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Stoke's Theorem: Definition, Formula, Proof, Examples

Stokes’ Theorem is a fundamental result in vector calculus that relates a surface integral over a closed surface to a line integral around its boundary. It is named after the Irish mathematician Sir George Stokes, who formulated it in the 19th century. Stokes’ Theorem states that the circulation (or line integral) of a vector field around a closed curve is proportional to the flux (or surface integral) of the vector field’s curl over the surface encompassed by the curve.

In this article, we will learn in detail about Stoke’s Theorem, its formula, its expression in different coordinate system and its application.

Table of Content

What is Stoke’s Theorem?

Stoke’s Theorem is a fundamental statement in multivariable calculus that relates the surface integral of the curl of a vector field over a surface to the line integral of the vector field around the boundary of the surface. This is a powerful tool that bridges gap between line integrals and surface integrals. Stokes’ Theorem is a higher-dimensional version of the two-dimensional Green’s Theorem, and it is important in many fields of physics and engineering, including fluid dynamics, electromagnetism, and differential geometry. It is an effective tool for evaluating line integrals and investigating the behavior of vector fields in three dimensions.

Stoke’s Theorem Formula

The general formula for Stoke’s Theorem in three dimensions is:

[Tex]\int\int_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}[/Tex]

Where:

  • [Tex]\nabla \times \mathbf{F}[/Tex] represents the curl of the vector field F.
  • [Tex]d\mathbf{S}[/Tex] is the vector area element of the surface S.
  • [Tex]C[/Tex] is the closed curve that is the boundary of S.
  • [Tex]d\mathbf{r}[/Tex] is the line element along C

Stoke’s Theorem Proof

According to Stoke’s Theorem, the line integral of vector field around a closed curve is equal to the surface integral of the curl of the vector field over the surface enclosed by the curve. To prove this let us denote the following

  • C is the Curve whose parameter is r(t) for a ≤ t ≤ b
  • S is the surface enclosed by C
  • D is the region in the xy plane projected from surface S
  • F(x, y, z) = (P(x, y, z), Q(x, y, z), R(x, y, z)) is the vector field defined on region S

Line integral around the closed curve C can be expressed as:

[Tex] \oint_{C}F.dr = \int_{a}^{b}F(r(t)).r'(t)dt[/Tex]

The surface integral of curl of F over S is expressed as

[Tex]\int \int_S (\bigtriangledown \times F).ndA[/Tex]

where n is the unit normal vector to the surface S, and dA is the area element on the surface S.

Stoke’s Theorem in Different Coordinate Systems

Stoke’s Theorem can be expressed in following different coordinate system

  • Cartesian Coordinate
  • Cylindrical Coordinate
  • Spherical Coordinate

Stoke’s Theorem in Cartesian Coordinates

In Cartesian coordinates, the curl and the surface integral are expressed in terms of i, j, k unit vectors and the differential elements [Tex]dx, dy, dz[/Tex].

[Tex]\nabla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} – \frac{\partial Q}{\partial z}\right) \mathbf{i} + \left(\frac{\partial P}{\partial z} – \frac{\partial R}{\partial x}\right) \mathbf{j} + \left(\frac{\partial Q}{\partial x} – \frac{\partial P}{\partial y}\right) \mathbf{k}[/Tex]

where [Tex]\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}[/Tex] is the vector field.

Stoke’s Theorem in Cylindrical Coordinates

Stoke’s Theorem in cylindrical coordinates involves the unit vectors [Tex]\mathbf{e}_\rho,\mathbf{e}_\phi, \mathbf{e}_z[/Tex] and the differentials [Tex]dρ, dφ, dz[/Tex].

