From the generalized Stokes’ theorem, which generalizes the fundamental theorem of calculus to higher dimensional differential forms on manifolds, one may derive a number of useful theorems of vector calculus, such as the gradient theorem, Kelvin-Stokes theorem (also frequently known as “Stokes’ theorem” or the “curl theorem”), the divergence theorem, and Green’s Theorem. One may also derive from it the formula for vector integration by parts: for a region *Ω* of with piecewise smooth boundary *Γ*, with outward surface normal , then for scalar function *φ*(**r**) and vector function **v**(**r**), then one has

,

or, rearranging,

,

or

.

Using the second form, and letting *φ*=1, we get

,

the divergence theorem.

Letting our vector field be the gradient of a scalar function, , in the first form, we obtain

,

which is Green’s first identity, often written as

,

and usually used in three dimensions:

,

Exchanging *φ* and *ψ*,

,

and subtracting this from the previous, the dot product of gradients terms cancel, giving Green’s second identity:

.

Taking Green’s first identity in the form

,

and setting *ψ*=*φ*, we get

.

Letting in the first form, we see

,

since the curl of a vector field always has zero divergence.

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Tags: Calculus, Divergence Theorem, Integration by Parts, Math, Monday Math, Stokes' Theorem

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