\limits_{00}^{13}
(8 + 4y + 2x) dy dx
Putting all the limits and integrating
\oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}=
42 +3
\oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}=
45
Example 2: Compute \oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}
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
\oiint\limits_S\overrightarrow{\rm P}.\overrightarrow{\rm n} .d\overrightarrow{\rm S}= \oiiint\limits_V(\overrightarrow{\rm \nabla}.\overrightarrow{\rm P})dv
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
\oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}= \iiint\limits_{000}^{111}
(2x + 5) dz dy dx
Integrate the above expression w.r.t x, y and z respectively.
\oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}= \iint\limits_{00}^{11}
(2x + 5) dy dx
Putting all the limits and integrating
\oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}=
12 + 5(1)
\oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}=
6
Example 3: Compute \oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}
where, F = (x + y + z, y2, x3 + z3) and 0 ≤ x ≤ 1, 0 ≤ y ≤ 2 and 0 ≤ z ≤ 2.
Solution:
According to Divergence Theorem
\oiint\limits_S\overrightarrow{\rm P}.\overrightarrow{\rm n} .d\overrightarrow{\rm S}= \oiiint\limits_V(\overrightarrow{\rm \nabla}.\overrightarrow{\rm P})dv
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
\oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}= \iiint\limits_{000}^{122}
(1 + 2y + 3z2) dz dy dx
Integrate the above expression w.r.t x, y and z respectively.
\oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}= \iint\limits_{00}^{12}
(2 + 4y + 8) dy dx
Putting all the limits and integrating
\oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}=
28(1)
\oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}=
28
Example 4: Compute \oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}
where, F = (2x + 3y + 4z, 2y2, 5x3 + z) and 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and 0 ≤ z ≤ 3.
Solution:
According to Divergence Theorem
\oiint\limits_S\overrightarrow{\rm P}.\overrightarrow{\rm n} .d\overrightarrow{\rm S}= \oiiint\limits_V(\overrightarrow{\rm \nabla}.\overrightarrow{\rm P})dv
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
\oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}= \iiint\limits_{000}^{113}
(3 + 4y) dz dy dx
Integrate the above expression w.r.t x, y and z respectively.
\oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}= \iint\limits_{00}^{11}
(9 + 12y) dy dx
Putting all the limits and integrating
\oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}=
15(1)
\oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}=
15
Practice Problems on Gauss Divergence Theorem
P1. By using the Divergence theorem, evaluate \oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}
where, \overrightarrow{\rm F}
= 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 \oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}
where, \overrightarrow{\rm F}
= 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 \oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}
where, F = (x4 + 4y, 5y2 - cot x2 , xe3z) and 0 ≤ x ≤ 2, 1 ≤ y ≤ 5 and -1 ≤ z ≤ 1.
P4. By using Gauss Divergence Theorem, compute \oiint\limits_S\overrightarrow{\rm F}.d\overrightarrow{\rm S}
where, F = (x2 + y2 + z2, 5y3 - z4 , xz5e3y) and -1 ≤ x ≤ 3, 1 ≤ y ≤ 3 and -2 ≤ z ≤ 2.
