MATH 251 Fall 2023 Section 16 1 Vector Fields De nition A vector eld on R2 is a function F that assigns to each point x y in a domain D R2 a two dimensional vector F x y We can write F x y P x y i Q x y j where P x y Q x y are functions that map each point x y to a scalar For example if F x y yi xj then P x y y and Q x y x To visualize a vector eld we draw the arrow representing the vector starting at the point x y In doing so note that a parallel shift of a vector is still considered the same vector De nition A vector eld on R3 is a function F that assigns to each point x y z in a domain E R3 a three dimensional vector F x y z We can express F into its components F x y z P x y z i Q x y z j R x y z k hP x y z Q x y z R x y z i 1 2 Example 1 Sketch the vector eld on R3 given by F x y z zk To match F with its vector eld choose sample points x y in each quadrant and use the direction and magnitude of F x y to choose the correct answer or eliminate wrong answers Example 2 Which of the following is the vector eld of F x y hy 1 xi Which is of F x y hx 2 x 1i Example 3 Which of the following is the vector eld of F x y z hx y 3i 3 4 function Recall from Chapter 14 the gradient of a scalar function of two variables f x y is the vector valued For a scalar function of three variables f x y z we have rf x y hfx x y fy x y i rf x y z hfx x y z fy x y z fz x y z i Example 4 Find the gradient of f x y ex2y 2y Example 5 Find the gradient of f x y z px2 y2 z2
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