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Convective Operator

Defined for a vector field A by (A·del ), where del is the gradient operator.

Applied in arbitrary orthogonal three-dimensional coordinates to a vector field B, the convective operator becomes

[(A·del )B]_j=sum_(k=1)^3[(A_k)/(h_k)(partialB_j)/(partialq_k)+(B_k)/(h_kh_j)(A_j(partialh_j)/(partialq_k)-A_k(partialh_k)/(partialq_j))],
(1)

where the h_is are related to the metric tensors by h_i=sqrt(g_(ii)). In Cartesian coordinates,

(A·del )B=(A_x(partialB_x)/(partialx)+A_y(partialB_x)/(partialy)+A_z(partialB_x)/(partialz))x^^+(A_x(partialB_y)/(partialx)+A_y(partialB_y)/(partialy)+A_z(partialB_y)/(partialz))y^^+(A_x(partialB_z)/(partialx)+A_y(partialB_z)/(partialy)+A_z(partialB_z)/(partialz))z^^.
(2)

In cylindrical coordinates,

(A·del )B=(A_r(partialB_r)/(partialr)+(A_phi)/r(partialB_r)/(partialphi)+A_z(partialB_r)/(partialz)-(A_phiB_phi)/r)r^^+(A_r(partialB_phi)/(partialr)+(A_phi)/r(partialB_phi)/(partialphi)+A_z(partialB_phi)/(partialz)+(A_phiB_r)/r)phi^^+(A_r(partialB_z)/(partialr)+(A_phi)/r(partialB_z)/(partialphi)+A_z(partialB_z)/(partialz))z^^.
(3)

In spherical coordinates,

(A·del )B=(A_r(partialB_r)/(partialr)+(A_theta)/r(partialB_r)/(partialtheta)+(A_phi)/(rsintheta)(partialB_r)/(partialphi)-(A_thetaB_theta+A_phiB_phi)/r)r^^ +(A_r(partialB_theta)/(partialr)+(A_theta)/r(partialB_theta)/(partialtheta)+(A_phi)/(rsintheta)(partialB_theta)/(partialphi)+(A_thetaB_r)/r-(A_phiB_phicottheta)/r)theta^^ +(A_r(partialB_phi)/(partialr)+(A_theta)/r(partialB_phi)/(partialtheta)+(A_phi)/(rsintheta)(partialB_phi)/(partialphi)+(A_phiB_r)/r+(A_phiB_thetacottheta)/r)phi^^.
(4)

See also

Convective Acceleration, Convective Derivative, Curvilinear Coordinates, Gradient

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Cite this as:

Weisstein, Eric W. "Convective Operator." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConvectiveOperator.html

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