The curl and the surface element are:

[Tex]\nabla \times \mathbf{F} = \left( \frac{1}{r}\frac{\partial R}{\partial \theta} – \frac{\partial Q}{\partial z} \right) \mathbf{e}r + \left( \frac{\partial P}{\partial z} – \frac{\partial R}{\partial r} \right) \mathbf{e}\theta + \frac{1}{r}\left( \frac{\partial (rQ)}{\partial r} – \frac{\partial P}{\partial \theta} \right) \mathbf{e}_z [/Tex]

[Tex]d\mathbf{S} = r dz d\theta \mathbf{e} _r + dr dz \mathbf{e}_\theta + r dr d\theta \mathbf{e}_z[/Tex]

Stoke’s Theorem in Spherical Coordinates

In spherical coordinates, the theorem uses the unit vectors [Tex]\mathbf{e}_r,\mathbf{e}_\theta,\mathbf{e}_\phi[/Tex] and the differentials [Tex]dr, dθ, dφ[/Tex].

The curl and the surface element are:

[Tex]\nabla \times \mathbf{F} = \frac{1}{\rho^2 \sin\phi}\left[ \frac{\partial}{\partial \phi} (\sin\phi R) – \frac{\partial Q}{\partial \theta} \right] \mathbf{e}\rho + \frac{1}{\rho \sin\phi}\left[ \frac{\partial P}{\partial \theta} – \frac{\partial}{\partial \rho}(\rho R) \right] \mathbf{e}\phi + \frac{1}{\rho}\left[ \frac{\partial}{\partial \rho}(\rho Q) – \frac{\partial P}{\partial \phi} \right] \mathbf{e}\theta[/Tex]

[Tex]d\mathbf{S} = \rho^2 \sin\phi d\phi d\theta \mathbf{e}\rho + \rho \sin\phi d\rho d\theta \mathbf{e}\phi + \rho d\rho d\phi \mathbf{e}\theta [/Tex]

In each coordinate system, the theorem connects the circulation of the vector field [Tex]\mathbf{F} [/Tex] along a closed curve C (the boundary of the surface S) with the flux of the curl of ( \mathbf{F} ) through the surface S. The specific form of the curl and the surface element [Tex]d\mathbf{S}[/Tex] will depend on the chosen coordinate system.

Gauss Divergence Theorem

The Gauss Divergence Theorem, also known as Gauss’s theorem, relates flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.
In mathematically, it is expressed as ;

[Tex]\iint_{\partial V} \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F}) \, dV[/Tex]

Here:

  • [Tex]V[/Tex] represents the volume enclosed by surface [Tex]S[/Tex]
  • [Tex]\nabla \cdot F[/Tex] is the divergence of vector field [Tex]F[/Tex]
  • dV is the volume element.
  • dS is the surface element.
  • n is the outward unit normal vector to the surface.

Stoke’s Theorem vs Gauss’s Theorem

While both theorems relate surface integrals to volume integrals, Stoke’s Theorem applies to surfaces (2D manifolds with boundary), where as Gauss Theorem applies to volumes (3D manifolds).

Aspect

Stokes’ Theorem

Gauss Divergence Theorem

Equation

[Tex]\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{S} (\nabla \times \mathbf{F}) \cdot d\mathbf{S}[/Tex]

[Tex]\iint_{\partial V} \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F}) \, dV[/Tex]

Integral

Relate Line integral & Surface Integral

Relate Surface integral & Volume Integral

Boundary Integral

Closed Curve

Closed Surface

Mathematical Focus

Curl of a Vector Field

Divergence of a Vector Field

Physical Interpretation

Circulation along a Curve

Net Flux through a Surface

Dimensionality

2-Dimensional within 3-D space

3-Dimensional

Applications of Stoke’s Theorem

Stoke’s Theorem has numerous applications in physics and engineering, particularly in electromagnetism and fluid dynamics, where it is used to simplify complex integrals. Here are some of its applications:

  • Electromagnetic field: Stoke ’s Theorem can be used to derive Maxwell equations which are fundamentals to understand electromagnetic field. It also helps us to relating the electric field in loop to magnetic field passing through loop as seen in Faraday ’s law of Induction.
  • Fluid Mechanics : The theorem is applied to study rotation and curl in fluid flow. It can be used to analyze circulation and vorticity in fluids which are very useful in aerodynamics and weather systems.
  • Computer Graphics: In computer graphics, Stoke’s Theorem generally use for rendering techniques like vector field visualization which is important for simulating realistic hair and fur movement, fluid flows and other complex dynamic systems.
  • Engineering : Engineers use this for various calculations including the design of electrical machinery analysis of aerodynamic surfaces and for study of stress and strain in materials.
  • Mathematics : Beyond its application in physics the theorem is also a powerful tool in mathematics for converting complex surface integral to more manageable line integrals in multivariable calculus.

