Compute the flux of F = xi + yj + zk

Question

It is through just the curved surface of the cylinder x²+y² = 16 bounded below by the plane x + y + z = 2, above the plane x + y + z =4, and oriented away from the z-axis

Answer 1

1. Identify the Vector Field and Surface

First, define the components of your vector field F and the geometry of the surface S.

1. Vector Field: F = <P,Q,R>
2. Surface Equation: Identify the cylinder's radius x² + y² = a²

2. Determine the Unit Normal Vector <n>

Since the flux is through the curved surface and oriented away from the z-axis, you need the outward-pointing unit normal vector.

1. Formula: For a cylinder x² + y² = a, the unit normal is:
2. n = ∇( x² + y²) / | ( x² + y²) |
3. Key Concept: Note whether n has a z-component. This will determine if the z-part of F affects the final integral.

3. Set Up the Flux Integral

The general formula for Surface Flux is:

Φ = ∫∫F • ndS

1. The Dot Product: Calculate F • n using your results from Step 1 and Step 2.
2. Surface Element (dS): In cylindrical coordinates, the differential area element for the side of a cylinder is dS = adzdθ

4. Define Limits of Integration

This is where the boundaries of the planes come into play.

1. Angle (θ): Since it is a full cylinder, determine the standard range for a full rotation.
2. Height (z): Set the limits based on the plane equations.
3. Solve the lower plane equation for z to get z_min(x, y).
4. Solve the upper plane equation for z to get z_max(x, y).
5. Transformation: Substitute x = acosθ and ysinθ into these bounds.

5. Evaluate the Iterated Integral

Combine everything into a double integral: for Φ with your first ∫ having your θ₁ and θ₂ constraints, and your second integral with the z_min and z_max constraints.

1. Tip: Look for symmetry or constants. If the integrand and the "height" (Δz = z_{max - z}min₎ simplify nicely, the calculation becomes much more straightforward.
2. Next step to help you solve: Start by calculating the dot product of F • n. Does the z variable remain in your integrand afer the dot product, or does it vanish?