F(x, y) represents a velocity field of a fluid over a surface S defined by z = 6 − 3x − 2y. If the magnitude of the velocity in the direction of the unit normal vector, n̂, on S is 3z⁄√14, compute the flux of F(x, y) over the surface S in the first octant oriented upward, using the projection of S on the xy - plane
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To compute the flux of F(x, y) over the surface S in the first octant, oriented upward, using the projection of S on the xy-plane, you can follow these steps:
Step 1: Find the unit normal vector n̂ on the surface S.
The surface S is defined by the equation z = 6 − 3x − 2y. To find the unit normal vector, we need to compute the partial derivatives of z with respect to x and y and then normalize the resulting vector.
∂z/∂x = -3, ∂z/∂y = -2
The unit normal vector n̂ is given by:
n̂ = ⟨∂z/∂x, ∂z/∂y, -1⟩ / √( (∂z/∂x)^2 + (∂z/∂y)^2 + 1 )
= ⟨-3, -2, -1⟩ / √( (-3)^2 + (-2)^2 + 1 )
= ⟨-3, -2, -1⟩ / √( 14 )
Step 2: Project surface S onto the xy-plane.
The projection of S onto the xy-plane will be a region in the xy-plane. In this case, since z = 6 − 3x − 2y, the projection can be found by setting z = 0.
0 = 6 - 3x - 2y
Simplifying the equation, we get:
3x + 2y = 6
Therefore, the projection of S on the xy-plane is the line 3x + 2y = 6.
Step 3: Compute the flux of F(x, y) over the surface S.
The flux of F(x, y) over the surface S is given by the following integral:
Flux = ∬S F(x, y) · n̂ dS
Since the flux is defined in terms of the unit normal vector n̂ and the magnitude of the velocity in the direction of n̂, we can rewrite the flux as:
Flux = ∬S (F(x, y) · n̂) * (3z / √14) dS
To evaluate the flux, we need to parameterize the surface S and compute the appropriate limits of integration. Since the projection of S onto the xy-plane is the line 3x + 2y = 6, we can parameterize it as:
r(x, y) = ⟨x, y, 6 - 3x - 2y⟩
The limits of integration will depend on the region enclosed by the projection of S in the first octant. In this case, the line 3x + 2y = 6 intersects the x and y axes at (2, 0) and (0, 3) respectively. So, we can choose the limits of integration as follows:
0 ≤ x ≤ 2
0 ≤ y ≤ 3 - (3/2)x
Finally, we can compute the flux using these parameters:
Flux = ∫∫R (F(x, y) · n̂) * (3z / √14) √( (∂z/∂x)^2 + (∂z/∂y)^2 + 1 ) dA
Here R represents the region enclosed by the projection of S on the xy-plane.
Evaluating this double integral will give you the flux of F(x, y) over the surface S in the first octant, oriented upward, using the projection of S on the xy-plane.