Calculus of Vector Fields

Line Integrals of Vector Fields

Line Integral 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

\[ \int_{\gamma} \mathbf{F} \cdot \mathrm{d}\mathbf{l} = \int_{a}^{b} \langle \mathbf{F}(\mathbf{x}(t)), \mathbf{x}'(t) \rangle dt. \]

Differential 1-form

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

Curl and Divergence of Plane Vector Fields

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

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 \(\Sigma\).

Surface Integrals of the Second Kind in Higher Dimensions

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 \(\mathbf{F}: \Sigma \to \mathbb{R}^{m+1}\),

where \(\mathrm{d}\sigma\) is the \(m\)-dimensional (undirected) volume element of the surface \(\Sigma\).

Wedge Product of Linear Functions and Differential 2-Forms

For two linear functions \(L_1, L_2\) and two vectors \(\mathbf{u}, \mathbf{v}\), we denote:

\[ (L_1 \wedge L_2)(\mathbf{u}, \mathbf{v}) = \det \begin{pmatrix} L_1(\mathbf{u}) & L_1(\mathbf{v}) \\ L_2(\mathbf{u}) & L_2(\mathbf{v}) \end{pmatrix}. \]

We denote the expression:

\[ \omega = X(x, y, z)\mathrm{d}y \wedge \mathrm{d}z + Y(x, y, z)\mathrm{d}z \wedge \mathrm{d}x + Z(x, y, z)\mathrm{d}x \wedge \mathrm{d}y, \]

which is called a differential 2-form (or second-order differential form).

Theorem 7.6 Let \(\Sigma\) be an oriented \(\mathscr{C}^1\) regular parametric surface in \(\mathbb{R}^3\), with parametric equation \(\mathbf{x} = \mathbf{x}(u_1, u_2)\), where \((u_1, u_2) \in D \subseteq \mathbb{R}^2\). Assume that the direction of the normal vector induced by the parametrization, \(\frac{\partial \mathbf{x}}{\partial u_1} \times \frac{\partial \mathbf{x}}{\partial u_2}\), is consistent with the given normal vector field \(\mathbf{n}\) of the surface.

For a given continuous vector field \(\mathbf{F} = (X, Y, Z)\), we construct the equivalent differential 2-form:

\[ \omega = X(x, y, z)\mathrm{d}y \wedge \mathrm{d}z + Y(x, y, z)\mathrm{d}z \wedge \mathrm{d}x + Z(x, y, z)\mathrm{d}x \wedge \mathrm{d}y \]

Then the surface integral of the second kind (flux integral) of the vector field \(\mathbf{F}\) over the surface \(\Sigma\) is equivalent to the integral of the differential 2-form \(\omega\) over \(\Sigma\). Utilizing the determinant definition of the wedge product, the calculation formula is expressed as:

\[ \int_{\Sigma} \langle \mathbf{F}, \mathbf{n} \rangle \mathrm{d}\sigma = \int_{\Sigma} \omega = \int_{D} \det\left(\mathbf{F}, \frac{\partial \mathbf{x}}{\partial u_1}, \frac{\partial \mathbf{x}}{\partial u_2}\right) \mathrm{d}u_1 \mathrm{d}u_2 \]

In this formula, the differential form is pulled back to the domain \(D\) via the parametrization, and the expansion of the determinant precisely corresponds to a linear combination of Jacobians:

\[ \int_{\Sigma} \omega = \int_{D} \left( X \frac{\partial(y,z)}{\partial(u_1,u_2)} + Y \frac{\partial(z,x)}{\partial(u_1,u_2)} + Z \frac{\partial(x,y)}{\partial(u_1,u_2)} \right) \mathrm{d}u_1 \mathrm{d}u_2 \]

Exterior Derivative

For a differentiable function (0-form) \(f\), the exterior derivative of \(f\) is defined as the first-order differential \(\mathrm{d}f\).

For a 1-form \(\omega = F_1\mathrm{d}x^1 + \dots + F_m\mathrm{d}x^m\), the exterior derivative of \(\omega\) is defined as:

\[ \mathrm{d}\omega = \mathrm{d}F_1 \wedge \mathrm{d}x^1 + \dots + \mathrm{d}F_m \wedge \mathrm{d}x^m. \]

Curl and Divergence of Spatial Vector Fields, Gauss’s Theorem and Stokes’s Theorem

Curl and Divergence

For a vector field \(\mathbf{F} = (X, Y, Z)^T\) in \(\mathbb{R}^3\) (under Cartesian coordinates), we denote:

\[ \operatorname{rot} \mathbf{F} = \operatorname{curl} \mathbf{F} = \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ X & Y & Z \end{vmatrix} = \begin{pmatrix} Z_y - Y_z \\ X_z - Z_x \\ Y_x - X_y \end{pmatrix}, \]

\[ \operatorname{div} \mathbf{F} = \nabla \cdot \mathbf{F} = \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} \cdot \begin{pmatrix} X \\ Y \\ Z \end{pmatrix} = X_x + Y_y + Z_z = \operatorname{tr} \frac{\partial(X, Y, Z)}{\partial(x, y, z)}, \]

These 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 solenoidal (or source-free) vector field.

