s12 - Solution to Set 12 1(a Σ is composed of two pieces...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Solution to Set 12 1. (a) Σ is composed of two pieces of smooth surfaces, a hemisphere and a flat disk. (i) The hemisphere is the graph of the function f ( x, y ) = p 1 − x 2 − y 2 . ∂ x f = − x p 1 − x 2 − y 2 ∂ y f = − y p 1 − x 2 − y 2 The projection of the sphere on the xy-plane is the unit disk D centered at the origin. The surface integral over the hemisphere is given by Z Z F · ndS = Z Z D − x∂ x f − y∂ y f + p 1 − x 2 − y 2 dA = Z Z D 1 p 1 − x 2 − y 2 dA = Z 2 π Z 1 rdrdθ √ 1 − r 2 = 2 π (ii) On the disk, the outward unit normal is − k and F · n = − z = 0. Therefore, the flux integral over this piece is 0. In total, Z Z Σ F · ndS = 2 π . (b) Σ is composed of three smooth pieces, a cylindrical surface and two flat disks. (i) The cylindrical surface To evaluate the flux integral over the cylindrical surface, one may cut the cylinder into two pieces and project them onto the xz-plane (or the yz-plane)....
View Full Document

Page1 / 2

s12 - Solution to Set 12 1(a Σ is composed of two pieces...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online