logo

Partial Integration of Expressions Containing the Del Operator 📂Mathematical Physics

Partial Integration of Expressions Containing the Del Operator

Formulas

The following expressions hold true for vector integration involving the del operator.

(a)

∫VA⋅(∇f)dτ=∮SfA⋅da−∫Vf(∇⋅A)dτ \int_{\mathcal{V}}\mathbf{A} \cdot (\nabla f)d\tau = \oint_{\mathcal{S}}f\mathbf{A} \cdot d \mathbf{a}-\int_{\mathcal{V}}f(\nabla \cdot \mathbf{A})d\tau

(b)

∫Sf(∇×A)A⋅da=∫S[A×(∇f)]⋅da+∮PfA⋅dl \int_{\mathcal{S}} f \left( \nabla \times \mathbf{A} \right)\mathbf{A} \cdot d \mathbf{a} = \int_{\mathcal{S}} \left[ \mathbf{A} \times \left( \nabla f \right) \right] \cdot d\mathbf{a} + \oint_{\mathcal{P}} f\mathbf{A} \cdot d\mathbf{l}

(c)

∫VB⋅(∇×A)dτ=∫VA⋅(∇×B)dτ+∮S(A×B)⋅da \int_{\mathcal{V}} \mathbf{B} \cdot \left( \nabla \times \mathbf{A} \right) d\tau = \int_{\mathcal{V}} \mathbf{A} \cdot \left( \nabla \times \mathbf{B} \right) d\tau + \oint_{\mathcal{S}} \left( \mathbf{A} \times \mathbf{B} \right) \cdot d \mathbf{a}

Explanation

Partial integration is a method that simplifies the integration of the product of a function (f or A(f\ or\ \mathbf{A} and the derivative of a function (∇f or ∇⋅A)(\nabla f\ or\ \nabla \cdot \mathbf{A}).

Partial Integration ddx(fg)=fdgdx+gdfdx\dfrac{d}{dx}\left( fg \right) = f\dfrac{dg}{dx}+g\dfrac{df}{dx} Integrating both sides yields

∫abddx(fg)=(fg)∣ab=∫abf(dgdx)dx+∫abg(dfdx)dx  ⟹  ∫abf(dgdx)dx=(fg)∣ab−∫abg(dfdx)dx \int_{a}^b \dfrac{d}{dx} \left(fg\right) = (fg)\Big|_{a}^b=\int_{a}^b f\left(\dfrac{dg}{dx}\right)dx+\int_{a}^bg\left(\dfrac{df}{dx}\right)dx \\ \implies \int_{a}^b f\left(\dfrac{dg}{dx}\right)dx = (fg)\Big|_{a}^b-\int_{a}^bg\left(\dfrac{df}{dx}\right)dx

Proof

(a)

Using Product Rule 3

∇⋅(fA)=A⋅(∇f)+f(∇⋅A) \nabla \cdot (f\mathbf{A}) = \mathbf{A} \cdot (\nabla f) + f(\nabla \cdot \mathbf{A})

Integrating both sides with respect to volume gives

∫V∇⋅(fA)dτ=∫VA⋅(∇f)dτ+∫Vf(∇⋅A)dτ \int_{\mathcal{V}} \nabla \cdot (f\mathbf{A})d\tau = \int_{\mathcal{V}}\mathbf{A} \cdot (\nabla f)d\tau + \int_{\mathcal{V}}f(\nabla \cdot \mathbf{A})d\tau

Applying the Divergence Theorem to the left hand side gives

∮SfA⋅da=∫VA⋅(∇f)dτ+∫Vf(∇⋅A)dτ \oint_{\mathcal{S}}f\mathbf{A} \cdot d \mathbf{a} = \int_{\mathcal{V}}\mathbf{A} \cdot (\nabla f)d\tau + \int_{\mathcal{V}}f(\nabla \cdot \mathbf{A})d\tau

Upon simplification, we get

∫Vf(∇⋅A)dτ=∮SfA⋅da−∫VA⋅(∇f)dτ \int_{\mathcal{V}}f(\nabla \cdot \mathbf{A})d\tau = \oint_{\mathcal{S}}f\mathbf{A} \cdot d \mathbf{a}-\int_{\mathcal{V}}\mathbf{A} \cdot (\nabla f)d\tau

Or equivalently,

∫VA⋅(∇f)dτ=∮SfA⋅da−∫Vf(∇⋅A)dτ \int_{\mathcal{V}}\mathbf{A} \cdot (\nabla f)d\tau = \oint_{\mathcal{S}}f\mathbf{A} \cdot d \mathbf{a}-\int_{\mathcal{V}}f(\nabla \cdot \mathbf{A})d\tau

■

(b)

(c)