Calculus of Vector Fields
Line Integrals of Vector Fields
The integral of a vector field \(\mathbf{F}: \mathbb{R}^m \to \mathbb{R}^m\) along a piecewise \(\mathscr{C}^1\) continuous path \(\mathbf{x}: [a, b] \to \mathbb{R}^m\) is defined as
\[ \int_{\gamma} \mathbf{F} \cdot \mathrm{d}\mathbf{l} = \int_{a}^{b} \langle \mathbf{F}(\mathbf{x}(t)), \mathbf{x}'(t) \rangle dt. \]
In a Cartesian coordinate system, let \(\mathbf{F} = (F_1, \dots, F_m)^T\) and \(\mathbf{x} = (x_1, \dots, x_m)^T\), then
\[ \int_a^b \langle \mathbf{F}(\mathbf{x}(t)), \mathbf{x}'(t) \rangle dt = \int_a^b F_1(\mathbf{x}(t))\mathrm{d}x_1(t) + \dots + F_m(\mathbf{x}(t))\mathrm{d}x_m(t). \]
We call
\[ \omega = F_1(\mathbf{x})\mathrm{d}x_1 + \dots + F_m(\mathbf{x})\mathrm{d}x_m \]
a differential 1-form.
Total Differentials, Potential Fields, Conservative Fields, and Irrotational Vector Fields
A differential 1-form \(\omega\) is called an exact differential (or total differential) if there exists a \(\mathscr{C}^1\) function \(f\) such that \(\omega = \mathrm{d}f\). In this case, \(f\) is called a primitive function of \(\omega\).
A vector field \(\mathbf{F}\) is called a potential field if there exists a \(\mathscr{C}^1\) function \(f\) such that \(\mathbf{F} = \nabla f\). In this case, \(f\) is called a potential function of \(\mathbf{F}\).
A vector field \(\mathbf{F}\) is called a conservative field if, for any path \(\gamma\), the value of the integral \(\int_{\gamma} \langle \mathbf{F}, \mathbf{T} \rangle \mathrm{d}l\) depends only on the starting and ending points of \(\gamma\), and is independent of \(\gamma\).
Theorem 7.1 For any \(\mathscr{C}^1\) function \(f: U \to \mathbb{R}\) and any \(\mathscr{C}^1\) path \(\gamma \subset U\),
\[ \int_{\gamma} \mathrm{d}f = \int_{\gamma} \langle \nabla f, \mathbf{T} \rangle \mathrm{d}l = f(B) - f(A), \]
where \(A\) and \(B\) are the starting and ending points of the path \(\gamma\) respectively, and \(\mathbf{T}\) is the unit tangent vector field of \(\gamma\). Therefore, gradient vector fields (i.e., potential fields) are conservative fields.
If \(\gamma\) is a closed curve, then \(\int_{\gamma} \mathrm{d}f = \int_{\gamma} \langle \nabla f, \mathbf{T} \rangle \mathrm{d}l = 0\).
Theorem 7.2 Every continuous conservative field in an open region (i.e., a path-connected open set) is a potential field.
Theorem 7.3 If a \(\mathscr{C}^1\) vector field \(\mathbf{F}: U \to \mathbf{R}^m\) is a potential field, then \[ \frac{\partial F_i}{\partial x_j} = \frac{\partial F_j}{\partial x_i}, \quad 1 \le i < j \le m. \]
A vector field satisfying the latter condition is called an irrotational vector field. Therefore, all potential fields are irrotational vector fields.
Curl and Divergence of Plane Vector Fields, Green’s Formula
For a plane \(\mathscr{C}^1\) vector field \[\mathbf{F}(x, y) = \begin{pmatrix} X(x, y) \\ Y(x, y) \end{pmatrix},\]
denote \[\operatorname{rot} \mathbf{F} = Y_x - X_y,\]
\[\operatorname{div} \mathbf{F} = X_x + Y_y = \operatorname{tr} \frac{\partial(X, Y)}{\partial(x, y)},\]
which are called the curl and divergence of \(\mathbf{F}\), respectively.
A vector field with zero curl is called an irrotational vector field, and a vector field with zero divergence is called a source-free (or solenoidal) vector field.
