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Engineering LibreTexts

B.3: Vector Identities

  • Page ID
    6379
  • [m0140_Vector_Identities]

    Algebraic Identities

    &A(B) = B(C) = C(A) & [m0140_eVI1]
    &A(B) = B(A) - C(A) [m0140_eVI2] &

    Identities Involving Differential Operators

    &(A) = 0 &
    &(f) = 0 &
    &(fA) = f() + (f) &
    &(A) = B()-A() &
    &(f) = ^2 f &
    & = ()-^2A &
    &^2 A = () - () &

    Divergence Theorem: Given a closed surface \({\mathcal S}\) enclosing a contiguous volume \({\mathcal V}\), \[\int_{\mathcal V} \left( \nabla \cdot {\bf A} \right) dv = \oint_{\mathcal S} {\bf A}\cdot d{\bf s}\] where the surface normal \(d{\bf s}\) is pointing out of the volume.

    Stokes’ Theorem: Given a closed curve \({\mathcal C}\) bounding a contiguous surface \({\mathcal S}\), \[\int_{\mathcal S} \left( \nabla \times {\bf A} \right) \cdot d{\bf s} = \oint_{\mathcal C} {\bf A}\cdot d{\bf l}\] where the direction of the surface normal \(d{\bf s}\) is related to the direction of integration along \({\mathcal C}\) by the “right hand rule.”