Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Calculus → Multivariable Calculus ↓


Divergence and Curl


Introduction

In multivariable calculus, we often deal with vector fields, which are functions that assign a vector to every point in space. Two important operations that help us understand vector fields are divergence and curl. These operations help investigate various properties of fields such as rate of change, direction, rotation, and more. Extensive knowledge about divergence and curl is required in fields such as physics, engineering, and computer graphics.

What is divergence?

Divergence measures how much a vector field spreads out or comes together at a point. If you can imagine fluid flowing through the field, the divergence at a point represents the net flow of fluid outward from the point.

Mathematically, for a vector field F = (F1, F2, F3) in three dimensions, the divergence is a scalar given by:

div F = ∇ • F = ∂F1/∂x + ∂F2/∂y + ∂F3/∂z

In this equation, ∇ (nabla) denotes the vector differential operator. Here, ∂ denotes a partial derivative, which measures the rate at which a function changes when one of its variables changes, while the others remain constant.

Visual example: Divergence in two dimensions

Let's consider a simple visual example of a vector field in two dimensions, where the vectors are radiating outward from the origin:

In this example, the divergence is positive, because the vector field expands as you move away from the origin.

Text examples of divergence

Example 1

Consider the vector field F = (x, y, z).

div F = ∂/∂x (x) + ∂/∂y (y) + ∂/∂z (z) = 1 + 1 + 1 = 3

Here the divergence is constant and positive, i.e. the field continues to expand.

Example 2

Take the vector field G = (x, -y, z).

div G = ∂/∂x (x) + ∂/∂y (-y) + ∂/∂z (z) = 1 – 1 + 1 = 1

The divergence in this area is lower than in the previous example, indicating less dispersion.

Physical interpretation of the divergence

Divergence has important real-world implications, especially in fluid dynamics and electromagnetism:

  • In fluid dynamics, divergence measures the rate at which the volumetric density of a fluid expands or compresses at a point. Positive divergence indicates a source of flow or expansion, while negative values suggest sink or compression.
  • In electromagnetism, Gauss's law, one of Maxwell's equations, uses divergence to relate the distribution of electric charge to the resultant electric field.

What is curl?

Curl, on the other hand, measures the tendency of a vector field to rotate around a point. If you think of a vector field as representing the flow of wind or water, curl represents the amount of rotation or whirling around a given point.

For a three-dimensional vector field F = (F1, F2, F3), the curl is a vector given by:

curl F = ∇ × F = (∂F3/∂y - ∂F2/∂z, ∂F1/∂z - ∂F3/∂x, ∂F2/∂x - ∂F1/∂y)

The curl is a vector whose components are determined by changes in the components of F. In essence, it provides a measure of the rotation of the field for each point.

Visual example: Curl in two dimensions (conceptual)

Imagine a vector field that forms a circle, where the vectors point on a circular path around the origin:

Here, the curl is not zero because the vectors are forming a circular pattern, which is indicative of a rotational or curved field.

Curl text examples

Example 1

Consider the vector field H = (-y, x, 0).

curl H = ∇ × H = (0, 0, (∂/∂x (x) - ∂/∂y (-y))) = (0, 0, 2)

Since the non-zero portion of the curl vector is in z direction, this suggests that the field rotates primarily around z axis.

Example 2

Take the vector field K = (yz, zx, xy).

curl K = ( ∂/∂y(xy) - ∂/∂z(zx), ∂/∂z(yz) - ∂/∂x(xy), ∂/∂x(zx) - ∂/∂y(yz) )
       = (0, z – x, z – y)

This example demonstrates the presence of rotation in the vector field about different axes.

Physical interpretation of curl

Curls can be observed in a variety of physical scenarios:

  • In fluid dynamics, curl indicates the rotational part of the velocity field of a fluid. If you think of water flowing down a drain, the curl would represent the rotating motion.
  • In electromagnetism, the curl is used in Faraday's law of induction to describe how a changing magnetic field induces an electric field.

Final thoughts

Divergence and curl are fundamental concepts used in the analysis of vector fields in calculus. They provide a precise mathematical framework for studying many physical phenomena, from fluid flow to electromagnetic fields. Understanding these concepts helps not only in theoretical mathematics but also in practical applications in various scientific and engineering domains.

Understanding these abstract concepts can be challenging at first. However, by using visual examples and breaking down each component, you can get a better intuitive sense of what divergence and curl represent. As you move into more advanced applications, these basic building blocks will serve you well in further understanding and effectively using multivariable calculus.


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