Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Bra–ket notation wikipedia , lookup
Matrix calculus wikipedia , lookup
Euclidean vector wikipedia , lookup
Field (mathematics) wikipedia , lookup
Cartesian tensor wikipedia , lookup
Laplace–Runge–Lenz vector wikipedia , lookup
Basis (linear algebra) wikipedia , lookup
Math 280 Homework Assignment 12 due date: Dec. 10, 2015 1. Let c be the top half of the unit circle, oriented from (−1, 0) to (1, 0), and F be the vector field F(x, y) = (ln(x + 5) + y 2 , 2xy − y 2 ). 2 Check that Curl(F) = Z 0 (the R curl), and use the flexibility theorem for curves to compute the integral F · ds. Flexing the curve to the line joining (−1, 0) and (1, 0) c is a possibility. 2. Let F be the vector field 2 F(x, y, z) = (yez − xexy , yexy + tan(z 2 + z + 1), x2 ). Check that Div(F) = 0, and use the flexibility theorem for surfaces to compute ZZ F · dS S where S is the top half of the unit sphere oriented outwards. (“Flexing” it to the unit disk seems like a good bet.) 3. Let c1 be the top half of the unit circle, oriented counterclockwise, c2 be the line segment joining (1, 0) to (−1, 0), oriented from (1, 0) to (−1, 0), and let F be the vector field 3 F(x, y) = (x2 − y, xy − arcsin(y) + ey ). Z Z F · ds. F · ds and (a) Use Green’s theorem to compute the difference between (b) Use (a) to compute Z F · ds (by computing the easier c1 Z c1 c2 F · ds, of course. . . ). c2 4. Questions about electricity and magnetism. (a) Could E(x, y, z) = (x2 − y 2, ey − 2z, z 2 + 3 sin(y)) be an electric field (in a static situation)? (b) Could B(x, y, z) = (sin(x), sin(y), z 2 ) be a magnetic field (in a static situation)? (c) Suppose that F and G are C 1 vector fields defined on all of R3 which have the following property. For every oriented surface S with oriented boundary curve c, we have the equality Z ZZ F · ds = G · dS. S c What relationship must hold between F and G, and why? 1 This homework assignment is due on or before Thursday, December 10th, at 4pm. The assigment can be handed in to my office, 507 Jeffery Hall. (There is a mailbox with my name on it to the right of my door. There is an even closer mailbox with someone else’s name on it to the left of my door. Do not be tempted by the closer mailbox!) 2