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Engineering LibreTexts

B.2: Vector Operators

  • Page ID
    6378
  • [m0139_Vector_Operators]

    This section contains a summary of vector operators expressed in each of the three major coordinate systems:

    • Cartesian (\(x\),\(y\),\(z\))

    • cylindrical (\(\rho\),\(\phi\),\(z\))

    • spherical (\(r\),\(\theta\),\(\phi\))

    Associated basis vectors are identified using a caret (\(\hat{~}\)) over the symbol. The vector operand \({\bf A}\) is expressed in terms of components in the basis directions as follows:

    • Cartesian: \({\bf A} = \hat{\bf x}A_x + \hat{\bf y}A_y + \hat{\bf z}A_z\)

    • cylindrical: \({\bf A} = \hat{\bf \rho}A_{\rho} + \hat{\bf \phi}A_{\phi} + \hat{\bf z}A_z\)

    • spherical: \({\bf A} = \hat{\bf r}A_r + \hat{\bf \theta}A_{\theta} + \hat{\bf \phi}A_{\phi}\)

    Gradient
    Gradient in Cartesian coordinates:

    f &= + + &

    Gradient in cylindrical coordinates:

    f &= + + &

    Gradient in spherical coordinates:

    f &= + + &

    Divergence
    Divergence in Cartesian coordinates:

    &= + + &

    Divergence in cylindrical coordinates: Divergence in spherical coordinates:

    Curl
    Curl in Cartesian coordinates:

    &=   ( - ) &
    &   +( - ) &
    &   +( - ) &

    Curl in cylindrical coordinates:

    &=   ( - ) &
    &   +( - ) &
    &   +&

    Curl in spherical coordinates:

    &=    &
    &   +&
    &   +&

    Laplacian
    Laplacian in Cartesian coordinates:

    ^2 f &= + + &

    Laplacian in cylindrical coordinates:

    ^2 f &= ( ) + + &

    Laplacian in spherical coordinates:

    ^2 f &=   (r^2 ) &
    &   +() &
    &   + &