Section 5.1 Vector Fields

5.1 Vector Fields

This chapter we focus on learning the vector fields and its applications. Vector fields have many applications because they can be used to model real fields such as electromagnetic or gravitational fields. A deep understanding of physics or engineering is impossible without an understanding of vector fields. Here is a web based calculator for drawing vector field. 

https://www.wolframalpha.com/input/?i=vector+field+plot

Definition:

 A vector field [latex]\mathbf{F}[/latex] in [latex]\mathbb{R}^{2}[/latex] is an assignment of a two-dimensional vector [latex]\mathbf{F}(x,y)[/latex] to each point [latex](x,y)[/latex] of a subset [latex]D[/latex] of [latex]\mathbb{R}^{2}[/latex]. The subset [latex]D[/latex] is the domain of the vector field. We often write [latex]\mathbf{F}(x,y)=\lt P(x,y),Q(x,y)\gt =P(x,y)\overrightarrow{i}+Q(x,y)\overrightarrow{j}[/latex].

A vector field [latex]\mathbf{F}[/latex] in [latex]\mathbb{R}^{3}[/latex] is an assignment of a three-dimensional vector [latex]\mathbf{F}(x,y,z)[/latex]to each point [latex](x,y,z)[/latex] of a subset [latex]D[/latex] of [latex]\mathbb{R}^{3}[/latex]. The subset [latex]D[/latex] is the domain of the vector field.[latex]\mathbf{F}(x,y,z)=\lt P(x,y,z),Q(x,y,z),Q(x,y,z)\gt =P(x,y,z)\overrightarrow{i}+Q(x,y,z)\overrightarrow{j}+R(x,yz)\overrightarrow{k}.[/latex]

Example 1: Sketch the vector field [latex]\mathbf{F}(x,y)=\lt y,-x\gt .[/latex]

Exercise 1: Sketch the vector field [latex]\mathbf{F}(x,y)=\lt x,y\gt .[/latex]

Example 2: Sketch the vector field [latex]\mathbf{F}(x,y)=\lt -x,2y\gt .[/latex]

Exercise 2: Sketch the vector field [latex]\mathbf{F}(x,y)=\lt -y,2x\gt .[/latex]

Example 3: Describe vector field [latex]\mathbf{F}(x,y,z)=\lt 1,1,z\gt[/latex] .

Exercise 3: Describe vector field [latex]\mathbf{F}(x,y,z)=\lt x,1,1\gt[/latex] .

Recall that if [latex]f[/latex] is a (scalar) function of [latex]x[/latex] and [latex]y[/latex], then the gradient of [latex]f[/latex] is grad[latex]f[/latex] [latex]=\nabla f=\lt f_{x}(x,y),f_{y}(x,y)\gt .[/latex]

Definition

A vector field [latex]\mathbf{F}[/latex] in [latex]\mathbb{R}^{2}[/latex] or in [latex]\mathbb{R}^{3}[/latex] is a gradient field if there exists a scalar function [latex]f[/latex] such that [latex]\nabla f=\mathbf{F}.[/latex] We call [latex]f[/latex] a potential function of [latex]\mathbf{F}[/latex] and we say [latex]\mathbf{F}[/latex] is conservative. 

Example 4: Is [latex]f(x,y,z)=xy^{2}z-\text{cos}(xy)[/latex] a potential function for vector field [latex]\mathbf{F}(x,y,z)=\lt y^{2}z+y\text{sin}(xy),2xyz+x\text{sin}(xy),xy^{2}\gt[/latex]?

Exercise 4: [latex]f(x,y,z)=x^{2}\text{sin}(yz)+y^{2}z[/latex] a potential function for [latex]\mathbf{F}(x,y,z)=\lt 2x\text{sin}(yz),x^{2}z\text{cos}(yz)+2yz,y^{2}\gt[/latex] ?

Theorem: Uniqueness of Potential Functions 

Let [latex]\mathbf{F}[/latex] be a conservative vector field on an open and connected domain and let [latex]f[/latex] and [latex]g[/latex] be functions such that [latex]\nabla f=\mathbf{F}[/latex] and [latex]\nabla g=\mathbf{F}.[/latex] Then, there is a constant [latex]C[/latex] such that [latex]f=g+C.[/latex]

Theorem: The Cross-Partial Property of Conservative Vector Fields

Let [latex]\mathbf{F}[/latex] be a vector field in two or three dimensions such that the component functions of [latex]\mathbf{F}[/latex] have continuous second-order mixed-partial derivatives on the domain of [latex]\mathbf{F}[/latex].  If [latex]\mathbf{F}(x,y)=\lt P(x,y),Q(x,y)\gt[/latex] is a conservative vector field in [latex]\mathbb{R}^{2}[/latex], then [latex]\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}[/latex]. If [latex]\mathbf{F}(x,y,z)=\lt P(x,y,z),Q(x,y,z),R(x,y,z)\gt[/latex] is a conservative vector field in [latex]\mathbb{R}^{3}[/latex], then 

\[\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x},\text{}\frac{\partial Q}{\partial z}=\frac{\partial R}{\partial y},\text{ and }\frac{\partial R}{\partial x}=\frac{\partial P}{\partial z}\]

Remark: The theorem can only help us to decide if a vector field is not conservative. We will learn how to find the potential function and decide if a vector field is conservative later.

Example 5: Is [latex]\mathbf{F}(x,y)=\lt y^{2},-xy\gt[/latex] conservative? 

Exercise 5: Show [latex]\mathbf{F}(x,y)=\lt xy,-x^{2}y\gt[/latex] is not conservative.

Example 6: Is [latex]\mathbf{F}(x,y)=\lt xy,2,z\gt[/latex] conservative? 

Exercise 6: Is [latex]\mathbf{F}(x,y)=\lt y,x,xyz\gt[/latex] conservative? 

Example 7: A particle moves in a velocity field [latex]V(x,y)=\lt xy,x^{2}-y^{2}\gt[/latex]. If it is at a position [latex](1,2)[/latex] at time [latex]t=2[/latex]. Estimate its location at time [latex]t=2.01[/latex]. 

Group work:

1. A particle moves in a velocity field [latex]V(x,y)=\lt y^{2},x^{2}+y\gt[/latex]. If it is at a position [latex](2,1)[/latex] at time [latex]t=1[/latex]. Estimate its location at time [latex]t=1.01[/latex]. 

2. Find the gradient vector field of each function [latex]f[/latex].

(a) [latex]f(x,y)=x\text{sin}(y)+\text{tan}(y)[/latex]

(b) [latex]f(x,y,z)=ze^{-x^{2}y}[/latex]

(c) [latex]f(x,y,z)=x^{2}y-xy^{2}+y^{2}z[/latex]

(d) [latex]f(x,y)=x^{2}\text{sin}(3y)[/latex]

(e) [latex]f(x,y,z)=x\text{sin}(\frac{z}{y})[/latex]