According to the Divergence Theorem, we know that the flux of a vector field F across a surface S is equal to the triple integral over the volume V bounded by the surface S of the divergence of F.
Here, the vector field F is given as F = e^(y*tan(z))i + (y^3 - x^2)j + x*sin(y)k. Firstly, we will find the divergence of F.
The divergence of a vector field F = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k is given by div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z.
In this case, P(x, y, z) = e^(y * tan(z)), Q(x, y, z) = y^3 - x^2, and R(x, y, z) = x * sin(y).
The partial derivative of P with respect to x is 0 because there is no x variable in P.
The partial derivative of Q with respect to y is 3 * y^2.
The partial derivative of R with respect to z is 0 because there is no z variable in R.
Therefore, the divergence of F, denoted as div F, is 0 + 3*y^2 + 0 = 3*y^2.
Next, we will take the triple integral over the volume V bounded by the surface S. The volume V is defined by -1 ≤ x, y ≤ 1 and 0 ≤ z ≤ 2 - x^4 - y^4.
We're going to write the triple integral in the following way:
∫ from -1 to 1 ∫ from -1 to 1 ∫ from 0 to 2 - x^4 - y^4 div F dz dy dx.
Substituting div F = 3*y^2 into the integral, we get:
∫ from -1 to 1 ∫ from -1 to 1 ∫ from 0 to 2 - x^4 - y^4 3*y^2 dz dy dx.
This is the final expression. You would typically perform a series of calculations to carry out these integrals but doing so competently requires knowledge of proper integration methods. Unfortunately, these calculations often get complex and are best handled with mathematical software or a calculator capable of symbolic computation.