Section 6.3 Curl

6.3 Curl

Recall that when $\mathbf{F}(x,y)=<P,Q>$, we can check if $\mathbf{F}$ is conservative by checking the equality $\frac{\partial Q}{\partial x}=\frac{\partial P}{\partial y}$. How about if $\mathbf{F}(x,y,z)=<P,Q,R>$ a 3-dimensional vector space?

The answer is curl of $\mathbf{F}$. 

Definition: Curl

If $\mathbf{F}(x,y,z)=<P,Q,R>$ is a vector field in ${\mathbb{R}}^3$, and partial derivative of $P$, $Q$, $R$ all exist, then the curl of $\mathbf{F}$ is defined by $$\text{curl}\mathbf{F}=<\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z},\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x},\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}>=\left|\begin{array}{ccc}\overrightarrow{i}& \overrightarrow{j}& \overrightarrow{k}\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ P& Q& R\end{array}\right|.$$

Example 1: Find the curl of $\mathbf{F}(x,y,z)=<x^{2}y,e^y+yz,xyz>.$

Exercise 1: Find the curl of $\mathbf{F}(x,y,z)=<x{y}^2,y{z}^2,x{z}^2>.$

Example 2: Find the curl of $\mathbf{F}(x,y,z)=xy{e}^z\overrightarrow{i}+yz{e}^x\overrightarrow{k}.$

Exercise 2: Find the curl of $\mathbf{F}(x,y,z)=x{e}^y\overrightarrow{j}+z{e}^y\overrightarrow{k}.$

Example 3: Find the curl of $\mathbf{F}(x,y,z)=<e^x\text{sin}(y),e^y\text{sin}(z),e^z\text{sin}(x)>.$

Exercise 3: Find the curl of $\mathbf{F}(x,y,z)=<e^y\text{cos}(z),e^x\text{cos}(y),e^z\text{cos}(x)>.$

Theorem:

If $f$ is a function of three variables that has continuous second order partial derivatives, then 

$$\text{curl}(\mathrm{\nabla }f)=\overrightarrow{0}.$$

The above theorem shows that if $\mathbf{F}$ is conservative then $\text{curl}\mathbf{F}=\overrightarrow{0}.$ This is not true if $\mathbf{F}$ is  not defined everywhere! 

Theorem:

If $\mathbf{F}$ is a vector field defined on all of ${\mathbb{R}}^3$ whose component functions have continuous partial derivatives and $\text{curl}\mathbf{F}=\overrightarrow{0},$ then $\mathbf{F}$ is a conservative vector field. 

Example 4: Decide if $\mathbf{F}=<y^2{z}^3,2xy{z}^3,3x{y}^2{z}^2>$ is conservative. If it is conservative, find $f$ such that $\mathrm{\nabla }f=\mathbf{F}$. 

Exercise 4: Decide if $\mathbf{F}=<2x{y}^{3}z,3x^2{y}^{2}z,x^2{y}^3>$ is conservative. If it is conservative, find $f$ such that $\mathrm{\nabla }f=\mathbf{F}$. 

Example 5: Decide if $\mathbf{F}=<e^{yz},xz{e}^{yz},xy{e}^{yz}>$ is conservative. If it is conservative, find $f$ such that $\mathrm{\nabla }f=\mathbf{F}$. 

Exercise 5: Decide if $\mathbf{F}=<y^{2}z{e}^{xz},2y{e}^{xz},x{y}^2{e}^{xz}>$ is conservative. If it is conservative, find $f$ such that $\mathrm{\nabla }f=\mathbf{F}$. 

Group work:

1. Decide if $\mathbf{F}=<xyz,y,x>$ is conservative. If it is conservative, find $f$ such that $\mathrm{\nabla }f=\mathbf{F}$. 

2. Decide if $\mathbf{F}=<e^{-xy},e^{xz},e^{yz}>$ is conservative. If it is conservative, find $f$ such that $\mathrm{\nabla }f=\mathbf{F}$.

3. Decide if $\mathbf{F}=e^x\text{sin}(y)\overrightarrow{i}+e^x\text{cos}(y)\overrightarrow{j}$ is conservative. If it is conservative, find $f$ such that $\mathrm{\nabla }f=\mathbf{F}$.

4. Decide if $\mathbf{F}=<3x^{2}y+3z,x^3,3x+3z^2>$ is conservative. If it is conservative, find $f$ such that $\mathrm{\nabla }f=\mathbf{F}$.

5. Decide if $\mathbf{F}=<e^x+3x^{2}y,x^3+4y^3,1>$ is conservative. If it is conservative, find $f$ such that $\mathrm{\nabla }f=\mathbf{F}$.