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

$$\text{curl} \overrightarrow{F} = \nabla \times \overrightarro... ...} & \frac {\partial}{\partial z} \\ M & N & P \end{vmatrix}$$

i.e.

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

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:

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

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

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