lect14_1

lect14_1 - Vector Fields - (14.1) 1. Vector Fields !...

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Vector Fields - (14.1) 1. Vector Fields Definition: A vector field in the plane is a vector-valued function F ! ! x , y " which maps points in R 2 to a vector space V 2 . A vector field in the plane is a vector-valued function F ! ! x , y , z " which maps points in R 3 to a vector in space V 3 . We denote F ! ! x , y " "# f 1 ! x , y " , f 2 ! x , y " $ and F ! ! x , y , z " "# f 1 ! x , y , z " , f 2 ! x , y , z " , f 3 ! x , y , z " $ . where f 1 , f 2 and f 3 are scalar functions. Graph of a vector field: For a given point ! x , y " in R 2 or ! x , y , z " in R 3 , F ! ! x , y " or F ! ! x , y , z " is a vector in V 2 or V 3 . Instead of sketch the vector F ! as a position vector (the initial point is the origin), ! x , y " is used as the initial point. Example Plot the following vector fields over grid points ! x , y " where x , y " ! 1, 0, 1 using ! x , y " as the initial point. a . F ! ! x , y " "# 1, y $ b . F ! ! x , y " "# x % y , y ! x $ -4 -2 0 2 4 y -4 -2 2 4 x F ! ! x , y " "# 1, y $ -4 -2 0 2 4 y -4 -2 2 4 x F ! ! x , y " "# x % y , y ! x $ Example Plot the vector field F ! ! x , y " "# y , ! x $ over the points on the curves: F ! " 0, 1, 2. -4 -2 0 2 4 y -4 -2 2 4 x F ! ! x , y " "# y , ! x $ 1
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Example Match the vector fields ! 1 " F ! ! x , y " "# y 2 , x ! 1 $ ! 2 " G ! ! x , y " "# y % 1, e x /6 $ ! 3 " H !
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lect14_1 - Vector Fields - (14.1) 1. Vector Fields !...

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