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# Lecture 24: Curl ## Curl The curl of a vector field $\vec{F} = \langle P, Q, R \rangle$ is the vector field: $\qquad \text{curl } \vec{F} = \langle R_y - Q_z, P_z - R_x, Q_x - P_y \rangle$ or $\qquad \text{curl } \vec{F} = \nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\...

# Lecture 24: Curl ## Curl The curl of a vector field $\vec{F} = \langle P, Q, R \rangle$ is the vector field: $\qquad \text{curl } \vec{F} = \langle R_y - Q_z, P_z - R_x, Q_x - P_y \rangle$ or $\qquad \text{curl } \vec{F} = \nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix} = \langle R_y - Q_z, P_z - R_x, Q_x - P_y \rangle$ ### Example 1 Find the curl of $\vec{F} = \langle xz, x y z, -y^2 \rangle$ $\qquad \text{curl } \vec{F} = \langle -2y - xy, x - 0, yz - z \rangle = \langle -2y - xy, x, yz - z \rangle$ ### Example 2 Find the curl of $\vec{F} = \langle x \sin z, y \sin x, z \sin y \rangle$ $\qquad \begin{aligned} \text{curl } \vec{F} &= \langle z \cos y - x \cos x, x \cos z - 0, y \cos x - 0 \rangle \\ &= \langle z \cos y - x \cos x, x \cos z, y \cos x \rangle \end{aligned}$ ## Theorem If $f(x, y, z)$ is a $C^2$ function then $\text{curl } (\nabla f) = \vec{0}$ ### Proof $\qquad \begin{aligned} \nabla f &= \langle f_x, f_y, f_z \rangle \\ \text{curl } (\nabla f) &= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ f_x & f_y & f_z \end{vmatrix} \\ &= \langle f_{zy} - f_{yz}, f_{xz} - f_{zx}, f_{yx} - f_{xy} \rangle \\ &= \langle 0, 0, 0 \rangle = \vec{0} \quad \text{by Clairaut's Theorem} \end{aligned}$ ## Theorem If $\vec{F}$ is a $C^2$ vector field, then $\text{div}(\text{curl } \vec{F}) = 0$ ### Proof $\qquad \begin{aligned} \vec{F} &= \langle P, Q, R \rangle \\ \text{curl } \vec{F} &= \langle R_y - Q_z, P_z - R_x, Q_x - P_y \rangle \\ \text{div}(\text{curl } \vec{F}) &= (R_y - Q_z)_x + (P_z - R_x)_y + (Q_x - P_y)_z \\ &= R_{yx} - Q_{zx} + P_{zy} - R_{xy} + Q_{xz} - P_{yz} \\ &= 0 \quad \text{by Clairaut's Theorem} \end{aligned}$ ## Physical Interpretation of Curl Imagine a cork placed in a fluid flow. The cork will rotate if the circulation around it is non-zero. The curl measures the tendency of the fluid to rotate around a point. If $\text{curl } \vec{F} = \vec{0}$, then the fluid flow is irrotational. ## Example 3 Is $\vec{F} = \langle xz, x y z, -y^2 \rangle$ irrotational? $\qquad \text{curl } \vec{F} = \langle -2y - xy, x, yz - z \rangle \neq \vec{0}$ No, $\vec{F}$ is not irrotational. ## Example 4 Is $\vec{F} = \langle y^2 z^3, 2xy z^3, 3xy^2 z^2 \rangle$ irrotational? $\qquad \begin{aligned} \text{curl } \vec{F} &= \langle 6xyz^2 - 6xyz^2, 3y^2 z^2 - 3y^2 z^2, 2yz^3 - 2yz^3 \rangle \\ &= \langle 0, 0, 0 \rangle = \vec{0} \end{aligned}$ Yes, $\vec{F}$ is irrotational. ## Theorem $\vec{F}$ is conservative if and only if $\text{curl } \vec{F} = \vec{0}$