Scalar and Vector Fields

1.1 Scalar and Vector Fields

A scalar field assigns \(f:\mathbb R^3\to\mathbb R\).

A vector field assigns \(\mathbf F:\mathbb R^3\to\mathbb R^3\).

Assume smooth fields throughout.

1.2 Flux and Circulation

TipSurface and Line Integrals

Flux measures flow through an oriented surface:

\[ \int_{\Sigma}\mathbf F\cdot\mathbf n\,\mathrm dS = \int_{\Sigma}\mathbf F\cdot\mathrm d\mathbf S. \]

Here \(\mathbf n\) is the unit normal.

Circulation measures flow along a closed curve: \[ \oint_L\mathbf F\cdot\mathrm d\mathbf l. \]

1.3 Exterior Derivative

TipDifferential Forms and Definition

A differential \(k\)-form on an open subset of \(\mathbb R^m\) has the form

\[ \omega=\sum_{i_1<\cdots<i_k}f_{i_1\cdots i_k}\, \mathrm dx_{i_1}\wedge\cdots\wedge\mathrm dx_{i_k}. \]

The exterior derivative maps \(k\)-forms to \((k+1)\)-forms:

\[ \mathrm d\omega=\sum_{i_1<\cdots<i_k}\mathrm df_{i_1\cdots i_k} \wedge\mathrm dx_{i_1}\wedge\cdots\wedge\mathrm dx_{i_k}, \qquad \mathrm df=\sum_{j=1}^{m}\frac{\partial f}{\partial x_j}\,\mathrm dx_j. \]

We have \(\mathrm d(\mathrm d\omega)=0\).

1.4 Gradient, Divergence, and Curl

TipDefinitions and Interpretation

\[ \begin{aligned} \nabla f&=(\partial_x f,\partial_y f,\partial_z f),\\ \nabla\cdot\mathbf F&=\partial_xF_x+\partial_yF_y+\partial_zF_z,\\ \nabla\times\mathbf F&=(\partial_yF_z-\partial_zF_y,\, \partial_zF_x-\partial_xF_z,\, \partial_xF_y-\partial_yF_x). \end{aligned} \]

These describe steepest increase, local outflow, and local rotation, respectively.

1.5 Stokes Theorem

Theorem 1.1 (Stokes Theorem) Let \(M\subseteq\mathbb R^m\) be a compact, oriented, smooth \(k\)-dimensional manifold with boundary, where \(1\le k\le m\). Let \(\omega\) be a smooth \((k-1)\)-form defined on an open neighborhood of \(M\).

Give \(\partial M\) the induced boundary orientation: an outward-pointing normal within \(M\), followed by a positively oriented basis of \(T_p(\partial M)\), forms a positively oriented basis of \(T_pM\). Then

\[ \int_M\mathrm d\omega=\int_{\partial M}\omega. \]

Corollary 1.1 (Green’s Theorem with \(m = 2, k = 2\))

For \(\omega=F_x\,\mathrm dx+F_y\,\mathrm dy\), \(\mathrm d\omega=(\partial_xF_y-\partial_yF_x)\,\mathrm dx\wedge\mathrm dy.\)

Let \(L=\partial\Sigma\) be positively oriented: traverse \(L\) with \(\Sigma\) on your left.

\[ \oint_{L}\mathbf F\cdot\mathrm d\mathbf l =\int_{\Sigma}(\partial_xF_y-\partial_yF_x)\,\mathrm dS. \]

Corollary 1.2 (Specific Case: Stokes Theorem with \(m = 3, k = 2\))  

Orient \(L\) by the right-hand rule relative to \(\mathbf n\).

\[ \oint_{L}\mathbf F\cdot\mathrm d\mathbf l =\int_{\Sigma}(\nabla\times\mathbf F)\cdot \mathrm d \mathbf S. \]

Corollary 1.3 (Gauss’s Theorem with \(m = 3, k = 3\))  

Orient \(\Sigma\) by the outward-pointing normal.

\[ \oint_{\Sigma}\mathbf F \cdot \mathrm d \mathbf S =\int_{\Omega}(\nabla\cdot\mathbf F)\,\mathrm dV. \]

1.6 Identities and the Laplacian

NoteKey Identities

\[ \nabla\times\nabla f=\mathbf0, \qquad \nabla\cdot(\nabla\times\mathbf F)=0. \]

Converse statements require domain assumptions; both hold on star-shaped domains.

\[ \Delta f=\nabla\cdot\nabla f =\partial_x^2f+\partial_y^2f+\partial_z^2f. \]

A field satisfying \(\Delta f=0\) is harmonic.

For a smooth vector field, the vector Laplacian satisfies

\[ \nabla^2\mathbf F =\nabla(\nabla\cdot\mathbf F)-\nabla\times(\nabla\times\mathbf F). \]

In Cartesian coordinates, \(\nabla^2\mathbf F=(\Delta F_x,\Delta F_y,\Delta F_z)\).