These applications show how this theorem bridges gap between theoretical mathematics and practical physical phenomena by providing a crucial link between abstract concepts and their physical interpretations.

Limitations of Stoke’s Theorem

Stoke’s Theorem is a powerful tool in vector calculus but it does have some limitations that are important to consider ;

  • Smoothness Requirement: The surface over which the theorem is applied must be smooth. If surface has sharp edges or corners or if it is not well-defined the theorem may not hold.
  • Orientation: The surface must have an orientation, meaning it must be possible to consistently define a normal vector at every point on the surface. For non-orientable surfaces like the Möbius strip, Stoke’s Theorem cannot be applied.
  • Boundary Definition: The boundary of the surface must be a simple, closed, piecewise smooth curve. Surfaces with boundaries that are not well-defined or integrable, such as fractal boundaries like the Koch snowflake, do not satisfy the conditions for Stoke’s Theorem.
  • Field Continuity: The vector field involved must have continuous partial derivatives over the surface and its boundary. If the field is not smooth or has discontinuities the theorem may not be applicable.
  • Applicability to Physical Problems: While Stoke ’s Theorem is used in various physical applications such as electromagnetism, it’s based on ideal conditions . In real-world scenarios, factors like turbulence, non-laminar flow or irregular particle shapes can limit the direct application of the theorem.

These limitations mean that while Stoke’s Theorem is a valuable theoretical tool, care must be taken when applying it to practical problems to ensure that the conditions for its use are met.

Also, Check

Solved Examples on Stoke’s Theorem

Example 1: Let’s consider a vector field F given by [Tex]\mathbf{F} = y\mathbf{i} – x\mathbf{j} + yx^3\mathbf{k}[/Tex] and let S be the portion of the sphere of radius 4 with [Tex]z \geq 0[/Tex] and the upwards orientation. Use Stoke’s Theorem to evaluate [Tex]\oint_C \mathbf{F} \cdot d\mathbf{r}[/Tex].

Solution:

Stoke’s Theorem relates the surface integral of the curl of a vector field over a surface S to the line integral of the vector field over the boundary curve C of S. The theorem states that:

[Tex]\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}[/Tex]

For the given vector field F and surface S, we first need to find the boundary curve C of S. In this case, C is the circle of radius 4 at ( z = 0 ).

First, we find curl of [Tex]( \mathbf{F} ): \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ y & -x & yx^3 \end{vmatrix}[/Tex]

This gives

[Tex]\nabla \times \mathbf{F} = \left( \frac{\partial}{\partial y}(yx^3) – \frac{\partial}{\partial z}(-x) \right)\mathbf{i} – \left( \frac{\partial}{\partial x}(yx^3) – \frac{\partial}{\partial z}(y) \right)\mathbf{j} + \left( \frac{\partial}{\partial x}(-x) – \frac{\partial}{\partial y}(y) \right)\mathbf{k}[/Tex]

On Simplifying

[Tex]\nabla \times \mathbf{F} = (x^3)\mathbf{i} – (3yx^2)\mathbf{j} – (2)\mathbf{k}[/Tex]. Now, we can set up the surface integral over hemisphere S the line integral over C : [Tex]\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}[/Tex]

Since S is the upper hemisphere [Tex]d\mathbf{S} [/Tex]will be[Tex] \mathbf{k} \cdot dS [/Tex]where ( dS ) is the area element of the hemisphere.The integral simplifies ;[Tex]\iint_S (x^3)\mathbf{i} \cdot \mathbf{k} \, dS – \iint_S (3yx^2)\mathbf{j} \cdot \mathbf{k} \, dS – \iint_S (2)\mathbf{k} \cdot \mathbf{k} \, dS[/Tex]

Since the dot product of perpendicular vectors is zero, the first two integrals vanish, and we are left with : [Tex]-2 \iint_S dS[/Tex] This integral represents the negative twice the area of the hemisphere of radius 4. The area of a sphere is[Tex]4\pi r^2 [/Tex] ,so the area of the hemisphere is [Tex]2\pi r^2 [/Tex]. Plugging in ( r = 4 ), we get

[Tex]-2 \times 2\pi (4)^2 = -128\pi[/Tex].

