Let

F(x, y, z) = z tan−1(y^2)i + z^3 ln(x^2 + 10)j + zk.
Find the flux of F across S, the part of the paraboloid
x^2 + y^2 + z = 19
that lies above the plane
z = 3
and is oriented upward.

Thanks! STudying for a quiz...
*tan−1 is tan^(-1)

To find the flux of the vector field F across the given surface S, we can use the surface integral formula:

Flux = ∬S F · dS

where F is the vector field, dS is the differential surface element on S, and ∬S represents the surface integral over S.

To begin, we need to parameterize the surface S. Since S is the part of the paraboloid x^2 + y^2 + z = 19 that lies above the plane z = 3, we can express S in terms of two variables, usually denoting two independent parameters u and v.

Let's choose a parameterization for S as follows:

x = √(19 - z) cos(u)
y = √(19 - z) sin(u)
z = v, where u and v vary over appropriate ranges.

Next, we'll calculate the partial derivatives of x, y, and z with respect to u and v:

∂x/∂u = -√(19 - z) sin(u)
∂x/∂v = 0
∂y/∂u = √(19 - z) cos(u)
∂y/∂v = 0
∂z/∂u = 0
∂z/∂v = 1

Now, we can calculate the cross product of the partial derivatives:

(∂r/∂u) × (∂r/∂v) = (-√(19 - z) sin(u), √(19 - z) cos(u), 0)

where r(u, v) is the position vector (x, y, z).

Normalize this vector by dividing it by its magnitude:

n(u, v) = ((-√(19 - z) sin(u), √(19 - z) cos(u), 0)) / √(19 - z)

Now, let's compute F · n:

F · n = (z tan^(-1)(y^2), z^3 ln(x^2 + 10), z) · ((-√(19 - z) sin(u), √(19 - z) cos(u), 0)) / √(19 - z)

Simplifying the dot product expression, we get:

F · n = -z tan^(-1)(y^2)√(19 - z) sin(u) + z^3 ln(x^2 + 10)√(19 - z) cos(u) + z*0 / √(19 - z)

Since the last component is zero, we can ignore it.

Now, the flux becomes:

Flux = ∬S F · dS = ∫∫R F · n * ||∂r/∂u × ∂r/∂v|| dA

where R represents the parameter domain, and dA is the differential area element in the parameter space (u, v).

Now, substitute the parameterization expressions and perform the integration to find the flux value.

The final result will be the flux of F across the surface S. Good luck with your quiz!