Final answer:
To find the flux of the vector field F across the surface S, we need to evaluate the surface integral ∬S F · dS. First, we rewrite the equation of the plane as z = 4 - 5x - 4y and write it in vector form as r(x, y) = (x, y, 4 - 5x - 4y). Next, we substitute the values of F and dS into the integral and simplify the expression (∂r/∂x) × (∂r/∂y). Finally, we evaluate the integral by multiplying the components of F and dS and integrating over the region in the xy-plane defined by the equation of the plane.
Step-by-step explanation:
To find the flux of the vector field across the surface S, we need to use the surface integral. The equation of the plane is given by 5x + 4y + z = 4. We can rewrite this equation as z = 4 - 5x - 4y to get a function that represents the plane in terms of x and y. The normal vector to the plane is (5, 4, 1), so we can write the equation of the plane in vector form as r(x, y) = (x, y, 4 - 5x - 4y).
The surface integral of a vector field F across a surface S is given by ∬S F · dS, where F is the vector field and dS is the differential of surface area. In this case, the vector field F is given by F = 1i + 2j + 3k. The differential of surface area dS is given by dS = (∂r/∂x) × (∂r/∂y) dA, where (∂r/∂x) and (∂r/∂y) are the partial derivatives of r with respect to x and y, and dA is the differential of area in the xy-plane.
To compute the flux, we need to evaluate the following surface integral: ∬S F · dS. Substituting the values of F and dS into the integral, we get ∬S (1i + 2j + 3k) · ((∂r/∂x) × (∂r/∂y)) dA. We can simplify this expression further by calculating (∂r/∂x) × (∂r/∂y), which is equal to the cross product of the vectors (1, 0, -5) and (0, 1, -4). The cross product is given by ((-5)(1) - (-4)(0))i - ((-5)(0) - (-4)(1))j + ((1)(1) - (0)(0))k = -5i + 4j + k.
Finally, we can evaluate the integral by multiplying the components of F and dS, and then integrating over the region in the xy-plane defined by the equation of the plane. The result will be the flux of the vector field across the surface S.