Section 6.3 Curl
6.3 Curl
Recall that when [latex]\mathbf{F}(x,y)=\lt P,Q\gt[/latex], we can check if [latex]\mathbf{F}[/latex] is conservative by checking the equality [latex]\frac{\partial Q}{\partial x}=\frac{\partial P}{\partial y}[/latex]. How about if [latex]\mathbf{F}(x,y,z)=\lt P,Q,R\gt[/latex] a 3-dimensional vector space?
The answer is curl of [latex]\mathbf{F}[/latex].
Definition: Curl
If [latex]\mathbf{F}(x,y,z)=\lt P,Q,R\gt[/latex] is a vector field in [latex]\mathbb{R}^{3}[/latex], and partial derivative of [latex]P[/latex], [latex]Q[/latex], [latex]R[/latex] all exist, then the curl of [latex]\mathbf{F}[/latex] is defined by \[\text{curl}\mathbf{F} =\lt \frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z},\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x},\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\gt =\left| \begin{array}{ccc} \overrightarrow{i} & \overrightarrow{j} & \overrightarrow{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R\end{array}\right|.\]
Example 1: Find the curl of [latex]\mathbf{F}(x,y,z)=\lt x^{2}y,e^{y}+yz,xyz\gt .[/latex]
Exercise 1: Find the curl of [latex]\mathbf{F}(x,y,z)=\lt xy^{2},yz^{2},xz^{2}\gt .[/latex]
Example 2: Find the curl of [latex]\mathbf{F}(x,y,z)=xye^{z}\overrightarrow{i}+yze^{x}\overrightarrow{k}.[/latex]
Exercise 2: Find the curl of [latex]\mathbf{F}(x,y,z)=xe^{y}\overrightarrow{j}+ze^{y}\overrightarrow{k}.[/latex]
Example 3: Find the curl of [latex]\mathbf{F}(x,y,z)=\lt e^{x}\text{sin}(y),e^{y}\text{sin}(z),e^{z}\text{sin}(x)\gt .[/latex]
Exercise 3: Find the curl of [latex]\mathbf{F}(x,y,z)=\lt e^{y}\text{cos}(z),e^{x}\text{cos}(y),e^{z}\text{cos}(x)\gt .[/latex]
Theorem:
If [latex]f[/latex] is a function of three variables that has continuous second order partial derivatives, then
\[\text{curl}(\nabla f)=\overrightarrow{0}.\]
The above theorem shows that if [latex]\mathbf{F}[/latex] is conservative then [latex]\text{curl}\mathbf{F}=\overrightarrow{0}.[/latex] This is not true if [latex]\mathbf{F}[/latex] is not defined everywhere!
Theorem:
If [latex]\mathbf{F}[/latex] is a vector field defined on all of [latex]\mathbb{R}^{3}[/latex] whose component functions have continuous partial derivatives and [latex]\text{curl}\mathbf{F}=\overrightarrow{0},[/latex] then [latex]\mathbf{F}[/latex] is a conservative vector field.
Example 4: Decide if [latex]\mathbf{F}=\lt y^{2}z^{3},2xyz^{3},3xy^{2}z^{2}\gt[/latex] is conservative. If it is conservative, find [latex]f[/latex] such that [latex]\nabla f=\mathbf{F}[/latex].
Exercise 4: Decide if [latex]\mathbf{F}=\lt 2xy^{3}z,3x^{2}y^{2}z,x^{2}y^{3}\gt[/latex] is conservative. If it is conservative, find [latex]f[/latex] such that [latex]\nabla f=\mathbf{F}[/latex].
Example 5: Decide if [latex]\mathbf{F}=\lt e^{yz},xze^{yz},xye^{yz}\gt[/latex] is conservative. If it is conservative, find [latex]f[/latex] such that [latex]\nabla f=\mathbf{F}[/latex].
Exercise 5: Decide if [latex]\mathbf{F}=\lt y^{2}ze^{xz},2ye^{xz},xy^{2}e^{xz}\gt[/latex] is conservative. If it is conservative, find [latex]f[/latex] such that [latex]\nabla f=\mathbf{F}[/latex].
Group work:
1. Decide if [latex]\mathbf{F}=\lt xyz,y,x\gt[/latex] is conservative. If it is conservative, find [latex]f[/latex] such that [latex]\nabla f=\mathbf{F}[/latex].
2. Decide if [latex]\mathbf{F}=\lt e^{-xy},e^{xz},e^{yz}\gt[/latex] is conservative. If it is conservative, find [latex]f[/latex] such that [latex]\nabla f=\mathbf{F}[/latex].
3. Decide if [latex]\mathbf{F}=e^{x}\text{sin}(y)\overrightarrow{i}+e^{x}\text{cos}(y)\overrightarrow{j}[/latex] is conservative. If it is conservative, find [latex]f[/latex] such that [latex]\nabla f=\mathbf{F}[/latex].
4. Decide if [latex]\mathbf{F}=\lt 3x^{2}y+3z,x^{3},3x+3z^{2}\gt[/latex] is conservative. If it is conservative, find [latex]f[/latex] such that [latex]\nabla f=\mathbf{F}[/latex].
5. Decide if [latex]\mathbf{F}=\lt e^{x}+3x^{2}y,x^{3}+4y^{3},1\gt[/latex] is conservative. If it is conservative, find [latex]f[/latex] such that [latex]\nabla f=\mathbf{F}[/latex].