Apply the Divergence Theorem to calculate the flux of the given vector field over the rectangular solid. Optimize the integral for maximum flux to find the greatest flux value.
1. Define the Rectangular Solid:
Consider the rectangular solid defined by the inequalities 0 < x < a, 0 < y < b, and 0 < z < 1. This establishes the boundaries for the solid in the x, y, and z directions.
2. Determine the Flux Vector Field:
The vector field F is given as (-x² - 4xy)i - 6yzj + 12zk. This represents the flux across each face of the rectangular solid.
3. Apply Divergence Theorem:
Use the Divergence Theorem, which states that the flux of a vector field across a closed surface is equal to the volume integral of the divergence of the field over the solid. In this case, the solid is the rectangular box.
4. Calculate Divergence:
Compute the divergence of F, which is the sum of the partial derivatives of its components. The divergence of F is div(F) = -2x - 6y.
5. Set Up Integral:
Set up the triple integral over the given rectangular solid for the divergence (-2x - 6y) with respect to x, y, and z.
6. Evaluate Integral:
Evaluate the integral to find the total flux. Use the given boundaries to integrate over the rectangular solid.
7. Optimize for Maximum Flux:
To find the maximum flux, optimize the integral by finding critical points. Take partial derivatives and set them equal to zero, then check the boundary values.
8. Final Answer for Maximum Flux:
The maximum flux and the corresponding dimensions of the rectangular solid can be determined from the optimized integral.