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The curl of a vector field in the space.

Let $\overrightarrow{F} (x,y,z)= M(x,y,z) \overrightarrow{i} +
N((x,y,z) \overrightarrow{j} + P(x,y,z) \overrightarrow{k} $ be a vector filed in the 3-dimensional space.

Definition 6.35   The curl of the vector field $\overrightarrow{F} $ is the symbolic cross-product

\begin{displaymath}\text{curl} \overrightarrow{F} = \nabla \times \overrightarro...
& \frac {\partial}{\partial z} \\
M & N & P


\begin{displaymath}\text{curl} \overrightarrow{F} =
\left( \frac {\partial P}{\p...
...- \frac {\partial M}{\partial y} \right)
\overrightarrow{k} .

Example 6.36   Let $\overrightarrow{F} (x,y,z) =\underbrace{x^2yz}_{M(x,y,z)}\overrightarrow{i} +\u...
..._{N(x,y,z)}\overrightarrow{j} +\underbrace{xyz^2}_{N(x,y,z)}\overrightarrow{k} $. Then:

\begin{align*}\frac {\partial M}{\partial y} = x^2z & \frac {\partial M}{\partia...
...tial P}{\partial x} = yz^2 & \frac {\partial P}{\partial y} = xz^2

\begin{displaymath}\text{curl} \overrightarrow{F} = (xz^2 - xy^2) \overrightarro...
...yz^2) \overrightarrow{j} + (y^2z - x^2z ) \overrightarrow{k} .

More on this topic will be studied in Section  6 about Stoke's theorem.

Noah Dana-Picard