Lecture 07: Divergence, Curl, and Integral Theorems
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Overview
Steven M. Wise defines divergence, tensor divergence, curl, and the scalar and vector Laplacians through coordinate-invariant operations, then derives their component formulas in an orthonormal frame. He states the divergence theorem and its tensor version, explains the orientation convention and circulation form of Stokes' theorem, and defers the localization theorem to the next lecture.
Key takeaways
- Defining divergence as tr(grad V) makes the operator coordinate-invariant, while an orthonormal frame gives the computable formula div V = Vᵢ,ᵢ.
- The tensor divergence definition (div S) · a = div(Sᵀa) yields the component formula (div S)ᵢ = Sᵢⱼ,ⱼ and is especially useful for stress tensors in elasticity.
- The curl of a vector field is the axial vector associated with the skew-symmetric tensor grad V − (grad V)ᵀ; its first component is ∂V₃/∂x₂ − ∂V₂/∂x₃.
- Defining both scalar and vector Laplacians as div grad leads to φ,ᵢᵢ for a scalar and Vᵢ,ⱼⱼ for a vector in an orthonormal frame.
- The divergence theorem converts boundary flux into a volume integral of divergence, while Stokes' theorem converts surface flux of curl into boundary circulation, provided orientations are consistent.
Chapters
- Steven M. Wise continues mechanics tensor calculus, moving from gradient definitions to divergence, curl, Laplacian, and integral theorems.
- He assumes fields have continuous derivatives to whatever order the calculations require.
- The planned topics include the divergence theorem, Stokes' theorem, and a localization result.
- For a differentiable vector field V, divergence is defined invariantly as tr(grad V), turning a vector field into a scalar field.
- In an orthonormal frame, grad V has components Vᵢ,ⱼ, so its trace is V₁,₁ + V₂,₂ + V₃,₃.
- The component formula is the familiar sum of partial derivatives, while the trace definition does not depend on a chosen coordinate system.
- For a second-order tensor field S, Wise defines div S through (div S) · a = div(Sᵀa) for every constant vector a.
- The definition reduces tensor order by one: div S is a vector field, just as div V is a scalar field.
- In an orthonormal frame, (div S)ᵢ = Sᵢⱼ,ⱼ, with the repeated j index summed.
- Wise checks product-rule formulas by verifying that every term has the same tensor type as the left-hand side.
- For grad(φV), the two terms are V ⊗ grad φ and φ grad V, both second-order tensors.
- He proves the rule by expanding components in an orthonormal frame, applying the ordinary scalar product rule, and rewriting the result invariantly.
- For a scalar field φ and vector field V, div(φV) = grad φ · V + φ div V.
- The component expansion differentiates φVᵢ with respect to xᵢ and applies the ordinary partial-derivative product rule.
- The resulting terms are scalar products or scalar multiples, confirming that both sides are scalar fields.
- Wise defines curl V through (curl V) × a = (grad V − (grad V)ᵀ)a for an arbitrary vector a.
- The difference grad V − (grad V)ᵀ is skew-symmetric, so the axial-vector result from the earlier tensor-calculus material guarantees a unique curl vector.
- Curl encodes local rotational behavior and will be useful later for describing flow fields.
- The curl construction uses the result that every skew-symmetric 3 × 3 tensor corresponds to a unique axial vector.
- Wise writes the skew tensor components using derivatives of V and applies the permutation symbol εᵢⱼₖ to extract the vector components.
- The construction establishes that the invariant definition of curl is well-defined before giving its familiar component formula.
- Swapping dummy indices in the permutation-symbol expression changes its sign, yielding the standard curl components.
- The first component is ∂V₃/∂x₂ − ∂V₂/∂x₃, with the other two obtained by cyclic permutation.
- Wise recalls the 3 × 3 determinant mnemonic with basis vectors, partial derivatives, and V₁, V₂, V₃.
- The scalar Laplacian is defined as div(grad φ), while the vector Laplacian is div(grad V).
- For φ, the component expression is φ,ᵢᵢ, or the sum of second partial derivatives with respect to x₁, x₂, and x₃.
- For V, the result remains a vector, with components Vᵢ,ⱼⱼ; both Laplacians combine two first-order operations.
- Wise derives the scalar formula by setting W = grad φ, so Wᵢ = φ,ᵢ, then applying div W = Wᵢ,ᵢ.
- Combining the derivatives gives φ,ᵢᵢ, with the repeated index indicating summation over the three spatial directions.
- The vector case follows the same sequence, applying divergence to the tensor grad V to obtain Vᵢ,ⱼⱼ.
- A regular region in E³ consists of finitely many disjoint, open, bounded components with a piecewise-smooth boundary.
- Regions may contain holes or multiple disconnected components, provided their boundaries remain suitably smooth and orientable.
- The boundary normal n is defined to point outward, including on the inner boundary of a hole.
- For a differentiable vector field V, the boundary flux integral ∫∂B V · n equals the volume integral ∫B div V.
- The surface integrand V · n is scalar, and the volume integrand div V is scalar, so both integrals produce numbers.
- For a second-order tensor S, the corresponding identity is ∫∂B S n = ∫B div S; both sides are vectors, as in applications to the stress tensor in elasticity.
- Stokes' theorem applies to an oriented surface γ with a piecewise-smooth boundary curve C.
- The boundary tangent T and surface normal n must follow the right-hand rule: curling fingers along C points the thumb along n.
- Wise illustrates the convention with a surface patch whose normal points out of the page and a counterclockwise boundary direction.
- Stokes' theorem states that ∫γ (curl V) · n equals ∫C V · T for a differentiable vector field V.
- The surface integral of the normal component of curl equals the circulation of V along the oriented boundary curve.
- Wise relates the identity to the fundamental theorem of calculus and postpones the localization theorem and further calculations to the next lecture.
Transcript excerpt
0:04 All right, welcome back to continue mechanics tensor calculus part two. I think we'll do three parts in this, maybe four. We didn't get to cover uh the divergence the def we we I think we just covered the definition of uh what I mean by the divergence operator last time. We did cover pretty well the gradient operators and the the definition of the derivatives in general terms. So I'll start here with divergence and the product rules. Um I'll I'll go through this material a little bit quickly. I'm only going to do one of the three or the uh examples that I pulled from exercises here and then we'll talk about the curl
0:53 leassian and maybe get into some of these uh integral theorems the divergence theorem Stokes theorem and uh if time permits we'll finish with something called the localization result. So how do we define divergence? So this is defined in a way that is um in keeping with everything else that we've done so far. We're going to define this in a component free way or in what's called an invariant way. So to do to do this, we just need to use other invariant uh quantities or definitions. So for every differentiable vector field, we're going to associate a scalar field called the divergence. And that
1:42 divergence dell.v or naba.v is defined as the trace of the gradient operator. The trace of the gradient operator. So this gradient operator for a vector uh field. So a vector field is just something which takes points in E3 in uklidian space and maps them to vectors. That's a vector field. We'll assume, by the way, I failed to say this last time clearly perhaps we're we're we're going to assume that we can take as many
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