Stokes' Theorem Explained: A Comprehensive Guide for Students

Stokes' Theorem is a fundamental result in vector calculus that relates the surface integral of the curl of a vector field to the line integral of the vector field along the boundary of the surface. It simplifies complex calculations by converting surface integrals into easier-to-compute line integrals and is widely used in physics and engineering, particularly in electromagnetism and fluid dynamics.

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1. Introduction to the Stokes' Theorem:

Stokes' Theorem is a central theorem in vector calculus, generalizing several theorems from lower-dimensional calculus into three dimensions. Named after Sir George Stokes, it connects surface integrals and line integrals by relating the integral of the curl of a vector field over a surface to the integral of the vector field along the boundary of that surface. This theorem is crucial in many physical applications, including fluid dynamics, electromagnetism, and mathematical physics. Whether you are solving problems involving vector fields or studying circulation and flux, Stokes' Theorem provides a powerful tool for simplifying calculations.

2. What is Stokes' Theorem:

Stokes' Theorem provides a bridge between the integral of a vector field over a surface and the integral of the same field along the curve that forms the boundary of the surface. In simpler terms, it converts a complex surface integral into a more manageable line integral and vice versa.

Stokes' Theorem is essentially an extension of Green’s Theorem to higher dimensions and is a generalization of the fundamental theorem of calculus. It connects the behavior of a vector field within a region to its behavior on the boundary of that region, making it a key concept in fields such as physics and engineering.

3. Stokes' Theorem Formula:

The formula for Stokes' Theorem relates a surface integral over a vector field to a line integral around the boundary of that surface. The mathematical expression is:

$\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$

Where:

• $C$ is the positively oriented boundary curve of the surface $S$,

• $\mathbf{F}$ is the vector field,

• $d\mathbf{r}$ is the differential vector along the boundary $C$,

• $\nabla \times \mathbf{F}$ is the curl of the vector field $\mathbf{F}$,

• $d\mathbf{S}$ is the surface element, a vector normal to the surface $S$.

Stokes' Theorem Statement

Stokes' Theorem states that the line integral of a vector field $\mathbf{F}$ around the boundary $C$ of a surface $S$ is equal to the surface integral of the curl of $\mathbf{F}$ over $S$. In simpler terms, it connects the circulation of a vector field along a curve to the rotation (curl) of the field over the surface bounded by that curve.

This theorem generalizes concepts from Green's Theorem to three dimensions and provides a valuable tool for simplifying complex integrals in physics and engineering.

4. Stokes' Theorem Solved Examples:

Question: 1.

Applying Stokes' Theorem to a Simple Vector Field

Use Stokes' Theorem to evaluate the line integral $\oint_C (yz dx zx dy xy dz)$, where $C$ is the boundary of the disk $x^2 y^2 \leq 1$, $z = 0$.

Solution:

1. Given vector field:
   $\mathbf{F} = (yz, zx, xy)$.

2. Apply Stokes' Theorem:
   Convert the line integral into a surface integral using Stokes' Theorem:
   $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$

3. Find the curl of $\mathbf{F}$:
   $\nabla \times \mathbf{F} = \left( \dfrac{\partial}{\partial y} (xy) - \dfrac{\partial}{\partial z} (zx), \dfrac{\partial}{\partial z} (yz) - \dfrac{\partial}{\partial x} (xy), \dfrac{\partial}{\partial x} (zx) - \dfrac{\partial}{\partial y} (yz) \right)$

   Simplifying:
   $\nabla \times \mathbf{F} = (x, y, z)$

4. Parametrize the surface $S$:
   Since $z = 0$, the curl reduces to $(x, y, 0)$. The surface is the unit disk $x^2 y^2 \leq 1$, so in polar coordinates:
   $x = r \cos \theta, \space y = r \sin \theta$

5. Compute the surface integral:
   The surface element is $d\mathbf{S} = (0, 0, r dr d\theta)$, and the curl is $(x, y, 0)$, so the dot product $(\nabla \times \mathbf{F}) \cdot d\mathbf{S} = 0$. Thus, the surface integral evaluates to $0$.

Final Answer: The value of the line integral is $0$.

Question: 2.

Using Stokes' Theorem to Evaluate a Line Integral

Evaluate the line integral $\oint_C (y \, dx - x \, dy)$, where $C$ is the boundary of the circle $x^2 y^2 = 1$ in the plane $z = 0$, oriented counterclockwise.

