• Calculating Curl

  

  • We can use this helpful cycle to remember how to calculate the cross product! For the $\hat i$ component you multiply first the $\hat j$ component of the first vector times the $\hat k$ component of the second vector minus the $\hat k$ component of the first vector times the $\hat j$ component of the second vector. We repeat for the $\hat j$ component of the cross product vector.
    • The equation for each component $(\text{curl}~F)_i$ can be obtained by exchanging each occurrence of a subscript 1, 2, 3 in cyclic permutation: 1 → 2, 2 → 3, and 3 → 1 (where the subscripts represent the relevant indices). So if we represent $\vec\nabla=(\partial_1,\partial_2,\partial_3)$ and $\vec F=(F_1,F_2,F_3)$ then $(\text{curl}~F)_i=\partial_j F_k-\partial _k F_j$ where $i\rightarrow j$ and $j\rightarrow k$
  • Cartesian:

  $$ \vec\nabla\times \vec f=\bigg(\frac{\partial f_z}{\partial y}-\frac{\partial f_y}{\partial z}\bigg)\hat i+\bigg(\frac{\partial f_x}{\partial z}-\frac{\partial f_z}{\partial x}\bigg)\hat j+\bigg(\frac{\partial f_y}{\partial x}-\frac{\partial f_x}{\partial y}\bigg)\hat k $$

  • Spherical

    $$ \vec\nabla\times \vec f=\frac{1}{r\sin\theta}\bigg(\frac{\partial (\sin\theta f_\phi)}{\partial \theta}-\frac{\partial f_\theta}{\partial \phi}\bigg)\hat e_r\\

    ~~~~~~+\\frac1r\\bigg(\\frac{\\partial (rf_r)}{\\partial r}-\\frac{\\partial f_r}{\\partial \\theta}\\bigg)\\hat e_\\phi
    $$
    
    

  • Cylindrical

    $$ \vec\nabla\times \vec f=\frac{1}{r}\bigg(\frac{\partial f_z}{\partial \theta}-\frac{\partial f_\theta}{\partial z}\bigg)\hat e_r+\bigg(\frac{\partial f_r}{\partial z}-\frac{\partial f_z}{\partial r}\bigg)\hat e_\theta+\frac1r\bigg(\frac{\partial (rf_\theta)}{\partial z}-\frac{\partial f_r}{\partial \theta}\bigg)\hat e_z $$

  • For any curvilinear orthogonal coordinates in $\R^3$

• Concept

  The curl is a measure of the circulation at a point. In essence it’s the circulation of a region in the limit of infinitesimal area. It’s an operation defined in $\R^3$ that takes a vector field $\vec F$ and gives a vector $\vec\nabla\times\vec F$ that measures the amount of circulation about a point. This gives us the amount that a vector field “rotates” about a point and points in the direction normal to this rotation

  Consider the circulation of a region

  $$ \oint_C\vec F\cdot d\vec r=\oint_CF_{||}dr\approx \braket{ F_{||}}L $$

  →The circulation can be thought of as a measure of the average amount of $F$ along $C$ where $L$ is the length of the path $C$ and $\braket{F_{||}}$ is the average amount of $F$ along $C$

  As we shrink the surface area enclosed by $C$ to an infinitesimal point we obtain the magnitude of curl. More explicitly, the curl, $\vec\nabla\times\vec F$, defined at a point $p$ is the vector whose projection along the vector $\hat u$ normal to the surface bounded by $C$ is the circulation:

  $$ (\vec\nabla\times\vec F)(p)\cdot\hat u\equiv\lim_{A\rightarrow 0}\frac1{|A|}\oint_C\vec F\cdot d\vec r $$

• Stoke’s Theorem

Just like the connection between divergence and flux integrals there is a connection between curl and circulation integrals

$$ \oint_C\vec F\cdot d\vec r=\int_S(\vec\nabla\times\vec F)\cdot d\vec A $$

Where $d\vec A$ is the area vector that points normal to the surface $S$ with a boundary defined by the closed path $C$ with a magnitude $d\vec A=|\frac{\partial \vec r}{\partial u}\times\frac{\partial\vec r}{\partial v}|dudv ~\hat n$ where $\vec r(u,v)$ is a parameterization of the surface $S$ and $\vec n=\pm\frac{\partial \vec r}{\partial u} \times\frac{\partial\vec r}{\partial v}$

Note: by convention we define a counterclockwise oriented closed path $C$ witha normal direction given by the RHR. If the path is oriented clockwise then we must introduce a negative sign

The statement of Stokes Theorem is in essence

$$ \braket{F_{||}}CL=\braket{(\vec\nabla\times\vec F)\perp} A\\ \braket{F_{||}}_CL=\braket{(\vec\nabla\times\vec F)\cdot \hat n} A $$

The average amount of curl in the normal direction of the surface bounded by $C$ times the area of the surface is equal to the average amount of $\vec F$ along $C$ times the length of the closed path $C$, $L$.

• Proof (Courtesy of Dr. Mewes)
• Conservative Fields

We know that for a conservative field, $\vec f$, for any closed path:

$$ \oint_C\vec f\cdot d\vec r=0 $$

By Stoke’s Theorem

$$ \oint_C\vec f\cdot d\vec r=\int_S(\vec\nabla\times\vec f)\cdot d\vec A=0 $$

Since $S$ is arbitrary, then for this to always be true than $\vec\nabla\times\vec f$ must be $0$

This provides a new condition for conservative fields:

<aside> 💡 If $\vec f$ is conservative $\iff$$\vec\nabla\times\vec f=0$

</aside>