MATH-M 312 Lecture Notes - Lecture 10: Cross Product, Parallelogram, Helicoid
Document Summary
M312 sections 7. 4 and 7. 5 notes- surface integrals of scalar-valued functions. 3-2-16 (cid:1856)(cid:1871); if (cid:1858): (cid:2871) and (cid:1855) given with an. Recall, for curves (1-dimensional objects in (cid:2871)), can define (cid:1516) (cid:1858)(cid:3030) orientation, then can define (cid:1516) (cid:1856)(cid:2201) (cid:3030) (cid:1856)(cid:1871)=(cid:1516) (cid:1858)((cid:1855)(cid:4666)(cid:1872)(cid:4667))|(cid:1855) (cid:4666)(cid:1872)(cid:4667)| =(cid:1516) ((cid:1855)(cid:4666)(cid:1872)(cid:4667)) (cid:1855) (cid:4666)(cid:1872)(cid:4667: can compute (cid:1516) (cid:1858)(cid:3030) (cid:1856)(cid:1872) and (cid:1516) (cid:1856)(cid:2201) (cid:3029)(cid:3028) (cid:3029)(cid:3028, parametric curve is function (cid:1855):[(cid:1853),(cid:1854)] (cid:2871), and curve it defines is its image; (cid:1855) must be differentiable. We know if is a gradient vector field (= (cid:1858) for (cid:1858): (cid:2871) ), then (cid:1516) (cid:1856)(cid:2201)=(cid:1858)((cid:1855)(cid:4666)(cid:1854)(cid:4667)) (cid:1858)((cid:1855)(cid:4666)(cid:1853)(cid:4667)) (cid:3030) Can do analogous things for integrals over surfaces (2-dimensional objects in (cid:2871): not every graph of a surface is a function, but every graph of a function (cid:1858): (cid:2870) (cid:3041) is a surface, two-sided (cid:862)pie(cid:272)e of tape(cid:863) Typical surfaces and 1-1 on its domain (cid:1856)(cid:1872) (cid:3030: bean-like hollow object.
