(Rate of change of circulation along a moving curve) (a)
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(Rate of change of circulation along a moving curve) (a) Let F.r;t/ be a smooth vector field in R3 depending on a parame
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(Rate of change of circulation along a moving curve) (a) Let F.r;t/ be a smooth vector field in R3 depending on a parameter t, and let G.s; t / D F r.s; t /; t D F x.s; t /; y.s; t /; z.x; t /; t ; where r.s; t / D x.s; t /i C y.s; t /j C z.s; t /k has continuous partial derivatives of second order. Show that @ @t G @r @s ? @ @s G @r @t D @F @t @r @s C .r F/ @r @t @r @s : Here, the curl r F is taken with respect to the position vector r. (b) For fixed t (which you can think of as time), r D r.s; t /, .a s b/, represents parametrically a curve Ct in R3 . The curve moves as t varies; the velocity of any point on Ct is vC .s; t / D @r=@t. Show that d dt Z Ct F dr D Z Ct @F @t dr C Z Ct .r F/ vC dr C F r.b; t /; t vC .b; t / ? F r.a; t /; t vC .a; t /: Hint: Write d dt Z Ct F dr D Z b a @ @t G @r @s ds D Z b a @ @s G @r @t C @ @t G @r @s ? @ @s G @r @t ds: Now use the result of (a).
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