5.2 Vector Fields

Vector Fields

Introduction

Examples of Vector Fields

Mathematical Vector Fields

Plotting Vector Fields

Gradient Vector Fields

It's common when discussing weather to talk about the wind speed, but as any student who has gotten this far in the text will know, this nomenclature is imprecise. It's not terribly helpful to tell someone the wind is blowing at 10 without telling them the direction in which the wind is blowing. If you're trying to make a decision based on what the wind is doing, you need to know about the direction as well. For instance, if you are taking off in a hot air balloon, the wind direction will determine which direction the chase team should start going to keep track of you. Because of the swirling nature of wind, it makes sense to give the wind velocity at each point in a region (two-dimensional or three-dimensional).

Suppose that given a point $(x,y)$ in the plane, you know that the wind velocity at that point is given by the vector $\langle y,x\rangle$. For example, we'd then know that at the point $(1,-1)$, the wind velocity is $\langle -1,1\rangle$. We will give the wind velocity as a function $\vF$, where $\vF(x,y) = \langle y,x\rangle$. In the table below, fill in the wind velocity vectors for the given points.

$(x,y)$ | $(2,1)$ | $(0,0)$ | $(-1,2)$ | $(3,-1)$ | $(-2,-1)$

$\vF(x,y)$ | | | | |

Starting with one of the vectors near the point $(2,0)$, sketch a curve that follows the direction of the vector field $\vF$. To help visualize what you are doing, it may be useful to think of the vector field as the velocity vector field for some flowing water and that you are imagining tracing the path that a tiny particle inserted into the water would follow as the water moves it around.

Repeat the previous step for at least two other starting points not on the curve you previously sketched.

What shape do the curves you sketched in the previous two steps form?

Verify that $\vF(x,y)$ is orthogonal to $\langle x,y\rangle$.

Calculate the gradient of the function $f(x,y) = x^2 + y^2$ and write a sentence comparing your result to the vector $x\vi + y\vj$.

Write a sentence describing the geometric relationship between $\vF(x,y)$ and a circle centered at the origin. What is the relationship between $\vecmag{\vF(x,y)}$ and the radius of that circle?

In there are three sets of axes showing level curves for functions $f$, $g$, and $h$, respectively. Sketch at least six vectors in the gradient vector field for each function. In making your sketches, you don't have to worry about getting vector magnitudes precise, but you should ensure that the relative magnitudes (and directions) are correct for each function independently.

Verify that $\vF(x,y) = \langle 6xy,3x^2+9\sqrt{y}\rangle$ is a gradient vector field by finding a function $f$ such that $\nabla f(x,y) = \vF(x,y)$. For reasons originating in physics, such a function $f$ is called a potential function for the vector field $\vF$.

Is the function $f$ found in part unique? That is, can you find another function $g$ such that $\nabla g(x,y)= \vF(x,y)$ but $f\neq g$?

Is the vector field $\vF(x,y) = 6xy\vi +(2x+9\sqrt{y})\vj$ a gradient vector field? Why or why not?