Verify That Stokes Theorem Is True For The Vector Field F(x, Y, Z) = Hx, Y, Zi, Where S Is The Part Of the surface in space, is a fundamental problem in vector calculus that illustrates the profound connection between surface integrals and line integrals. Stokes' Theorem provides a powerful tool for converting a surface integral of the curl of a vector field into a line integral around the boundary of the surface, simplifying complex calculations and enhancing our understanding of vector fields in three-dimensional space. In this article, we will explore the process of verifying the theorem for the specified vector field \( \mathbf{F}(x, y, z) = \langle x, y, z \rangle \), considering a given surface \( S \), and demonstrate step-by-step how the theorem holds true through detailed calculations and geometric interpretations.
Understanding Stokes' Theorem
Statement of Stokes' Theorem
Stokes' Theorem states that for a smooth surface \( S \) with boundary curve \( C \), and a vector field \( \mathbf{F} \), the following equality holds:\[
\iintS (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \ointC \mathbf{F} \cdot d\mathbf{r}
\]
Where:
- \( \nabla \times \mathbf{F} \) is the curl of \( \mathbf{F} \),
- \( d\mathbf{S} \) is the vector surface element,
- \( C \) is the boundary curve of \( S \),
- \( d\mathbf{r} \) is the differential element along the curve \( C \).
This theorem links the circulation of the vector field around the boundary \( C \) to the flux of its curl across the surface \( S \).
Defining the Vector Field \( \mathbf{F}(x, y, z) \)
Given the vector field:
\[
\mathbf{F}(x, y, z) = \langle x, y, z \rangle
\]
This is a simple, radial vector field pointing outward from the origin, with components directly proportional to the position coordinates. Its curl, divergence, and other properties are straightforward to compute, making it an ideal candidate for verifying Stokes' Theorem.
Calculating the Curl of \( \mathbf{F} \)
The curl of \( \mathbf{F} \), \( \nabla \times \mathbf{F} \), in component form is:
\[
\nabla \times \mathbf{F} = \left|
\begin{array}{ccc}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
x & y & z
\end{array}
\right|
\]
Computing the determinant:
\[
\nabla \times \mathbf{F} = \left( \frac{\partial z}{\partial y} - \frac{\partial y}{\partial z} \right) \mathbf{i} - \left( \frac{\partial z}{\partial x} - \frac{\partial x}{\partial z} \right) \mathbf{j} + \left( \frac{\partial y}{\partial x} - \frac{\partial x}{\partial y} \right) \mathbf{k}
\]
Since all derivatives of variables with respect to different variables are zero, we get:
\[
\nabla \times \mathbf{F} = (0 - 0) \mathbf{i} - (0 - 0) \mathbf{j} + (0 - 0) \mathbf{k} = \mathbf{0}
\]
Thus, the curl of \( \mathbf{F} \) is zero everywhere:
\[
\nabla \times \mathbf{F} = \mathbf{0}
\]
This simplifies the verification process significantly because the surface integral of the curl over \( S \) must be zero.
Choosing the Surface \( S \) and Its Boundary \( C \)
To proceed with the verification, we need to specify the surface \( S \) and its boundary \( C \). For simplicity, assume \( S \) is a standard surface, such as a disk or a portion of a plane, with a well-defined boundary.
Example: \( S \) as a Disk in the XY-Plane
Suppose \( S \) is the disk of radius \( R \) lying in the \( xy \)-plane, centered at the origin:\[
S = \{ (x, y, 0) \mid x^2 + y^2 \leq R^2 \}
\]
The boundary \( C \) is the circle:
\[
C = \{ (x, y, 0) \mid x^2 + y^2 = R^2 \}
\]
The orientation of \( C \) is counterclockwise when viewed from above (positive \( z \)-direction).
Verifying the Surface Integral of \( \nabla \times \mathbf{F} \) over \( S \)
Since \( \nabla \times \mathbf{F} = \mathbf{0} \), the surface integral simplifies:
\[
\iintS (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \iintS \mathbf{0} \cdot d\mathbf{S} = 0
\]
This confirms that the flux of the curl over the surface is zero, regardless of the surface's shape.
Calculating the Line Integral of \( \mathbf{F} \) over Boundary \( C \)
According to Stokes' Theorem, this line integral should also be zero.
Parameterization of \( C \)
The boundary circle \( C \) can be parameterized as:\[
\mathbf{r}(t) = \langle R \cos t, R \sin t, 0 \rangle, \quad t \in [0, 2\pi]
\]
The differential \( d\mathbf{r} \) is:
\[
d\mathbf{r} = \langle -R \sin t, R \cos t, 0 \rangle dt
\]
The vector field along \( C \):
\[
\mathbf{F}(\mathbf{r}(t)) = \langle R \cos t, R \sin t, 0 \rangle
\]
Compute the line integral:
\[
\ointC \mathbf{F} \cdot d\mathbf{r} = \int0^{2\pi} \mathbf{F}(\mathbf{r}(t)) \cdot \frac{d\mathbf{r}}{dt} dt
\]
Calculating the dot product:
\[
\mathbf{F}(\mathbf{r}(t)) \cdot \frac{d\mathbf{r}}{dt} = \langle R \cos t, R \sin t, 0 \rangle \cdot \langle -R \sin t, R \cos t, 0 \rangle
\]
\[
= R \cos t \times (-R \sin t) + R \sin t \times R \cos t + 0
\]
\[
= - R^2 \cos t \sin t + R^2 \sin t \cos t = 0
\]
Integrating over \( t \):
\[
\int_0^{2\pi} 0 \, dt = 0
\]
This confirms that the line integral around \( C \) is zero.
Conclusion: Verifying Stokes' Theorem
Since both the surface integral of the curl and the line integral of the vector field over the boundary are zero, we have verified that Stokes' Theorem holds for the specified vector field \( \mathbf{F}(x, y, z) = \langle x, y, z \rangle \) over the chosen surface \( S \).
Summary of key points:
- The curl of \( \mathbf{F} \) is zero everywhere.
- The surface integral of the curl over any surface \( S \) is therefore zero.
- The line integral of \( \mathbf{F} \) around the boundary \( C \) of \( S \) is also zero.
- The equality prescribed by Stokes' Theorem is satisfied, confirming its validity for this case.
Further Considerations and Generalizations
While this example demonstrates a straightforward verification with a simple vector field and surface, more complex scenarios involve non-zero curl fields and surfaces with more intricate boundary curves. In such cases, the process involves:
- Computing the curl of \( \mathbf{F} \),
- Choosing an appropriate surface \( S \),
- Parameterizing the boundary \( C \),
- Calculating both the surface and line integrals explicitly,
- Confirming their equality as per Stokes' Theorem.
This rigorous approach is essential in fields such as electromagnetism, fluid dynamics, and differential geometry, where understanding the interplay between circulation and flux is crucial.
