MATH-M 312 Lecture Notes - Lecture 11: Triple Product, Electric Current, Parametric Surface
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M312 section 7. 6 notes- surface integrals of vector fields. 3-9-16 (note: in section 7. 2 also represents vector fields: multiply by area of each polygon, add all up. When you look at riemann sum definitions, they are virtually identical. Suppose for points (cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667) on , function (cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667): (cid:2871) (cid:2871) gives the unit normal vector to at (cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667: notice that (cid:1517) (cid:1856) vector, other uses unit normal to approximated faces on. Recall that cross product implicitly gives orientation; to get opposite, take (cid:2202)(cid:2204) (cid:2202)(cid:2203) =(cid:1517) (cid:4672)( (cid:4666)(cid:1873),(cid:1874)(cid:4667)) (cid:4666)(cid:2202)(cid:3048) (cid:2202)(cid:3049)(cid:4667)(cid:4673) (cid:1856)(cid:1873)(cid:1856)(cid:1874) because magnitude is scalar that can be pulled out of. Given 3 vectors (cid:2183),(cid:2184),(cid:2185): (cid:2871), their scalar triple product is (cid:2183) (cid:4666)(cid:2184) (cid:2185)(cid:4667: (cid:2183) (cid:4666)(cid:2184) (cid:2185)(cid:4667)=|(cid:1853)(cid:2869) (cid:1853)(cid:2870) (cid:1853)(cid:2871) notice this is generalization of |(cid:2841) (cid:2842) (cid:1854)(cid:2869) (cid:1854)(cid:2871) (cid:1854)(cid:2870) (cid:1855)(cid:2871)| (cid:1855)(cid:2871)| (cid:1854)(cid:2869) (cid:1854)(cid:2870) (cid:1854)(cid:2871) (cid:1855)(cid:2869) (cid:1855)(cid:2870) (cid:1855)(cid:2870) (cid:1855)(cid:2869) Suppose is constant vector field, ie. (cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667)=(cid:2185) for all (cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667).
