Stokes' Theorem and Green's Theorem

Name: Stokes' Theorem and Green's Theorem
Uploaded: 2022-05-20T10:00:09.000Z
Duration: 46 min 54 s

Stokes's Theorem and Green's Theorem Explained

Introduction to Stokes's and Green's Theorems

Stokes's theorem and Green's theorem are vector calculus formulas that relate surface integrals of the curl of a vector field over a surface to contour integrals around an enclosing contour.

These theorems, similar to Gauss’s divergence theorem, can express physical quantities in partial differential equations.

Understanding the Surface and Notation

The discussion focuses on an open surface in three dimensions, denoted as S , with a boundary referred to as partial S .

In manifold theory, the boundary of a surface is one dimension lower than the surface itself; hence, for a 2D surface, its boundary is a 1D curve.

Tangent and Normal Vectors

At each point along the boundary curve partial S , there exists a tangent vector denoted as dmathbfs , which has x and y components.

On every patch of the surface S , there is also a normal vector with area given by dA , pointing in the normal direction.

Stokes's Theorem Formulation

Stokes’s theorem states that the integral of the curl of a vector field over a surface equals the integral of that vector field along its boundary.

A general ambient vector field mathbfF = (f_1, f_2, f_3) may be considered when discussing these integrals.

Application and Interpretation

Examples include atmospheric flow dynamics or fluid motion on surfaces like soap bubbles where understanding how vectors behave on surfaces is crucial.

To apply Stokes’s theorem, compute the curl at every patch on the surface and integrate those contributions while considering their direction relative to dA .

Equating Surface Integrals to Contour Integrals

The integral of curl over the surface relates directly to path or contour integrals along its perimeter:

[

int_S (nabla times mathbfF) cdot dmathbfA = int_partial S mathbfF cdot dmathbfs

]

This relationship allows for practical computation by measuring how much of F 's tangential component circulates around this closed orbit.

Conclusion on Stokes's Theorem Significance

Tracking how much flow aligns with tangential directions provides insights into circulation or flow characteristics around boundaries.

Understanding Green's Theorem

Introduction to Green's Theorem

Green's theorem is presented as a simplified version of Stokes' theorem, specifically applied to flat surfaces.

The discussion emphasizes the importance of orienting the curve counterclockwise, adhering to the right-hand rule for proper application.

Mathematical Formulation

Green's theorem can be expressed simply in terms of a 2D vector field with two components, f_1 and f_2 .

The curl of a 2D vector field is defined as partial f_2 / partial x - partial f_1 / partial y , which points out of the plane (z-direction).

Integral Representation

The surface integral over the curl simplifies to integrating this expression across the area patch defined by dx times dy .

Green’s theorem equates this surface integral to a line integral around the perimeter, represented as f_1 dx + f_2 dy .

Conceptual Understanding

This relationship allows one to compute circulation along the perimeter by summing contributions from each component of the vector field.

Similarities are drawn between Green’s theorem and Gauss’s divergence theorem, transitioning from larger integrals over surfaces to smaller ones over perimeters.

Physical Intuition and Visualization

A visual representation using boxes illustrates how local curls cancel each other out within an area, leaving only contributions along the perimeter.

Understanding Stokes' Theorem and Its Applications

Smooth Vector Fields and Curl

A smooth vector field, such as fluid flows (excluding shock waves), exhibits continuous behavior. This implies that components on different edges of a small box cancel each other out, leaving only the tangential components aligned with the perimeter.

Green's Theorem Explained

The surface area of the curl is equivalent to a path integral of the vector field dotted into its tangential direction. This provides an intuitive understanding of why Green's theorem holds true.

Generalization to Stokes' Theorem

Stokes' theorem generalizes Green's theorem by extending it from two-dimensional surfaces to three-dimensional surfaces, maintaining similar cancellation properties for inner walls when integrating over a surface.

Application in Fluid Dynamics

In practical scenarios like analyzing hurricanes, summing up the curl across a region can yield insights into circulation along a perimeter, allowing for calculations related to vorticity and flow dynamics.

Importance in Aerodynamics

Stokes' theorem is crucial in aerodynamics for calculating lift over wings by determining circulation around them. It connects angular momentum conservation with fluid flow analysis.

Using Stokes' Theorem for Area Calculation

Historical Context of Land Measurement

Before modern technology like GPS, surveyors used methods involving walking around property boundaries to compute areas of irregular land shapes using principles from Stokes’ theorem.

Area Calculation Formula

The area A of an irregular shape S can be calculated using:

[

A = 1/2 int_textperimeter (x dy - y dx)

]

This formula allows one to count directional steps while traversing the boundary.

Connection Between Curl and Area

Understanding Area Calculation via Vector Fields

Integral Around the Perimeter

The integral around the perimeter of a vector field f can be expressed as the dot product of f with the tangent direction of the perimeter. This tangent direction is represented as (-y, x) .

By integrating this quantity around the perimeter, one can derive that it equates to twice the area of the enclosed shape.

Practical Application for Irregular Shapes

To measure the area of irregular shapes, one can walk along their perimeter and compute this integral quantity, effectively determining their area.

An example includes calculating areas for complex shapes like hyper hypocycloids through parametrization and perimeter integration.

Theorems in Context

Stokes's theorem and Green's theorem are highlighted as tools for quantifying conservation principles such as angular momentum or vorticity in rotational flows.

These concepts parallel Gauss's theorem, which addresses mass conservation using divergence.

Higher-Dimensional Generalizations

There exists a mathematical generalization of Stokes's and Gauss's theorems applicable to n-dimensional manifolds, although this topic extends beyond basic vector calculus.

Understanding these higher-dimensional concepts offers intriguing insights into advanced calculus applications on n-dimensional surfaces.

Future Topics