Grassmann algebra and deRham cohomology (1:47:18)
by Frederic P. Schuller (#12, 2015年09月21日).
Implicit differentiation (15:33)
Grant Sanderson (3Blue1Brown, 2017年05月03日).
Vector Calculus Overview (1:12:16)
by Peyam Tabrizian (2018年12月12日).
In the context of a topological vector space E over a field K, a form over E is simply a continuous(*) linear application from E to K.
(*) All linear applications from a finite-dimensional vector space into another are continous, but this isn't necessarily so when considering spaces with infinitely many dimensions (like those which routinely occur in functional analysis). Therefore, the above requirement for continuity isn't superfluous.
Intuitively, the differential du of a form u represents the infinitesimal change u would undergo if other forms underwent together some specified infinitesimal changes. It's thus a linear combination of all the differential forms corresponding to a complete set of quantities which fully specify u
[画像: Come back later, we're still working on this one... ]
Saddlepoints of a multivariate function. One equation for each variable.
A stunning generalization of the fundamental theorem of calculus states that the integral of a form's derivative dw over an oriented manifold W is the integral of that form over the border ¶W. This is called Stokes' Theorem : Joseph-Louis Lagrange (1736-1813) Carl Friedrich Gauss (1777-1855) Lord Kelvin 1824-1907 Sir George Stokes (1819-1903)
Stokes' TheoremIn a way, Nicolas Bourbaki and this result are due to each other. The urge to elucidate the latter gave birth to the former, in 1935 (as reported by Bourbaki founder André Weil, see: The Many Faces of Nicolas Bourbaki).
Ostrogradski's theorem was independently discovered by Lagrange in 1762, by Gauss in 1813 and by George Green in 1825. A rigorous proof was given by Mikhail Vasilevich Ostrogradski (1801-1861) in 1831.
Applied to U = p grad q
(and/or U = q grad p )
that theorem yields two relations known as Green's formulas :
òòòV
( p D q
+ grad p . grad q )
dV
=
òòS
p grad q . dS
òòòV
( p D q
- q D p )
dV
=
òòS
( p grad q
- q grad p ) . dS
Applied to U = [P,Q,0] the third cartesian component of the Kelvin-Stokes formula yields the following elementary result, dubbed Green's theorem, which relates the counterclockwise line integral around the border (C) of a planar region (S) to a surface integral on that region:
òòS ( ¶Q / ¶x - ¶P / ¶y ) dx dy = òC P dx + Q dy
According to E.T. Whittaker (A History of the Theories of Aether and Electricity) the Kelvin-Stokes formula (commonly called the Stokes formula) was first stated without proof by Kelvin in 1850. George G. Stokes assigned the proof of that formula for the 1854 Smith's Prize exam, in which James Clerk Maxwell was sitting. It was Maxwell who later traced that question back to Stokes and saw fit to give him credit for the answer... In the end, the generalized theorem (elucidated by the Bourbakists around 1935) was named after Stokes too!
Among the above formulae, the least popular is surely the one involving an irreducible nabla (as tabulated above in terms of cartesian coordinates). I believe it has never been given a special name:
òòS [ div U dS - (U.Ñ) dS ] = òC U ´ dr
The next section features this formula and demonstrates what's involved in elementary proofs of such, when more elegant general reasoning is shunned.
Highbrow Poincaré duality is a loosely related topic.
A twisted loop, with color-coded apparent area. This apparent area is a signed quantity which is positive for an observer looking at the north side of the loop (readily identified if the loop is not too twisted).
The area vector, or surface vector is an axial vector, defined by a contour integral around the oriented loop:
S = ½ òC+ r ´ dr
For a closed loop C+ this defining integral does not depend on the choice of origin for the position vector r. Anyone encountering this for the first time is encouraged to work out S explicitely for a circle of radius R, with the following parametric equations (0 < q < 2p).
x = a + R cos q ; y = b + R sin q ; z = c
For simple planar loops, the magnitude of S is simply the usual surface area enclosed by the loop (the vector is perpendicular to the plane and points to whichever direction is implied by the orientation of the loop). This definition is consistent with the general integration formulas tabulated above. (HINT: With U = r you'll end subtracting two very simple integrands: 3 dS and dS.)
The apparent area surrounded by a simple planar curve is proportional to the cosine of its tilt to the observer. This establishes the advertised property in terms of scalar products, for all simple oriented planar curves, including triangles.
The same is true of a non-planar quadrilateral, because such a polygon always has the same apparent area as two triangles sharing an edge (one of the quadrilateral's diagonal) if they are oriented in such a way that this hinge is traveled in opposite directions. The quadrilateral contour is effectively equivalent to that of the triangles (as the hinge would be counted once positively and once negatively if the triangles were considered individually).
This argument may be extended by induction to any polygon and, by continuity, to any smooth enough curve. So, the above formula for S and its relevance to apparent areas hold for all rectifiable 3D curves. QED
An elementary proof of the Kelvin-Stokes formula could proceed similarly.
Note that the surface area of a surface bounded by a given loop has little to do with the above vectorial area. For example, an hemisphere of radius R has a surface area of 2pR2, which is twice the magnitude of the equator's vectorial area... The vectorial area of an entire sphere is zero !
All our operators are additive (e.g., div (A+B) = div A + div B ) and they commute with the Laplacian operator D, defined in the second line below.
The Fundamental Theorem of Vector Calculus states that any smooth vector field (decaying rapidly enough at large distances) is the sum of an irrotational field (of zero rotational) and a solenoidal one (of zero divergence). Such a sum is called a Helmholtz decomposition.
Hodge decomposition is the generalization of this to all differential forms.