Therefore, the line integral :

[Tex] \oint_C \mathbf{F} \cdot d\mathbf{r}=-128\pi[/Tex]

Please note that this is a simplified explanation that the orientation of ( S ) is such that the normal vector points outward. For a more detailed solution, one would need to consider the parametrization of ( C ) and ( S ), and ensure that the orientations are consistent with the right-hand rule.

Example 2: Consider a vector field

[Tex]\mathbf{F} = (3yx^2 + z^3)\mathbf{i} + y^2\mathbf{j} + 4yx^2\mathbf{k}[/Tex]

and let C be the triangle with vertices at (0,0,3), (0,2,0), and (4,0,0). If C has a counterclockwise rotation when viewed from above, use Stoke’s Theorem to evaluate [Tex]\oint_C \mathbf{F} \cdot d\mathbf{r}[/Tex].

Solution:

Stoke’s Theorem allows us to convert the line integral over curve C to a surface integral over the surface S that C bounds. The theorem states: [Tex]\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}[/Tex]

First, we find the curl of F:

[Tex]\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ 3yx^2 + z^3 & y^2 & 4yx^2 \end{vmatrix} = (0 – 0)\mathbf{i} – (12yx – 0)\mathbf{j} + (2y – 3x^2)\mathbf{k} = -12yx\mathbf{j} + (2y – 3x^2)\mathbf{k}[/Tex]

Next, we need to parametrize the surface S. Since S is a triangular portion of the plane, we can use the vertices to define the plane equation and parametrize S accordingly.

After parametrization, we can evaluate the surface integral:

[Tex]\iint_S (-12yx\mathbf{j} + (2y – 3x^2)\mathbf{k}) \cdot d\mathbf{S}[/Tex].

The dot product and integration will yield the final result, which is the value of the line integral over C.

Practice Questions on Stoke’s Theorem

Q1: Consider the vector field

[Tex]\mathbf{F} = (z\sin(x), yz, x^2 + y^2)\mathbf{i}[/Tex] and let S be the upper hemisphere of the sphere [Tex]x^2 + y^2 + z^2 = a^2[/Tex] with radius a and centered at the origin. Then verify Stoke theorem.

Q2: Paraboloid Surface Let’s take a vector field

[Tex]\mathbf{F} = (xy, e^z, z\cos(y))[/Tex] and consider S to be the surface of the paraboloid

[Tex]z = 1 – x^2 – y^2[/Tex] capped by the plane z = 0. Then evaluate [Tex]\oint_C \mathbf{F} \cdot d\mathbf{r}[/Tex].

Frequently Asked Questions on Stoke’s Theorem

What is the physical interpretation of Stoke ’s Theorem ?

Stoke’s Theorem can be interpreted as a way to relate the rotation of a fluid within a surface to the flow along the boundary of the surface.

How does Stoke’s Theorem simplify calculations in multivariable calculus ?

By converting a complex surface integral into a simpler line integral, calculations become more manageable and simple.

Can Stoke’s Theorem be applied to any surface ?

Stoke ’s Theorem can be applied to any surface that is smooth and has a well-defined boundary.

What is the difference between the curl of a vector field and its divergence?

Curl measures rotations at a point while the divergence measures how much a vector field spreads out or converges at a point.

Yes, Stoke’s Theorem is higher dimensional analog to Fundamental Theorem of Calculus relating derivatives to integrals .


 


Reffered: https://www.geeksforgeeks.org

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