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 ↓


Stokes' Theorem


Stokes' theorem is an important result in vector calculus that connects the surface integral of a vector field over a surface to the line integral of the field along the boundary of that surface. This theorem serves as a powerful tool for converting problems involving complicated surface integrals into simple line integrals and vice versa, leading to easier calculations and a deeper understanding of the behavior of fields and surfaces.

The theorem states: If S is an oriented, smooth surface with an oriented, closed boundary curve C, and F is a vector field whose components have continuous partial derivatives over an open region containing S, then:

∮_c f • dr = ∬_s curl f • ds

Here, ∮_C indicates the line integral of the vector field F along the curve C, curl F denotes the curl of F, and dS is the vector field on the surface S

Understanding the notation

To better understand Stokes' theorem, let's take a moment to understand the notation and terms involved:

  • Vector field (F): A function that assigns a vector to every point in space. For example, F(x, y, z) = (P, Q, R) where P, Q, and R are functions of x, y, and z.
  • Curl (curl F): A vector that describes the infinitesimal rotation of a 3D vector field. It is calculated curl F = ( ∂R/∂y - ∂Q/∂z, ∂P/∂z - ∂R/∂x, ∂Q/∂x - ∂P/∂y )
  • Surface integral (denoted as ∬): An integral where we integrate over a surface. For Stokes' theorem, the surface integral accounts for the component of the curl perpendicular to the surface at each point.
  • Line integral (noted as ∮): An integral that involves integration over a curve or line.
  • Surface (S): A two-dimensional manifold, which is essentially a 2D shape like a piece of paper or a sea wave. In mathematics, these manifolds can be infinitely large and can be described by equations.
  • Boundary curve (C): The closed curve that outlines the surface S

Now that we understand the main components and variables involved in the theorem, the main purpose of Stokes' theorem is to equate these two types of integrals - the surface integral over the surface S and the line integral around the boundary C

Analogies and visualizations

To understand Stokes' theorem intuitively, consider the following analogy: Imagine a field of wind flowing over a surface such as a piece of cloth, where each vector in the field represents both the direction and speed of the wind at that point. The curve C represents the boundary of the cloth. Stokes' theorem tells us that the "swirl" effect of the wind across the surface of the cloth (represented by curl F) is equal to the cumulative effect measured around the boundary.

Let's use a visual example to make this clear. Consider a rectangular piece of fabric whose edges form a border curve:

C(limit curve)

In the illustration, the red arrows indicate a vector field acting on the surface S The line integral along C will involve the sum of the effects of this vector field along the path of the boundary. However, computing the curl on S allows for the simplification described by Stokes' theorem: turning this 3D complexity into a 2D boundary construction.

Application of Stokes theorem: An example

Let us work through an example to further strengthen our understanding:

Suppose we have a vector field F = (xz, yz, z), and we are dealing with a surface S that is shaped as the part of the sphere x² + y² + z² = 1 in the first octant, that is, only where x, y, z are non-negative and where z = 0 is a circular disk of radius 1.

The limit curve C is the circle at the vertex when z = 1 We are to evaluate the line integral ∮_C F • dr using Stokes' theorem.

First, let's find curl F :

F = (xz, yz, z)
curl F = (∂/∂y (z) - ∂/∂z (yz), ∂/∂z (xz) - ∂/∂x (z), ∂/∂x (yz) - ∂/∂y (xz))
       = (0 – y, x – 0, y – x)
       = (-y, x, y – x)

The main component contributing to our surface integral is the component perpendicular to the surface (-y, x, yx).

For the surface z = 1, we evaluate the double integral of k component (yx) over the surface.

∬_S curl F • dS = ∬_S (y-x) dS

Let's parametrize x² + y² = 1 in this surface with polar coordinates: x = r cosθ, y = r sinθ, where r goes from 0 to 1 and θ from 0 to π/2.

∬_s(yx)ds 
= ∫_(0)^(π/2)∫_(0)^(1) (r sinθ - r cosθ) r dr dθ
= ∫_(0)^(π/2)∫_(0)^(1) (r² sinθ - r² cosθ) dr dθ

By computing the inner integral and then the outer integral, we find that the area integral is equivalent to the line integral around C, thereby verifying Stokes' theorem.

Key concepts and implications

  • Stokes theorem helps in converting surface integrals to line integrals and vice versa, which is important when calculating one type of integral is simpler.
  • The vector field curl provides important information about rotational behavior within the field, which is often applied in physics problems.
  • This theorem generalizes several fundamental theorems of vector calculus, including both Green's theorem and the divergence theorem.

From understanding weather patterns to the behavior of electromagnetic fields, Stokes' theorem provides valuable insights and connects the three-dimensional understanding of the nature of vector fields to two-dimensional intuitive concepts.


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