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MAT 21D Lecture Notes - Lecture 17: Dot Product, Cross Product, Curve

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UC-Davis
Department
Mathematics
Course
MAT 21D
Professor
Shiqian Ma

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MAT 21D Lecture 16: MAT 21D – Lecture 16 – Line Integrals

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MAT 21D Lecture Notes - Lecture 17: Dot Product, Cross Product, Curve

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MAT 21D Lecture Notes - Lecture 18: Dot Product, Cross Product, Curve

Document Summary

Mat 21d lecture 17 work and circulation flux. =(cid:2869: the vector, =(cid:1839)(cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667)(cid:2835) +(cid:1840)(cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667)(cid:2836) +(cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667)(cid:1863) and the curve, (cid:1870) (cid:4666)(cid:1872)(cid:4667)= (cid:4666)(cid:1872)(cid:4667)(cid:2835) + (cid:4666)(cid:1872)(cid:4667)(cid:2836) +(cid:1863)(cid:4666)(cid:1872)(cid:4667)(cid:1863) for (cid:1853) (cid:1872) (cid:1854). (cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667) (cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667) (cid:1871), deriving the work formula: = (cid:1875) . Take the limit as (cid:1866) such that (cid:1867)(cid:1870)(cid:1863)= (cid:1856)(cid:1871)= (cid:4666) (cid:4666)(cid:1870) (cid:4666)(cid:1872)(cid:4667)(cid:4667) (cid:3045) (cid:3047) (cid:4667)(cid:1856)(cid:1872). (cid:3029)(cid:3028: example: given =(cid:4666)(cid:1877) (cid:1876)(cid:2870)(cid:4667)(cid:2835) +(cid:4666)(cid:1878) (cid:1877)(cid:2870)(cid:4667)(cid:2836) +(cid:4666)(cid:1876) (cid:1878)(cid:2870)(cid:4667)(cid:1863) and (cid:1870) (cid:4666)(cid:1872)(cid:4667)=(cid:1872)(cid:2835) +(cid:1872)(cid:2870)(cid:2836) + (cid:1872)(cid:2871)(cid:1863) for (cid:882) (cid:1872) (cid:883), find the work done from (0, 0, 0) to (1, 1, 1). Based on the curve given by (cid:1870) (cid:4666)(cid:1872)(cid:4667), (cid:1876)=(cid:1872), (cid:1877)=(cid:1872)(cid:2870), and (cid:1878)=(cid:1872)(cid:2871). The dot product of ((cid:1870) (cid:4666)(cid:1872)(cid:4667)) and (cid:3045) (cid:3047) is ((cid:1870) (cid:4666)(cid:1872)(cid:4667)) (cid:3045) (cid:3047)= (cid:884)(cid:1872)(cid:4666)(cid:1872)(cid:2871) (cid:1872)(cid:2872)(cid:4667)+(cid:885)(cid:1872)(cid:2870)(cid:4666)(cid:1872) (cid:1872)(cid:2874)(cid:4667)=(cid:884)(cid:1872)(cid:2872) (cid:884)(cid:1872)(cid:2873)+(cid:885)(cid:1872)(cid:2871) (cid:885)(cid:1872)(cid:2876). =(cid:2870)(cid:2873)(cid:1872)(cid:2873) (cid:2869)(cid:2871)(cid:1872)(cid:2874)+(cid:2871)(cid:2872)(cid:1872)(cid:2872) (cid:2869)(cid:2871)(cid:1872)(cid:2877)|(cid:3047)=(cid:2868)(cid:3047)=(cid:2869)=(cid:2870)(cid:2873) (cid:2870)(cid:2871)+(cid:2871)(cid:2872)=(cid:2870)(cid:2877)(cid:2874)(cid:2868). (cid:2869)(cid:2868: equivalent expressions for work: = (cid:1856)(cid:1871)= (cid:4666) (cid:4666)(cid:1870) (cid:4666)(cid:1872)(cid:4667)(cid:4667) (cid:3045) (cid:3047) (cid:4667)(cid:1856)(cid:1872)= (cid:3029)(cid:3028) If (cid:1827)= (cid:1828), then the flow is called circulation around the curve: example: suppose =(cid:4666)(cid:1876) (cid:1877)(cid:4667)(cid:2835) +(cid:1876)(cid:2836) and the circle (cid:1870) (cid:4666)(cid:1872)(cid:4667)=(cid:4666)(cid:1855)(cid:1867)(cid:1871)(cid:1872)(cid:4667)(cid:2835) +(cid:4666)(cid:1871)(cid:1866)(cid:1872)(cid:4667)(cid:2836) for (cid:882) (cid:1872) (cid:884).

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