Theorem 7.4 Suppose the boundary of a bounded closed planar region \(\Omega\) is a piecewise \(\mathscr{C}^1\) curve, and let
\[ \mathbf{F}(x,y) = \begin{pmatrix} X(x,y) \\ Y(x,y) \end{pmatrix} \]
be a \(\mathscr{C}^1\) vector field on \(\Omega\). Then
\[ \int_{\partial\Omega} X(x,y)\mathrm{d}x + Y(x,y)\mathrm{d}y = \iint_{\Omega} (Y_x(x,y) - X_y(x,y))\mathrm{d}x\mathrm{d}y, \]
Written in vector field form:
\[ \int_{\partial\Omega} \langle \mathbf{F}, \mathbf{T} \rangle \mathrm{d}l = \iint_{\Omega} \mathrm{rot} \mathbf{F} \, \mathrm{d}x\mathrm{d}y, \]
\[ \int_{\partial\Omega} \langle \mathbf{F}, \mathbf{n} \rangle \mathrm{d}l = \iint_{\Omega} \mathrm{div} \mathbf{F} \, \mathrm{d}x\mathrm{d}y, \]
where \(\partial\Omega\) is the boundary of the region \(\Omega\), and its orientation is determined by the left-hand rule with respect to the outward normal of the region, meaning that as you move forward along \(\partial\Omega\), the region is always located on your left side.
The Concept of Surface Integrals of the Second Kind
Let \((\Sigma, \mathbf{n})\) be an oriented surface. For a continuous vector field \(\mathbf{F}: \Sigma \to \mathbf{R}^3\), the integral \[ \int_{\Sigma} \langle \mathbf{F}, \mathbf{n} \rangle d\sigma \]
is called the integral of the vector field \(\mathbf{F}\) along the normal vector field \(\mathbf{n}\) of the surface \(\Sigma\), also known as the flux of the vector field \(\mathbf{F}\) across the oriented surface \(\Sigma\).
Theorem 7.5 For a \(\mathscr{C}^1\) regular parametric surface \(\Sigma\):
\[ \begin{cases} x = x(u_1, u_2), \\ y = y(u_1, u_2), & (u_1, u_2) \in D \subseteq \mathbb{R}^2, \\ z = z(u_1, u_2), \end{cases} \]
satisfying that the direction of \(\frac{\partial \mathbf{x}}{\partial u_1} \times \frac{\partial \mathbf{x}}{\partial u_2}\) is consistent with the normal vector field \(\mathbf{n}\). Then for any continuous vector field \(\mathbf{F}: \Sigma \to \mathbb{R}^3\),
\[ \int_{\Sigma} \langle \mathbf{F}, \mathbf{n} \rangle \mathrm{d}\sigma = \int_{D} \det\left(\mathbf{F}, \frac{\partial \mathbf{x}}{\partial u_1}, \frac{\partial \mathbf{x}}{\partial u_2}\right) \mathrm{d}u_1 \mathrm{d}u_2, \]
where \(\mathrm{d}\sigma\) is the (undirected) surface area element of the surface \(\Sigma\).
For an \(m\)-dimensional \(\mathscr{C}^1\) regular parametric surface \(\Sigma \subset \mathbb{R}^{m+1}\):
\[ \mathbf{x} = \mathbf{x}(u_1, u_2, \ldots, u_m), \quad (u_1, u_2, \ldots, u_m) \in D \subseteq \mathbb{R}^m, \]
and a continuous vector field \(\mathbf{F}: \Sigma \to \mathbb{R}^{m+1}\),
\[ \int_{\Sigma} \langle \mathbf{F}, \mathbf{n} \rangle \mathrm{d}\sigma = \int_{D} \det\left(\mathbf{F}, \frac{\partial \mathbf{x}}{\partial u_1}, \frac{\partial \mathbf{x}}{\partial u_2}, \ldots, \frac{\partial \mathbf{x}}{\partial u_m}\right) \mathrm{d}u_1 \mathrm{d}u_2 \cdots \mathrm{d}u_m, \]
where \(\mathrm{d}\sigma\) is the \(m\)-dimensional (undirected) volume element of the surface \(\Sigma\).