MAT235Y1 Midterm: Term Test 3 - Fall 2017

• School: UTSG
• Department: Mathematics
• Course: MAT235Y1
• Professor: Shinji Sasaki

Document Summary

= integration of a function f(x) and (cid:1516) (cid:1858)((cid:1876)) (cid:1856)(cid:1859)((cid:1876)) (cid:3029) (cid:3028) The puchline is what we are integrating is not just functions, but what we are really integrating 1- forms of the form f(x) dx or p(x,y) dx + q(x,y) dy etc. 1-form: (cid:1858)((cid:1876), (cid:1877)) (cid:1856)(cid:1876) + (cid:1859)((cid:1876), (cid:1877)) (cid:1856)(cid:1877) 3-form: (cid:1858)((cid:1876), (cid:1877), (cid:1878)) (cid:1856)(cid:1876) (cid:1856)(cid:1877) (cid:1856)(cid:1878) (cid:1856)(cid:1876) (cid:1856)(cid:1877) is a wedge product (cid:1516) (cid:1858)((cid:1876), (cid:1877)) (cid:1856)(cid:1827) (cid:3005) = (cid:1858)((cid:1876), (cid:1877)) (cid:1856)(cid:1876) (cid:1856)(cid:1877) (cid:3005) da is a the infinitesimal change in area while (cid:1878) = (cid:1858)((cid:1876), (cid:1877)) Recall that the (cid:1827)(cid:1870)(cid:1857)(cid:1853) (cid:1867)(cid:1858)(cid:1868)(cid:1853)(cid:1870)(cid:1853)(cid:1864)(cid:1864)(cid:1857)(cid:1864)(cid:1867)(cid:1859)(cid:1870)(cid:1853)(cid:1865) = |(cid:1873)(cid:4652)(cid:1318) (cid:1875)(cid:4652)(cid:4652)(cid:1318)| Recall from the one of the properties for cross product is (cid:1873)(cid:4652)(cid:1318) (cid:1875)(cid:4652)(cid:4652)(cid:1318) = (cid:1875)(cid:4652)(cid:4652)(cid:1318) (cid:1873)(cid:4652)(cid:1318) The following represents: (cid:1830) represents the n-dimensional region (cid:2034)(cid:1830) represents (n-1) dimensional boundary (cid:1875) represents the (n-1) form (cid:1856)(cid:1875) represents the differential of (cid:1875) Stokes say: instead, you can calculate over some other boundary. In particular, the shape of the membrane only matters up to the boundary.