Similar Reads
Engineering Mathematics Tutorials Engineering mathematics is a vital component of the engineering discipline, offering the analytical tools and techniques necessary for solving complex problems across various fields. Whether you're designing a bridge, optimizing a manufacturing process, or developing algorithms for computer systems,
3 min read
Linear Algebra
MatricesMatrices are key concepts in mathematics, widely used in solving equations and problems in fields like physics and computer science. A matrix is simply a grid of numbers, and a determinant is a value calculated from a square matrix.Example: \begin{bmatrix} 6 & 9 \\ 5 & -4 \\ \end{bmatrix}_{2
3 min read
Row Echelon FormRow Echelon Form (REF) of a matrix simplifies solving systems of linear equations, understanding linear transformations, and working with matrix equations. A matrix is in Row Echelon form if it has the following properties:Zero Rows at the Bottom: If there are any rows that are completely filled wit
4 min read
Eigenvalues and EigenvectorsEigenvalues and eigenvectors are fundamental concepts in linear algebra, used in various applications such as matrix diagonalization, stability analysis and data analysis (e.g., PCA). They are associated with a square matrix and provide insights into its properties.Eigen value and Eigen vectorTable
10 min read
System of Linear EquationsA system of linear equations is a set of two or more linear equations involving the same variables. Each equation represents a straight line or a plane and the solution to the system is the set of values for the variables that satisfy all equations simultaneously.Here is simple example of system of
5 min read
Matrix DiagonalizationMatrix diagonalization is the process of reducing a square matrix into its diagonal form using a similarity transformation. This process is useful because diagonal matrices are easier to work with, especially when raising them to integer powers.Not all matrices are diagonalizable. A matrix is diagon
8 min read
LU DecompositionLU decomposition or factorization of a matrix is the factorization of a given square matrix into two triangular matrices, one upper triangular matrix and one lower triangular matrix, such that the product of these two matrices gives the original matrix. It is a fundamental technique in linear algebr
6 min read
Finding Inverse of a Square Matrix using Cayley Hamilton Theorem in MATLABMatrix is the set of numbers arranged in rows & columns in order to form a Rectangular array. Here, those numbers are called the entries or elements of that matrix. A Rectangular array of (m*n) numbers in the form of 'm' horizontal lines (rows) & 'n' vertical lines (called columns), is calle
4 min read
Sequence & Series
Calculus
Limits, Continuity and DifferentiabilityLimits, Continuity, and Differentiation are fundamental concepts in calculus. They are essential for analyzing and understanding function behavior and are crucial for solving real-world problems in physics, engineering, and economics.Table of ContentLimitsKey Characteristics of LimitsExample of Limi
10 min read
Cauchy's Mean Value TheoremCauchy's Mean Value theorem provides a relation between the change of two functions over a fixed interval with their derivative. It is a special case of Lagrange Mean Value Theorem. Cauchy's Mean Value theorem is also called the Extended Mean Value Theorem or the Second Mean Value Theorem.According
7 min read
Taylor SeriesA Taylor series represents a function as an infinite sum of terms, calculated from the values of its derivatives at a single point.Taylor series is a powerful mathematical tool used to approximate complex functions with an infinite sum of terms derived from the function's derivatives at a single poi
8 min read
Inverse functions and composition of functionsInverse Functions - In mathematics a function, a, is said to be an inverse of another, b, if given the output of b a returns the input value given to b. Additionally, this must hold true for every element in the domain co-domain(range) of b. In other words, assuming x and y are constants, if b(x) =
3 min read
Definite Integral | Definition, Formula & How to CalculateA definite integral is an integral that calculates a fixed value for the area under a curve between two specified limits. The resulting value represents the sum of all infinitesimal quantities within these boundaries. i.e. if we integrate any function within a fixed interval it is called a Definite
8 min read
Application of Derivative - Maxima and MinimaDerivatives have many applications, like finding rate of change, approximation, maxima/minima and tangent. In this section, we focus on their use in finding maxima and minima.Note: If f(x) is a continuous function, then for every continuous function on a closed interval has a maximum and a minimum v
6 min read
Probability & Statistics
Mean, Variance and Standard DeviationMean, Variance and Standard Deviation are fundamental concepts in statistics and engineering mathematics, essential for analyzing and interpreting data. These measures provide insights into data's central tendency, dispersion, and spread, which are crucial for making informed decisions in various en
10 min read
Conditional ProbabilityConditional probability defines the probability of an event occurring based on a given condition or prior knowledge of another event. It is the likelihood of an event occurring, given that another event has already occurred. In probability, this is denoted as A given B, expressed as P(A | B), indica
12 min read
Bayes' TheoremBayes' Theorem is a mathematical formula used to determine the conditional probability of an event based on prior knowledge and new evidence. It adjusts probabilities when new information comes in and helps make better decisions in uncertain situations.Bayes' Theorem helps us update probabilities ba
13 min read
Probability Distribution - Function, Formula, TableA probability distribution is a mathematical function or rule that describes how the probabilities of different outcomes are assigned to the possible values of a random variable. It provides a way of modeling the likelihood of each outcome in a random experiment.While a Frequency Distribution shows
13 min read
Covariance and CorrelationCovariance and correlation are the two key concepts in Statistics that help us analyze the relationship between two variables. Covariance measures how two variables change together, indicating whether they move in the same or opposite directions. Relationship between Independent and dependent variab
6 min read
Practice Questions