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14 - interval t where F ’ is continuous and F ’...

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f F t = f F t = f t F t (scalar function times vector function) F x G t = F t x G t F G t = blah blah blah...all operations pretty much work You can even stick limits on both sides. Continuity is defined the same way as before. Thm 14.2.3: If the nonzero vector F (t) is differentiable and has constant length, then F (t) is orthogonal to F’ (t). (If you have a unit vector, it is necessarily of constant length.) F t d t = f 1 t d t , f 2 t d t , f 3 t d t C , where C = C 1 , C 2 , C 3 . Since T is the unit tangent vector d T d s only measures the change in direction. Direction of motion is the unit velocity vector. Graph of the position vector is the trajectory. Note velocity V t = speed direction = V V V . A graph of a vector function is smooth on an
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Unformatted text preview: interval t where F ’ is continuous and F ’ t (except possibly at endpoints of the interval). Newton’s Second Law : F t = m a t Consider coordinate system with the sun at the origin (ignore all other planets): you get Newton’s Law of Gravitiation F = GMm r 3 r = GMm r 2 u , where u is a unit vector and r = r , F is the force vector of the planet, and r is the position vector of the planet. Thm 14.4.1 A polar eqn of form r = ed 1 e cos or r = ed 1 e sin represents a conic section. The conic section is an ellipse if e 1, parabola if e = 1 and hyperbola if e 1....
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