The image illustrates the use of Stokes' theorem to evaluate $\iint_S \text{curl } \mathbf{F} \cdot \mathbf{dS}$ using the equivalence $\iint_S \text{curl } \mathbf{F} \cdot \mathbf{dS} = \int_C \mathbf{F} \cdot \mathbf{dr}$.

Explanation of the Process:

1. Stokes' Theorem Application:

   • The surface integral of $\text{curl } \mathbf{F}$ is converted to a line integral over the boundary curve $C$.

   $\iint_S \text{curl } \mathbf{F} \cdot \mathbf{dS} = \int_C \mathbf{F} \cdot \mathbf{dr}$

2. Vector Dot Product Expansion:

   • The line integral is calculated by taking the dot product of $\mathbf{F}$ and the tangent vector to the curve parameterized by $t$.
   • Here, $\mathbf{F}(t)$ is given as $\langle 16\cos^2(t), 16\sin^2(t), 0 \rangle$, and $\mathbf{dr}$ involves components like $\langle -4\sin(t), 4\cos(t), 0 \rangle dt$.

3. Integral Splitting:

   • The expression is expanded into two terms with integrals of $\cos^2(t)$ and $\sin^2(t)$ scaled by coefficients and trigonometric functions ($\cos(t)$, $\sin(t)$).

4. Symmetry Argument:

   • The integrals like $\int_0^{2\pi} \cos^3(t) dt$ and $\int_0^{2\pi} \sin^3(t) dt$ evaluate to zero due to the symmetry of $\cos^3(t)$ and $\sin^3(t)$ over one full period of $2\pi$.

5. Final Evaluation:

   • The remaining terms simplify, and all integrals sum to $0$, which satisfies the conclusion.

Would you like a step-by-step derivation or details about any part of the process?

5 Related Questions:

1. What are the conditions for applying Stokes' theorem?
2. Why do certain integrals of odd powers of $\sin(t)$ or $\cos(t)$ over $[0, 2\pi]$ equal zero?
3. How does parameterizing the curve simplify surface integral computations?
4. Can you generalize this approach to any vector field $\mathbf{F}$?
5. What other applications of Stokes' theorem are common in physics or engineering?

Tip: Always check for symmetry in trigonometric integrals to simplify computation.