Stokes’ Theorem Verifier
The Stokes’ Theorem Verifier verifies Stokes’ theorem for a vector field \( \mathbf{F} \) over a surface \( S \), computing both the surface integral of the curl and the line integral over the boundary, with steps displayed using MathJax.
Stokes’ Theorem Verifier
The Stokes’ Theorem Verifier checks that \( \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} \) for a vector field \( \mathbf{F} \) and a surface \( S \). Input the vector field, surface parameterization, and parameter bounds to see the computed integrals with MathJax-rendered steps. Results are copyable, with sharing and embedding options for vector calculus students.
Example 1: Stokes’ Theorem for a Plane
Vector Field: \( \mathbf{F} = (y, -x, z) \).
Surface: \( \mathbf{r}(u,v) = (u, v, 0) \), \( u \in [0,1], v \in [0,1] \).
Step 1: Compute the curl.
\( \nabla \times \mathbf{F} = (-1, -1, -2) \).
Step 2: Surface integral.
\( \iint_S (\nabla \times \mathbf{F}) \cdot (\mathbf{r}_u \times \mathbf{r}_v) \, du \, dv = \int_0^1 \int_0^1 -2 \, du \, dv = -2 \).
Step 3: Line integral over boundary.
\( \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = -2 \).
Step 4: Conclusion.
Both integrals equal \(-2\), verifying Stokes’ theorem.
Example 2: Stokes’ Theorem for a Hemisphere
Vector Field: \( \mathbf{F} = (-y, x, 0) \).
Surface: \( \mathbf{r}(\theta,\phi) = (\sin\theta \cos\phi, \sin\theta \sin\phi, \cos\theta) \), \( \theta \in [0,\pi/2], \phi \in [0,2\pi] \).
Step 1: Compute the curl.
\( \nabla \times \mathbf{F} = (0, 0, 2) \).
Step 2: Surface integral.
\( \iint_S (\nabla \times \mathbf{F}) \cdot (\mathbf{r}_\theta \times \mathbf{r}_\phi) \, d\theta \, d\phi = 2\pi \).
Step 3: Line integral over boundary.
\( \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = 2\pi \).
Step 4: Conclusion.
Both integrals equal \( 2\pi \), verifying Stokes’ theorem.