Question

Use Stokes' theorem to evaluate ( iint_{5} ) curi( ( (F)=05 ). ( F(x, y, z)=e³x y}left(1+e{x z} i+x²2 kright. ), sis the haif of the elspsold ( 4 x²y²4 x²) w 4 that lies

Answer 1

To evaluate the given integral using Stokes' theorem, compute the curl of the vector field F and find the surface integral over the given hemisphere. First, compute the curl of F. Then, parametrize the surface of the hemisphere using spherical coordinates. Next, compute the surface normal vector. Finally, evaluate the surface integral using Stokes' theorem and integrate over the projection of the surface onto the xy-plane.

Step-by-step explanation:

To evaluate the given integral using Stokes' theorem, we need to compute the curl of the vector field F and find the surface integral over the given hemisphere.

1. Compute the curl of F: curl(F) = (3e³x, 1, 2x²).
2. Note that the given surface is the upper hemisphere of the ellipsoid x²/4 + y²/4 + z²/9 = 1 that lies above the xy-plane.
3. Parametrize the surface of the hemisphere using spherical coordinates: x = 2r sin(θ) cos(φ), y = 2r sin(θ) sin(φ), z = 3r cos(θ).
4. Compute the surface normal vector: n = (-∂x/∂θ, -∂y/∂θ, ∂x/∂r × ∂y/∂r).
5. Evaluate the surface integral using Stokes' theorem: iint_S curl(F) · dS = iint_D (curl(F) · n) dA, where D is the projection of the surface onto the xy-plane.
6. Integrate over D to find the final result.