Question

Let V=⟨X³cosz,3−3x²ycosz−3yz²sinx,Z³sinx⟩ Be The Velocity Field Of A Fluid. Compute The Flux Of V Across The Surface X2+Y+Z2=4 Where Y >0 and the surface is oriented away from the origin.

Answer 1

To compute the flux of the velocity field across the given surface, we can use the surface integral of the dot product between the velocity field V and the outward unit normal vector.

Step-by-step explanation:

To compute the flux of the velocity field across the given surface, we can use the surface integral of the dot product between the velocity field V and the outward unit normal vector n. However, before doing that, we need to find the unit normal vector to the surface. The equation of the surface given is x² + Y + z² = 4, which is a paraboloid. The normal vector to this surface at any point can be found by taking the gradient of the equation of the surface, which gives us n = (2x, 1, 2z).

Next, we need to find the dot product of V and n, and evaluate it at each point on the surface. Plugging in the components of V and n, and evaluating the dot product, we get V.n = 2x^4cosz + 3 - 3x²ycosz - 6xyzsinx + 2z^3sinx.

Finally, we can compute the flux by integrating V.n over the surface. Since the surface is oriented away from the origin, the flux will be positive. However, to find the bounds of integration, we need to parameterize the surface. Since Y > 0, we can solve the equation of the surface for Y and get Y = 4 - x² - z². We can then rewrite the surface integral as a double integral in the xz-plane, with appropriate bounds for x and z.