Theorem 7.7 (Gauss’s Theorem) Let \(\Omega \subset \mathbb{R}^3\) be a bounded closed region, and the boundary \(\partial \Omega\) of \(\Omega\) be a piecewise \(\mathscr{C}^1\) surface, oriented outwards from \(\Omega\). Then:

For any \(\mathscr{C}^1\) vector field \(\mathbf{F}: \Omega \to \mathbb{R}^3\),

\[ \int_{\partial \Omega} \langle \mathbf{F}, \mathbf{n} \rangle \mathrm{d}\sigma = \int_{\Omega} \operatorname{div} \mathbf{F} \, \mathrm{d}\mu, \]

which means the net outward flux of the vector field along the region’s boundary is equal to the integral of the divergence inside the region. Here, \(\mu\) is the three-dimensional volume.

For any \(\mathscr{C}^1\) differential 2-form \(\omega = P(x,y,z)\mathrm{d}y \wedge \mathrm{d}z + Q(x,y,z)\mathrm{d}z \wedge \mathrm{d}x + R(x,y,z)\mathrm{d}x \wedge \mathrm{d}y\), the generalized Stokes’ theorem \(\int_{\partial \Omega} \omega = \int_{\Omega} \mathrm{d}\omega\) takes the explicit expanded form:

\[ \int_{\partial \Omega} P \, \mathrm{d}y \wedge \mathrm{d}z + Q \, \mathrm{d}z \wedge \mathrm{d}x + R \, \mathrm{d}x \wedge \mathrm{d}y = \int_{\Omega} \left( \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \right) \mathrm{d}x \wedge \mathrm{d}y \wedge \mathrm{d}z. \]

Generalized Gauss’s Theorem

Let \(\Omega \subset \mathbb{R}^{m+1}\) be a bounded closed region, and the boundary \(\partial \Omega\) of \(\Omega\) be a piecewise \(\mathscr{C}^1\) \(m\)-dimensional surface, oriented outwards from \(\Omega\). Then:

For any \(\mathscr{C}^1\) vector field \(\mathbf{F}: \Omega \to \mathbb{R}^{m+1}\),

\[ \int_{\partial \Omega} \langle \mathbf{F}, \mathbf{n} \rangle \mathrm{d}\sigma = \int_{\Omega} \operatorname{div} \mathbf{F} \, \mathrm{d}\mu, \]

which means the net outward flux of the vector field along the region’s boundary is equal to the integral of the divergence inside the region. Here, \(\mu\) is the \((m+1)\)-dimensional volume.

For any \(\mathscr{C}^1\) \(m\)-form

\[ \omega = \sum_{k=1}^{m+1} (-1)^{\#\sigma(k)} X_k \mathrm{d}x_1 \wedge \cdots \wedge \mathrm{d}x_{k-1} \wedge \mathrm{d}x_{k+1} \wedge \cdots \wedge \mathrm{d}x_{m+1}, \]

(where \(\#\sigma(k)\) is the parity of the permutation of \(k, 1, 2, \dots, k-1, k+1, \dots, m+1\) relative to \(1, 2, \dots, m+1\)), it holds that

\[ \int_{\partial \Omega} \omega = \int_{\Omega} \mathrm{d}\omega, \]

where

\[ \mathrm{d}\omega = \operatorname{tr} \frac{\partial(X_1, X_2, \dots, X_{m+1})}{\partial(x_1, x_2, \dots, x_{m+1})} \mathrm{d}x_1 \wedge \mathrm{d}x_2 \wedge \cdots \wedge \mathrm{d}x_{m+1}. \]

Theorem 7.8 (Stokes’s Theorem) Let \(\Sigma \subset \mathbb{R}^3\) be a bounded, piecewise \(\mathscr{C}^1\) oriented surface, and the boundary \(\partial \Sigma\) of \(\Sigma\) be a piecewise \(\mathscr{C}^1\) oriented curve, such that when standing in the direction of the specified normal of the surface and walking along \(\partial \Sigma\), the surface is always on the left-hand side. Then:

For any \(\mathscr{C}^1\) vector field \(\mathbf{F}: \Sigma \to \mathbb{R}^3\),

\[ \int_{\partial \Sigma} \langle \mathbf{F}, \mathbf{T} \rangle \mathrm{d}l = \int_{\Sigma} \langle \operatorname{rot} \mathbf{F}, \mathbf{n} \rangle \mathrm{d}\sigma, \]

which means the circulation of the vector field along the surface boundary is equal to the integral of the curl over the surface.

For any \(\mathscr{C}^1\) differential 1-form \(\omega = P(x,y,z)\mathrm{d}x + Q(x,y,z)\mathrm{d}y + R(x,y,z)\mathrm{d}z\), the generalized Stokes’ theorem \(\int_{\partial \Sigma} \omega = \int_{\Sigma} \mathrm{d}\omega\) takes the explicit expanded form:

\[ \int_{\partial \Sigma} P \, \mathrm{d}x + Q \, \mathrm{d}y + R \, \mathrm{d}z = \int_{\Sigma} \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathrm{d}y \wedge \mathrm{d}z + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathrm{d}z \wedge \mathrm{d}x + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathrm{d}x \wedge \mathrm{d}y. \]