Solution:

1. Given vector field:
   $\mathbf{F} = (y, -x, 0)$.

2. Apply Stokes' Theorem:
   Convert the line integral into a surface integral using Stokes' Theorem:
   $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$

3. Find the curl of $\mathbf{F}$:
   $\nabla \times \mathbf{F} = \left( \dfrac{\partial}{\partial y} (0) - \dfrac{\partial}{\partial z} (-x), \dfrac{\partial}{\partial z} (y) - \dfrac{\partial}{\partial x} (0), \dfrac{\partial}{\partial x} (-x) - \dfrac{\partial}{\partial y} (y) \right)$

   Simplifying:
   $\nabla \times \mathbf{F} = (0, 0, -2)$

4. Parametrize the surface:
   The surface is the unit disk in the $xy$-plane, so:
   $d\mathbf{S} = (0, 0, r \, dr \, d\theta)$

5. Compute the surface integral:
   The dot product $(\nabla \times \mathbf{F}) \cdot d\mathbf{S} = -2 \cdot (0, 0, r)$. Therefore, the surface integral becomes:
   $\iint_S -2 \, dA = -2 \times \text{Area of the circle}$

   The area of the unit circle is $\pi$, so:
   $\iint_S -2 \, dA = -2\pi$

Final Answer: The value of the line integral is $-2\pi$.

Question: 3.

Using Stokes' Theorem to Evaluate a Line Integral

Evaluate the line integral $\oint_C (y^2 \, dx x^2 \, dy)$, where $C$ is the boundary of the circle $x^2 y^2 = 4$, using Stokes' Theorem.

Solution:

1. Given vector field:
   $\mathbf{F} = (y^2, x^2, 0)$.

2. Apply Stokes' Theorem:
   $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$

   Here, $C$ is the boundary of the disk with radius $2$.

3. Find the curl of $\mathbf{F}$:
   $\nabla \times \mathbf{F} = \left( \dfrac{\partial (x^2)}{\partial z} - \dfrac{\partial 0}{\partial y}, \dfrac{\partial 0}{\partial x} - \dfrac{\partial (y^2)}{\partial z}, \dfrac{\partial (x^2)}{\partial y} - \dfrac{\partial (y^2)}{\partial x} \right)$

   $= (0, 0, -2x 2y)$

4. Set up the surface integral:
   In polar coordinates, the surface element is $d\mathbf{S} = r \, dr \, d\theta$, and the curl simplifies to $\nabla \times \mathbf{F} = (0, 0, -2)$.

5. Evaluate the surface integral:
   $\iint_S (-2) \, dA = -2 \times \text{Area of the circle}$

   The area of the circle with radius $2$ is $4\pi$, so:
   $\iint_S (-2) \, dA = -8\pi$

Final Answer: The value of the line integral is $-8\pi$.

Question: 4.

Circulation of a Vector Field

Use Stokes' Theorem to evaluate the line integral $\oint_C (z \, dx x \, dy y \, dz)$, where $C$ is the boundary of the unit disk in the plane $z = 1$.

Solution:

1. Given vector field:
   $\mathbf{F} = (z, x, y)$.

2. Apply Stokes' Theorem:
   $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$

3. Find the curl of $\mathbf{F}$:
   $\nabla \times \mathbf{F} = \left( \dfrac{\partial y}{\partial y} - \dfrac{\partial x}{\partial z}, \dfrac{\partial z}{\partial x} - \dfrac{\partial z}{\partial y}, \dfrac{\partial x}{\partial y} - \dfrac{\partial y}{\partial x} \right)$

   $= (1, 0, 0)$

4. Set up the surface integral:
   The surface element is $d\mathbf{S} = (0, 0, r \, dr \, d\theta)$ since the surface is in the $z = 1$ plane.

5. Evaluate the surface integral:
   The curl is $(1, 0, 0)$ and the surface element is perpendicular to the $z = 1$ plane. Thus, the dot product is zero: $\iint_S (1, 0, 0) \cdot (0, 0, r) \, dr \, d\theta = 0$

Final Answer: The value of the line integral is $0$.

Question: 5.

Stokes' Theorem with Parametrized Surface

Evaluate the line integral $\oint_C (yz \, dx zx \, dy xy \, dz)$, where $C$ is the boundary of the square with vertices $(0, 0, 0), (1, 0, 0), (1, 1, 0), (0, 1, 0)$.

Solution:

1. Given vector field:
   $\mathbf{F} = (yz, zx, xy)$.

2. Apply Stokes' Theorem:
   $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$

3. Find the curl of $\mathbf{F}$:
   $\nabla \times \mathbf{F} = \left( \dfrac{\partial (xy)}{\partial y} - \dfrac{\partial (zx)}{\partial z}, \dfrac{\partial (yz)}{\partial z} - \dfrac{\partial (xy)}{\partial x}, \dfrac{\partial (zx)}{\partial x} - \dfrac{\partial (yz)}{\partial y} \right)$

   $= (x, y, z)$

4. Set up the surface integral:
   The surface is a flat square in the $z = 0$ plane, so the surface element is $d\mathbf{S} = (0, 0, 1) \, dx \, dy$, and the curl reduces to $(x, y, 0)$.

