Vector Fields

In plane $R^2$, $\vec{F}(x,y)=<P(x,y),Q(x,y)>=P\hat{i}+Q\hat{j}$

In plane $R^3$, $\vec{F}(x,y)=<P(x,y,z),Q(x,y,z),R(x,y,z)>=P\hat{i}+Q\hat{j}+R\hat{k}$

Example 1 §

Sketch a vector field: $\vec{F}(x,y)=y\hat{i}-x\hat{j}$

$$\begin{array}{rl}\vec{F}(0,0)& =<0,0>\\ \vec{F}(1,0)& =<0,-1>\\ \vec{F}(-1,0)& =<0,1>\\ \vec{F}(0,1)& =<1,0>\end{array}$$

It’s a rotational field.

Example 2 §

Sketch the vector field: $\vec{F}=-x\hat{i}-y\hat{j}$

$$\begin{array}{r}\vec{F}(0,0)=<0,0>\\ \vec{F}(-1,0)=<1,0>\\ \vec{F}(1,0)=<-1,0>\\ \vec{F}(1,1)=<-1,-1>\end{array}$$

Radial field that points to center.

Conservative Vector Field §

Conservative vector fields are basically graidents.

Given $f(x,y)=x^2+y^2$, vector field is $\vec{F}=\nabla f(x,y)=<2x,2y>$

Definition §

A vector field $\vec{F}$ is a conservative vector field if there exists a differential function $f$ such that $\nabla f=\vec{F}$, where $f$ is a potential function

Example 3 §

Given $f(x,y)=3x^{2}y+4x{y}^2$, find the conservative vector field.

$$\begin{array}{c}\vec{F}=\nabla f=<f_x,f_y>=<6xy+4y^2,3x^2+8xy>\\ f_{xy}=6x+8y=f_{yx}=6x+8y\end{array}$$

Test for Conservative Vector Field §

$\vec{F}(x,y)=P\hat{i}+Q\hat{j}$ is a conservative vector field $⟺$ their cross partial derivatives are equal.

$$\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}$$

Example 4 §

Determine whether the following is conservative:

$$\begin{array}{rl}\vec{F}(x,y)& =2xy\hat{i}+(2x^2-y)\hat{j}\\ \frac{\partial P}{\partial y}& =2x\\ \frac{\partial Q}{\partial x}& =4x\end{array}$$

$2x\ne 4x⟹$ not conservative.

3-D vector fields §

In $R^3$, if $\vec{F}(x,y,z)$ = $<P,Q,R>$ is conservative, then

$$\begin{array}{c}\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x},\frac{\partial Q}{\partial z}=\frac{\partial R}{\partial y},\frac{\partial R}{\partial x}=\frac{\partial P}{\partial z}\end{array}$$

Finding potential functions §

Example 5 §

Find potential function for $\vec{F}(x,y)=2xy\hat{i}+(x^2-y)\hat{j}$

$$\begin{array}{c}\int f_{x}dx=\int 2xydx⟹f(x,y)=x^{2}y+g(y)\\ \int f_{y}dy=\int x^2-ydy⟹f(x,y)=x^{2}y-\frac{y^2}{2}+h(x)\\ ⟹g(y)=\frac{y^2}{2},C=h(x)\\ ⟹f(x,y)=x^2-\frac{y^2}{2}+c\end{array}$$

We can also use partial derivative of final answer and see if we get original field to check our answer.

Example 6 §

Find potential function of $\vec{F}(x,y)=<4x-y,6y-x>$

$$\begin{array}{c}\int f_{x}dx=\int 4x-ydx=2x^2-xy+g(y)\\ \int f_{y}dy=\int 6y-xdy=3y^2-xy+h(x)\\ ⟹g(y)=3y^2,h(x)=2x^2\\ ⟹2x^2+3y^2-xy+C\end{array}$$