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After all, the Fundamental Theorem of Calculus, the Curved Fundamental Theorem
of Calculus, Green’s Theorem, Stokes Theorem, and the Divergence Theorem, are
instances of the Generalized Stokes Theorem.
To state it, we need to talk about forms, and to talk about forms, we need
determinant-like functions called \(k\)-forms. Let’s recall a couple of properties of
determinants:
The determinants satisfy the following properties:
(multilinearity) If all but one row is fixed, the determinant depends linearly
on the non-fixed row.
Let \(k, n \in \mathbb {N}\) be natural numbers. A simple \(k\)-form in \(\mathbb {R}^n\) is a continuous function \(\omega : (\mathbb {R}^n)^k \to \mathbb {R}\) that takes \(k\)
vectors in \(\mathbb {R}^n\) (also known as a \(k \times n\) matrix) and returns a number, satisfying the following
properties:
(multilinearity) When all but one of the vectors are fixed, \(\omega \) depends linearly
on the non-fixed vector:
\begin{gather*} P \, dx + Q \, dy \\ 3xy \, dx + x^2 \cos y \, dy \\ 5y^2 e^x \, dx \\ (\sqrt {x} + \cos y ) \, dx + \frac {1}{x} \, dy \end{gather*}
Smooth \(1\)-forms in \(\mathbb {R}^3\):
\begin{gather*} P \, dx + Q \, dy + R \, dz \\ y^2 \cos x \,dx + x \sin z \, dy + z x^2 \, dz \\ 2xz \, dy + ze^y\cos x \,dz \\ (\sin y + \sin z ) \, dx + e^x \cos y \, dy + (\cos x - 1 ) \, dz \end{gather*}
Smooth \(2\)-forms in \(\mathbb {R}^2\):
\begin{gather*} f(x,y) \, dx \wedge dy \\ (e ^x \sin y + 3xy) \, dx \wedge dy\\ (5y - x \sqrt {y})\, dx \wedge dy \\ (e^x - x \cos y )\, dx \wedge dy \end{gather*}
Smooth \(2\)-forms in \(\mathbb {R}^3\):
\begin{gather*} P \, dy \wedge dz + Q \, dz \wedge dx + R \, dx \wedge dy \\ (5x^2 + e^z) \, dx \wedge dy + (2x \sin z + 3) \, dx \wedge dz \\ ( 3 \sqrt {x} + \cos y )\, dx \wedge dy - z\, dy \wedge dz \\ (7y + y^2 e^x)\, dy \wedge dz + (2e^y - \cos x) \, dx \wedge dy \end{gather*}
Smooth \(3\)-forms in \(\mathbb {R}^3\):
\begin{gather*} f(x,y,z) \, dx \wedge dy \wedge dz \\ (e^x - y^2 z + \cos z) \, dx \wedge dy \wedge dz \\ (3xyz + y^2 - \sqrt {z}) \, dx \wedge dy \wedge dz \end{gather*}
General form of a
in \(\mathbb {R}\)
in \(\mathbb {R}^2\)
in \(\mathbb {R}^3\)
\(0\)-form
\(f(x)\)
\(f(x,y)\)
\(f(x,y,z)\)
\(1\)-form
\(f(x) \, dx\)
\(P\,dx + Q \, dy\)
\(P\, dx + Q \, dy + R \, dz\)
\(2\)-form
\(0\)
\(f(x,y)\,dx \wedge dy\)
\(P\, dy \wedge dz + Q \, dz \wedge dx + R \, dx \wedge dy\)
\(3\)-form
\(0\)
\(0\)
\(f(x,y,z ) \, dx \wedge dy \wedge dz \)
\(4\)-form
\(0\)
\(0\)
\(0 \)
4 Manifolds
In general, \(k\)-forms can be integrated over \(k\)-dimensional manifolds. \(0\)-dimensional
manifolds are points, \(1\)-dimensional manifolds are curves, \(2\)-dimensional manifolds are
surfaces.
A parametrized \(k\)-manifold in \(\mathbb {R}^n\) is a smooth map \(\varphi : D \to \mathbb {R}^n\) with \(D \subset \mathbb {R}^k\) a region. We often also call \(M : = \varphi (D) \subset \mathbb {R}^n\) the
manifold.
The integral of a smooth \(k\)-form \(\omega \) with domain \(U \subset \mathbb {R}^n\) over a parametrized \(k\)-manifold \(\varphi : D \to M \subset U\) is given
by
MTH 255H