Mathematical Methods for Physics and Engineering - Matematica.NET

10.7 VECTOR OPERATORS10.7.3 Curl of a vector fieldThe curl of a vector field a(x, y, z) is defined by( ∂azcurl a = ∇×a =∂y − ∂a ) (y ∂axi +∂z ∂z − ∂a ) (z ∂ayj +∂x ∂x − ∂a )xk,∂ywhere a x , a y and a z are the x-, y- andz- components of a. The RHS can bewritten in a more memorable form as a determinant:∣ i j k ∣∣∣∣∣∣∣ ∇×a =∂ ∂ ∂, (10.35)∂x ∂y ∂z∣ a x a y a zwhere it is understood that, on expanding the determinant, the partial derivativesin the second row act on the components of a in the third row. Clearly, ∇×a isitself a vector field. Any vector field a for which ∇×a = 0 is said to be irrotational.◮Find the curl of the vector field a = x 2 y 2 z 2 i + y 2 z 2 j + x 2 z 2 k.The curl of a is given by∣ i j k ∣∣∣∣∣∣∣ ∂ ∂ ∂∇φ == −2 [ y 2 zi +(xz 2 − x 2 y 2 z)j + x 2 yz 2 k ] . ◭∂x ∂y ∂z∣x 2 y 2 z 2 y 2 z 2 x 2 z 2For a vector field v(x, y, z) describing the local velocity at any point in a fluid,∇×v is a measure of the angular velocity of the fluid in the neighbourhood ofthat point. If a small paddle wheel were placed at various points in the fluid thenit would tend to rotate in regions where ∇×v ≠ 0, while it would not rotate inregions where ∇×v = 0.Another insight into the physical interpretation of the curl operator is gainedby considering the vector field v describing the velocity at any point in a rigidbody rotating about some axis with angular velocity ω. Ifr is the position vectorof the point with respect to some origin on the axis of rotation then the velocityof the point is given by v = ω × r. Without any loss of generality, we may takeω to lie along the z-axis of our coordinate system, so that ω = ω k. The velocityfield is then v = −ωy i + ωx j. The curl of this vector field is easily found to bei j k∇×v =∂ ∂ ∂∂x ∂y ∂z=2ωk =2ω. (10.36)∣ −ωy ωx 0 ∣353