Math251_Fall2023_section16_1

© Amy Austin, November 6, 2023 Section 16.1 Vector Fields Definition: A vector field , F , in two dimension is a function F that assigns to each point ( x, y ) in D ⊂ IR 2 a two dimensional vector, F ( x, y ), as shown below. Definition: A vector field , F , in three dimension is a function F that assigns to each point ( x, y, z ) in D ⊂ IR 3 a three dimensional vector, F ( x, y, z ), as shown below. Example 1: Which of the following vector fields matches the plot below? a.) F ( x, y ) = ⟨− x, − y ⟩ b.) F ( x, y ) = ⟨ x, y ⟩ c.) F ( x, y ) = x x 2 + 1 , y x 2 + 1 d.) F ( x, y ) = * − x p x 2 + y 2 , − y p x 2 + y 2 + 1

© Amy Austin, November 6, 2023 Example 2: Which of the following plots can be eliminated as the vector field for F ( x, y ) = y i + 1 2 j ? A common process of elimination: Choose a point in all four quadrants to test the direction F . 2