Vector Fields

Vector Fields Let D be a plane region. A vector field on 𝐑 2 is a vector function 𝐅 𝑥,𝑦 =𝑃 𝑥,𝑦 𝐢+𝑄 𝑥,𝑦 𝐣= 𝑃 𝑥,𝑦 ,𝑄(𝑥,𝑦) that assigns to each point 𝑥,𝑦 in D the vector 𝐅(𝑥,𝑦). Example: Wind map (velocity vector field) (here we have a map of Japan) Example: Ocean currents ( velocity vector field)

Vector Fields We can illustrate a vector field by plotting at each point a vector with its tail at (𝑥,𝑦). Example: A constant field : 𝐅 𝑥,𝑦 =2𝐢+𝐣 Example: A radial field : 𝐅 𝑥,𝑦 =𝑥𝐢+𝑦𝐣 Note that not all vectors are graphed and the vectors are scaled to fit the picture.

Vector Fields A circulating vector field: 𝐅 𝑥,𝑦 = 𝑦,−𝑥 In the first quadrant x ≥ 0, y ≥ 0, thus F = <+, −> and the vectors have the general orientation: In the second quadrant x ≤ 0, y ≥ 0, thus F = <+, +> and the vectors have the general orientation: In the third quadrant x ≤ 0, y ≤ 0, thus F = <−, +> and the vectors have the general orientation: In the fourth quadrant x ≥ 0, y ≤ 0, thus F = <−, − > and the vectors have the general orientation: Magnitude of 𝐅= 𝑥 2 + 𝑦 2 The magnitude is constant along circles. For instance, all the vectors starting along the circle x2 +y2 =16 have magnitude 4. The vector 𝑦,−𝑥 is orthogonal to the position vector 𝐫= 𝑥,𝑦 , so all the vectors in the vector field are tangent to the corresponding circles.

Vector Fields Match the vector field with its graph: Graph A Graph B Graph C Graph D Answer: 1- C, 2- D, 3-A, 4-B

Vector Fields When we take the gradient of a function 𝑓(𝑥,𝑦), we obtain a gradient field. The vectors in this field are perpendicular to the level curves 𝑓 𝑥,𝑦 =𝑐. Example: 𝑓 𝑥,𝑦 =𝑦− 𝑥 2 . The gradient field is 𝐹 𝑥,𝑦 =𝛻𝑓 𝑥,𝑦 = −2𝑥,1 Level curves: y = x2 + c We will graph the level curves for c = 0 and c =2 together with a few gradient vectors. NOTE: The vectors in the gradient field are orthogonal to the level curves. The vectors are long where the level curves are close to each other and short where they are far apart. This is because the length of the gradient field is the value of the directional derivative of f and close level curves indicate a steep graph.

Vector Fields Let 𝐅 𝑥,𝑦 =𝛻𝑓(𝑥,𝑦) for some function f, that is, let F be the gradient field of some function f. We call f a potential function for F and we say F is a conservative vector field. For instance, 𝑓 𝑥,𝑦 =𝑦− 𝑥 2 is a potential function for 𝐅 𝑥,𝑦 = −2𝑥,1 and F is a conservative vector field. Not all vector fields are conservative, that is, not all vector fields are gradient fields.

Vector Fields Match the gradient field of a function 𝑓(𝑥,𝑦) to the contour plot of the function Graph A Graph B Graph C Contour I Contour II Contour III

Vector Fields Gradients are orthogonal to level curves. Answer: A-II, B-III, C-I Graph A Graph B Graph C Contour I Contour II Contour III

Vector Fields A vector field in 𝐑 3 is a vector function 𝐅 𝑥,𝑦,𝑧 =𝑃 𝑥,𝑦,𝑧 𝐢+𝑄 𝑥,𝑦,𝑧 𝐣+𝑅 𝑥,𝑦,𝑧 𝐤 that assigns to each point (x, y, z) the vector 𝐅(𝑥,𝑦,𝑧). Example: 𝐅 𝑥,𝑦,𝑧 =−𝑧𝐤 The magnitude of the arrows increases with the distance from the 𝑥𝑦-plane. All the vectors point toward the 𝑥𝑦-plane.