Math 21a. Line IntegralsVector FieldsT. JudsonHarvard UniversitySpring 2008Learning Objectives1• To understand and be able to apply the definition of vector field in R2and R3.• To understand and be able to apply the definition of gradient field.• To understand and be able to apply the definition of conservative field.1Section 13.1 in J. Stewart. Multivariable Calculus: Concepts and Contexts, third edition. Brooks/Cole,Belmont CA, 2005.1Testing Your Knowledge1. Sketch the vector field F(x, y) = yi + (1/2)j.2. Sketch the vector fieldF(x, y) =yi + xjpx2+ y2.3. Find the gradient field of f(x, y) = ln(x + 2y).4. Match each function f with the plot of its gradient field.(a) f(x, y) = xy(b) f(x, y) = x2+ y2(c) f(x, y) = x2− y2(d) f(x, y) =px2+ y2fZ-Zg m Plot the gradient vecto, nAO of / together with acon-.our map of /. Explain how they are related to each other.17, f(x,y) : sin.r * sinY28. f(x,y) : sin(x * y)W-geM Match th" f*rtion, /with the plots of their gradientvector fields (labeled I-IV). Give reasons for your choices.-429. f(x,y): xyffi .f(r, y): x2 + y2l 4-44/ / t l/ / / tr ' r ' / t- € -I \ \ \\ \ \ \\ \ \ \\ \ \\ \ \Itt| / / ?| / / . r 'I t / /-430. f(x,y): x2 - y232. f(x,y): .rF + Y'[ I I 44\ \ \ 1\ \ \ \\ \ \ \t | / , /t / / . /t . / r ' 2, t a JF e -r ' r ' / t/ / / t/ / / tI \ \ \I \ \ \\ \ \ \-44\ \ - . . -\ \ - . -\ \ \ \L t r ,* . a , , / /- . r / /2 . / / |. t t lI/ / t -/ / r -l ll t/ /! \ \ 1\ \ \ \* f \ \* q - \ \-44-4IVilI-433.A particle *ou., in u n"*n, n"fa V(", y) : (r), * i ,'7.'If it is at position (2, L) at time r : 3, estimate its locationat time t :3.0L.4\ \ \ \\ \ \ 1* - - f \ \e-+-C-\t / r z| / t z| ,l,nn,y' -+-u--+<-a-{ {rs/ I"// / I/ / / I\.-.e--a--o\ \-"-.\ \ r ;! \ r
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