Curl Microscopic rotation or spin of a vector field at a specific point. It tells how much, and in what direction, the field is swirling around that exact location.
Positive or negative curl: Rotational
Zero curl: Irrotational
Let a vector:
V ⃗ = V x e ^ x + V y e ^ y + V z e ^ z \vec{V} = V_x\,\hat{e}_x + V_y\,\hat{e}_y + V_z\,\hat{e}_z V = V x e ^ x + V y e ^ y + V z e ^ z ∴ ∇ × V ⃗ \therefore \nabla \times \vec{V} ∴ ∇ × V = ( ∂ ∂ x e ^ x + ∂ ∂ y e ^ y + ∂ ∂ z e ^ z ) × ( V x e ^ x + V y e ^ y + V z e ^ z ) = \left(\frac{\partial}{\partial x}\,\hat{e}_x + \frac{\partial}{\partial y}\,\hat{e}_y + \frac{\partial}{\partial z}\,\hat{e}_z\right) \times \left(V_x\,\hat{e}_x + V_y\,\hat{e}_y + V_z\,\hat{e}_z\right) = ( ∂ x ∂ e ^ x + ∂ y ∂ e ^ y + ∂ z ∂ e ^ z ) × ( V x e ^ x + V y e ^ y + V z e ^ z ) = ( ∂ ∂ y V z − ∂ ∂ z V y ) e ^ x + ( ∂ ∂ z V x − ∂ ∂ x V z ) e ^ y + ( ∂ ∂ x V y − ∂ ∂ y V x ) e ^ z = \left(\frac{\partial}{\partial y} V_z - \frac{\partial}{\partial z} V_y\right)\hat{e}_x + \left(\frac{\partial}{\partial z} V_x - \frac{\partial}{\partial x} V_z\right)\hat{e}_y + \left(\frac{\partial}{\partial x} V_y - \frac{\partial}{\partial y} V_x\right)\hat{e}_z = ( ∂ y ∂ V z − ∂ z ∂ V y ) e ^ x + ( ∂ z ∂ V x − ∂ x ∂ V z ) e ^ y + ( ∂ x ∂ V y − ∂ y ∂ V x ) e ^ z = ∣ e ^ x e ^ y e ^ z ∂ ∂ x ∂ ∂ y ∂ ∂ z V x V y V z ∣ = \begin{vmatrix} \hat{e}_x & \hat{e}_y & \hat{e}_z \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ V_x & V_y & V_z \end{vmatrix} = e ^ x ∂ x ∂ V x e ^ y ∂ y ∂ V y e ^ z ∂ z ∂ V z Curl of Central Force Field Central force field:
F ⃗ = f ( r ) r ^ \vec{F} = f(r)\,\hat{r} F = f ( r ) r ^ = f ( r ) r ( x e ^ x + y e ^ y + z e ^ z ) = \frac{f(r)}{r} \left(x\,\hat{e}_x + y\,\hat{e}_y + z\,\hat{e}_z\right) = r f ( r ) ( x e ^ x + y e ^ y + z e ^ z ) Curl: ∇ × F ⃗ \text{Curl: } \nabla \times \vec{F} Curl: ∇ × F
= ( ∂ ∂ x e ^ x + ∂ ∂ y e ^ y + ∂ ∂ z e ^ z ) × ( x e ^ x + y e ^ y + z e ^ z ) f ( r ) r = \left(\frac{\partial}{\partial x}\,\hat{e}_x + \frac{\partial}{\partial y}\,\hat{e}_y + \frac{\partial}{\partial z}\,\hat{e}_z\right) \times \left(x\,\hat{e}_x + y\,\hat{e}_y + z\,\hat{e}_z\right) \frac{f(r)}{r} = ( ∂ x ∂ e ^ x + ∂ y ∂ e ^ y + ∂ z ∂ e ^ z ) × ( x e ^ x + y e ^ y + z e ^ z ) r f ( r ) = ( ∂ ∂ y z f ( r ) r − ∂ ∂ z y f ( r ) r ) e ^ x + ( ∂ ∂ z x f ( r ) r − ∂ ∂ x z f ( r ) r ) e ^ y + ( ∂ ∂ x y f ( r ) r − ∂ ∂ y x f ( r ) r ) e ^ z = \left(\frac{\partial}{\partial y}\frac{z\,f(r)}{r} - \frac{\partial}{\partial z}\frac{y\,f(r)}{r}\right)\hat{e}_x + \left(\frac{\partial}{\partial z}\frac{x\,f(r)}{r} - \frac{\partial}{\partial x}\frac{z\,f(r)}{r}\right)\hat{e}_y + \left(\frac{\partial}{\partial x}\frac{y\,f(r)}{r} - \frac{\partial}{\partial y}\frac{x\,f(r)}{r}\right)\hat{e}_z = ( ∂ y ∂ r z f ( r ) − ∂ z ∂ r y f ( r ) ) e ^ x + ( ∂ z ∂ r x f ( r ) − ∂ x ∂ r z f ( r ) ) e ^ y + ( ∂ x ∂ r y f ( r ) − ∂ y ∂ r x f ( r ) ) e ^ z Let's work on e ^ x \hat{e}_x e ^ x
∂ ∂ y z f ( r ) r − ∂ ∂ z y f ( r ) r \frac{\partial}{\partial y}\frac{z\,f(r)}{r} - \frac{\partial}{\partial z}\frac{y\,f(r)}{r} ∂ y ∂ r z f ( r ) − ∂ z ∂ r y f ( r ) = f ( r ) r ∂ z ∂ y + z ∂ ∂ y ( f ( r ) r ) − f ( r ) r ∂ y ∂ z − y ∂ ∂ z ( f ( r ) r ) = \frac{f(r)}{r}\frac{\partial z}{\partial y} + z \frac{\partial}{\partial y}\left(\frac{f(r)}{r}\right) - \frac{f(r)}{r}\frac{\partial y}{\partial z} - y \frac{\partial}{\partial z}\left(\frac{f(r)}{r}\right) = r f ( r ) ∂ y ∂ z + z ∂ y ∂ ( r f ( r ) ) − r f ( r ) ∂ z ∂ y − y ∂ z ∂ ( r f ( r ) ) = z ⋅ ∂ ∂ y [ f ( r ) r ] ⋅ y r − y ⋅ ∂ ∂ z [ f ( r ) r ] ⋅ z r = z \cdot \frac{\partial}{\partial y}\left[\frac{f(r)}{r}\right] \cdot \frac{y}{r} - y \cdot \frac{\partial}{\partial z}\left[\frac{f(r)}{r}\right] \cdot \frac{z}{r} = z ⋅ ∂ y ∂ [ r f ( r ) ] ⋅ r y − y ⋅ ∂ z ∂ [ r f ( r ) ] ⋅ r z By symmetry, other components are also zero.
∴ ∇ × f ( r ) r ^ = 0 \therefore \nabla \times f(r)\,\hat{r} = 0 ∴ ∇ × f ( r ) r ^ = 0