Final answer:
To evaluate the surface integral ∫ ∫ F·dA for a given vector field F and surface S, we need to define the normal vector to the surface and parametrize the surface. In this case, we find the normal vector N and parametrize the hemisphere surface S. Then, we substitute the vector field F and the normal vector N into the surface integral equation and integrate over the parametrized surface S to find the value of the surface integral.
Step-by-step explanation:
To evaluate the surface integral ∫ ∫ F·dA for a given vector field F and surface S, we need to define the normal vector to the surface and parametrize the surface to represent it in terms of two variables. In this case, the vector field F is defined as F(x, y, z) = x+y, and the surface S is the hemisphere z=(1−x^2−y^2)^(1/2).
To find the normal vector N, derivatives of the surface equation are required. By differentiating z=(1−x^2−y^2)^(1/2) with respect to x and y, we get:
dz/dx = -x/(1−x^2−y^2)^(1/2)
dz/dy = -y/(1−x^2−y^2)^(1/2)
Then, the normal vector N is given by N = (dz/dx, dz/dy, -1).
Next, we parametrize the surface S. As S is a hemisphere, we can parametrize it as:
x = r cos(θ)
y = r sin(θ)
z = (1−r^2)^(1/2)
where r is the radius of the hemisphere and θ ranges from 0 to 2π.
Now, we can compute the surface integral by substituting F and N into the equation ∫ ∫ F·dA:
∫ ∫ F·dA = ∫ ∫ (x + y) · N · r dz dr dθ
Integrating over the parametrized surface S and applying the appropriate limits of integration, we can find the value of the surface integral.