Monday Math 128

We have previously discussed the Fourier transform (here and here, especially). In this post, we noted that (using the symmetric angular convention) the space transform for an n dimensional space is

and the inverse is
.
We can also do the same for a vector field:

and
.
We note from vector calculus then that for a vector field, the components of the transform are the transforms of the components:
.

We also used integration by parts here to show that for a one-dimensional function f(x), with , the derivative has Fourier transform:
.
Similarly, we can use vector integration by parts for our multi-dimensional transforms. Working in three dimensions from here:
First, one form of the divergence theorem states:
,
where S is the boundary of the volume V, with outward normal.
Letting f=φψ, and using the gradient product rule ,

.
Letting , we see
.
and since , we have
.
Now, if as , φ(x) goes to zero faster than , then, as we expand the volume V to cover all space, the surface integral will go to zero, and we obtain
,
which means
,
in analogy to our one-dimensional rule.

Monday Math 126

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.

Physics Friday 96

Suppose, as in this post, we have a fluid of density ρ, with some arbitrary three-dimensional object immersed in the fluid, occupying a volume V. This object has surface ∂V. We saw that for any point on the surface, the pressure is , and the force on an area element is:
. Thus, the torque on that element is
,
where r is the coordinate vector to the element. Recalling that for vectors v and w and scalar a,
, so since P(x,y,z) is a scalar, we see

and so the total torque (about the origin) is
;
and one form of the divergence theorem tells us that for vector field A,
.
Thus

Now, the product rule for the curl of the product of a scalar field ψ and a vector field a is
.
Now, applying that to , and noting that (as it is a radial vector field), and , we see
.
Now, since is a constant vector, it can be “factored out” of the integral:
.
We recall that the total buoyant force on the object is ; plugging this into the above, we see
.

Now, recall that the center of mass of a region with density function ρ(x,y,z) is . If the density is a constant, it factors out of the integrals, and one gets . Thus, if we consider the volume as if it were filled with the fluid; that is to say, the volume of fluid displaced, its center of mass would be , and then we see that
,
which is equivalent to the torque if the buoyant force acted entirely on the point r_b; this point is called the center of buoyancy. Just as the force of gravity on an extended object can be treated as if it acts entirely on the center of mass, the buoyant force can be treated as if it acts entirely on the center of buoyancy.