Changing limits of integration

[imagemania]

Homework Statement

A surface S is defined by z = 1-x^2-y^2 between 0≤ z ≤1
I need to calculate the flux of the vector field F = (y)i + (z)j through S.

Homework Equations

Cylindrical polar coordinates, Normal etc

The Attempt at a Solution

By changing the variables using cylindrical polar coordinates:
[tex]
\iint{\vec{F}.d\vec{S}}=\iint{\vec{F}.\vec{n}d{S}}
\vec{n} = \vec{N}/|{\vec{N}|}[/tex]

[tex]\vec{N} = \nabla g{(\rho,\phi,z)}[/tex]
Where \rho,\phi,z are the cyldrical coordinate symbols.
Then
[tex]ds = \frac{dA}{\vec{n}\vec{k}} = \frac{d\rho d\phi}{\vec{n}\vec{k}}[/tex]Doing the above i get to (after cancellations between the fraction and the the absolute value within unit vector)

[tex]\iint 2{\rho}^{2}sin{\phi} d\rho d\phi[/tex]

Im not 100% sure if this is correct, but it doesn't seem too complicated. Though i am not sure how to deal with the limits of integration from this points onwards.

Hope you can help me :)

 

[anathema]

Hello,

Your approach to solving this problem is correct. To find the flux, you need to integrate the dot product of the vector field F and the unit normal vector n over the surface S.

To find the unit normal vector n, you can use the gradient of the function g(x,y,z) = z - 1 + x^2 + y^2. This function represents the surface S, so its gradient will give you the normal vector at any point on the surface.

Using cylindrical polar coordinates, the function g can be written as g(ρ, φ, z) = z - 1 + ρ^2. Taking the gradient of this function, we get:

∇g = (∂g/∂ρ, ∂g/∂φ, ∂g/∂z) = (2ρ, 0, 1)

To find the unit normal vector n, we need to divide ∇g by its magnitude:

|∇g| = √(4ρ^2 + 1)

Thus, the unit normal vector n is given by:

n = (2ρ/√(4ρ^2 + 1), 0, 1/√(4ρ^2 + 1))

Now, to find the flux, we need to integrate the dot product of F and n over the surface S.

∫∫ F⋅n dS = ∫∫ (y, z)⋅(2ρ/√(4ρ^2 + 1), 0, 1/√(4ρ^2 + 1)) dρ dφ

= ∫∫ (2ρy/√(4ρ^2 + 1) + z/√(4ρ^2 + 1)) dρ dφ

= ∫∫ (2ρsinφ/√(4ρ^2 + 1) + z/√(4ρ^2 + 1)) ρ dρ dφ

= ∫∫ (2ρ^2sinφ/√(4ρ^2 + 1) + ρz/√(4ρ^2 + 1)) dρ dφ

Now, you can use the limits of integration for ρ and φ to evaluate this integral. Since the surface S is defined