講者： 許永昌 老師 1

Contents 2

Line integrals In Riemann integrals ( ), line integrals can be written ashttp://mathworld.wolfram.com/RiemannIntegral.html  C V  dr=lim  r i  0  i V(r i )  r i.  C V  dr=  V x dx+  V y dy+  V z dz =  C V // dr =  C Vdr //. NOTE: Physical applications: work, potential, etc. 3

Example P59e (P56) Path-Dependent Work Geometric: The line integrals of path OA & OB =0 because of F(r i )  r i. Algebraic: 4 Force field Path 1 Path 2 O A BC How about a curved path?

Surface Integrals 5

In Riemann integral, ê i  d  = ê i  ndA=dAcos . The projection of the surface.  J  d  =  J x d  x +  J y d  y +  J z d  z. d  x =sign( ) | dydz |. d  y =? d  z =? 6 Example: on the xy plane. êiêi dd

Surface integrals   J z dxdy= 7

Example: electric flux Assume, what is the electric flux through the green surface x 2 +y 2 +z 2 =1 & z  0. 8

Volume Integrals 9 dd

Integral Definitions of Gradient, Divergence and Curl Point  Line  Area  Volume. 10 dd

Integral Definitions of Gradient, Divergence and Curl 11 d  z >0 d  z <0

Summary Line integrals: Surface integrals: Volume integrals: Symbols: dr, d  and d  have told you the dimension of the integration element, you can use only one  to represent the integration. Closed loop: Closed surface: Gradient, divergence and curl can be defined by integrations. P66e (P58). 12

Homework 1.9.3e (1.10.5) 1.9.4e (1.10.6) Essential 那一本其實在內文與 exercises 都有不一樣且 不錯的題目可以練習。 13

1.9 nouns 14