| Symbol |
Description |
Location |
| \(\R^n\) |
\(n\)-dimensional Euclidean space. |
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| \(\vu,\vv\) |
Generic vectors in Euclidean space. |
Definition 1.1.2 |
| \(\vz\) |
The zero vector. |
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| \(\ui,\uj,\uk\) |
The standard basis vectors in \(\R^3\text{.}\)
|
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| \(D_{\vu}f(\va)\) |
The directional derivative of \(f\) at \(\va\) in the direction \(\vu\text{.}\)
|
Definition 2.4.1 |
| \(D_i f(\va)\) |
The \(i\)-th partial derivative of \(f\) at \(\va\text{.}\)
|
Definition 2.4.4 |
| \(\nabla f(\va)\) |
The gradient of \(f\) at \(\va\text{.}\)
|
Definition 2.4.5 |
| \(\lambda\) |
A Lagrange multiplier. |
Definition 2.6.5 |
| \(\int_D f\,dV\) |
The integral of a scalar field over a region \(D \subseteq \R^n\text{.}\)
|
Definition 3.1.1 |
| \(dV\) |
The volume element. |
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| \(\iint_D f\,dA\) |
A double integral over a planar region \(D\text{.}\)
|
Definition 3.1.2 |
| \(\iiint_E f\,dV\) |
A triple integral over a solid region \(E\text{.}\)
|
Definition 3.1.2 |
| \(\Phi\) |
A coordinate map or change-of-variables map. |
Definition 3.2.1 |
| \(J_{\Phi}(\vu)\) |
The Jacobian determinant of \(\Phi\) at \(\vu\text{.}\)
|
Definition 3.2.1 |
| \((r,\theta)\) |
Polar coordinates in the plane. |
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| \((r,\theta,z)\) |
Cylindrical coordinates in \(\R^3\text{.}\)
|
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| \((\rho,\theta,\phi)\) |
Spherical coordinates in \(\R^3\text{.}\)
|
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| \(\mathbf{F}\) |
A vector field. |
Definition 3.3.1 |
| \(\nabla \cdot \mathbf{F}\) |
The divergence of the vector field \(\mathbf{F}\text{.}\)
|
Definition 3.3.3 |
| \(\nabla \times \mathbf{F}\) |
The curl of the vector field \(\mathbf{F}\) in \(\R^3\text{.}\)
|
Definition 3.3.3 |
| \(\Omega^p(U)\) |
The space of smooth \(p\)-forms on the open set \(U\text{.}\)
|
Definition 3.3.6 |
| \(\omega\) |
A typical \(1\)-form on \(\R^3\text{.}\)
|
Definition 3.3.6 |
| \(\eta\) |
A typical \(2\)-form on \(\R^3\text{.}\)
|
Definition 3.3.6 |
| \(\mu\) |
A typical \(3\)-form on \(\R^3\text{.}\)
|
Definition 3.3.6 |
| \(d\omega\) |
The exterior derivative of the differential form \(\omega\text{.}\)
|
Definition 3.3.8 |
| \(\Phi^*\omega\) |
The pullback of a differential form \(\omega\) by the map \(\Phi\text{.}\)
|
Definition 3.3.12 |
| \(c\) |
A singular \(n\)-cube in the set \(A\text{.}\)
|
Definition 3.4.1 |
| \(\iota_n\) |
The standard \(n\)-cube. |
Definition 3.4.1 |
| \(\Gamma\) |
A chain, written as an integer linear combination of singular cubes. |
Definition 3.4.2 |
| \(\partial c\) |
The boundary of the singular cube \(c\text{.}\)
|
Definition 3.4.3 |
| \(\int_c \omega\) |
The integral of a differential form \(\omega\) over a chain or singular cube \(c\text{.}\)
|
Definition 3.4.11 |
| \(\int_{\gamma} g\,ds\) |
The line integral of the scalar field \(g\) with respect to arclength along \(\gamma\text{.}\)
|
Definition 3.4.17 |
| \(\int_{\gamma}\mathbf{F}\cdot d\mathbf{r}\) |
The line integral of the vector field \(\mathbf{F}\) along the oriented curve \(\gamma\text{.}\)
|
Definition 3.4.22 |
| \(\int_{\sigma} g\,dS\) |
The scalar surface integral of \(g\) over the parameterized surface \(\sigma\text{.}\)
|
Definition 3.4.27 |
| \(\eta_{\mathbf{F}}\) |
The \(2\)-form associated to the vector field \(\mathbf{F}\) for flux integrals. |
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| \(\int_{\sigma}\eta_{\mathbf{F}}\) |
The flux of \(\mathbf{F}\) across the oriented surface \(\sigma\text{.}\)
|
Definition 3.4.30 |