5. Evaluate the surface integral:
   $\iint_S (x, y, 0) \cdot (0, 0, 1) \, dx \, dy = 0$

Final Answer: The value of the line integral is $0$.

5. Practice Questions on Stokes' Theorem:

Q:1. Use Stokes' Theorem to evaluate the line integral $\oint_C (z \, dx x \, dy y \, dz)$, where $C$ is the boundary of the unit disk $x^2 y^2 \leq 1$, in the plane $z = 1$.

Q:2. Evaluate the line integral $\oint_C (x^2 \, dx xy \, dy)$ using Stokes' Theorem, where $C$ is the circle $x^2 y^2 = 4$.

Q:3. Find the flux of the curl of the vector field $\mathbf{F}(x, y, z) = (2y, -x, z)$ through the surface of the hemisphere $x^2 y^2 z^2 = 1$, $z \geq 0$.

Q:4. Apply Stokes' Theorem to find the circulation of the vector field $\mathbf{F}(x, y, z) = (z, y, x)$ around the curve $C$, which is the boundary of the square with vertices $(0, 0, 0)$, $(1, 0, 0)$, $(1, 1, 0)$, and $(0, 1, 0)$.

Q:5. Using Stokes' Theorem, evaluate the line integral $\oint_C (y^2 \, dx - x^2 \, dy)$, where $C$ is the ellipse $\dfrac{x^2}{9} \dfrac{y^2}{4} = 1$.

6. FAQs on Stokes' Theorem:

What does Stokes' Theorem state?

Stokes' Theorem states that the line integral of a vector field along a closed curve equals the surface integral of the curl of the vector field over the surface bounded by that curve.

What is the difference between Green’s Theorem and Stokes' Theorem?

Green’s Theorem applies to two-dimensional regions and relates a line integral to a double integral over a plane region. Stokes' Theorem generalizes this to three dimensions and relates line integrals to surface integrals.

How is Stokes' Theorem used in physics?

Stokes' Theorem is widely used in electromagnetism and fluid dynamics, particularly in calculating the circulation and flux of vector fields such as electric and magnetic fields.

What are the conditions for applying Stokes' Theorem?

The surface must be smooth, oriented, and bounded by a simple, closed curve. The vector field should also have continuous partial derivatives in the region.

Can Stokes' Theorem be used in any dimension?

No, Stokes' Theorem applies in three dimensions. However, it is part of a broader family of theorems, including Green’s Theorem in two dimensions and the Divergence Theorem in higher dimensions.

What does the curl of a vector field represent in Stokes' Theorem?

The curl of a vector field represents the rotation or circulation of the field. In Stokes' Theorem, the curl helps measure how much the field is "circulating" over the surface.

How does Stokes' Theorem simplify calculations?

By converting complex surface integrals into simpler line integrals (or vice versa), Stokes' Theorem often makes it easier to compute circulation and flux in vector fields.

What is the significance of the orientation of the surface in Stokes' Theorem?

The surface's orientation determines the surface's normal direction, which affects the direction in which the line integral is calculated. Consistent orientation is crucial for correct results.

7. Real-life Application of Stokes' Theorem:

Stokes' Theorem has numerous applications in physics and engineering, particularly in fields where circulation and flux are important:

• Electromagnetism: In electromagnetism, Stokes' Theorem helps calculate the electric and magnetic flux and understand the behavior of electric and magnetic fields around loops and surfaces. Maxwell’s equations, which describe the behavior of electric and magnetic fields, rely on Stokes' Theorem.

• Fluid Dynamics: Stokes' Theorem is used in fluid mechanics to analyze fluid flow circulation around objects, helping design aerodynamic systems and understanding vorticity in fluid flow.

• Aerodynamics: In aerodynamics, Stokes' Theorem helps engineers calculate how air flows over wings and other surfaces, optimizing aircraft design to reduce drag and improve efficiency.

8. Conclusion:

Stokes' Theorem is a powerful and versatile vector calculus theorem that connects surface and line integrals. It plays a significant role in fluid dynamics, electromagnetism, and engineering, making it an essential concept for understanding circulation and flux in three-dimensional spaces. Mastering Stokes' Theorem simplifies complex calculations and provides a deeper understanding of the behavior of vector fields and their boundaries.

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