tangent vector

(0.001 seconds)

4 matching pages

1: 21.1 Special Notation

$g, h$	positive integers.
…
$a\circ b$	intersection index of $a$ and $b$ , two cycles lying on a closed surface. $a\circ b=0$ if $a$ and $b$ do not intersect. Otherwise $a\circ b$ gets an additive contribution from every intersection point. This contribution is $1$ if the basis of the tangent vectors of the $a$ and $b$ cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is $-1$ .
…

…

2: 1.6 Vectors and Vector-Valued Functions

… ► …

3: 3.11 Approximation Techniques

… ►The pair of vectors

\{\mathbf{f},\mathbf{a}\}

… ►The slope of the curve at

(x_{0},y_{0})

is tangent to the line between

(x_{0},y_{0})

and

(x_{1},y_{1})

; similarly the slope at

(x_{3},y_{3})

is tangent to the line between

x_{2},y_{2}

and

x_{3},y_{3}

. …

4: Errata

… ►

Chapter 1 Additions

The following additions were made in Chapter 1:

Section 1.2

New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).
Section 1.3

The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).
Section 1.4

In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).
Section 1.8

In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).
Section 1.10

New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).
Section 1.13

New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).
Section 1.14(i)

Another form of Parseval’s formula, (1.14.7_5).
Section 1.16

We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.
Section 1.17

Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.
Section 1.18

An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

… ►

Subsection 19.11(i)

A sentence and unnumbered equation

R_{C}\left(\gamma-\delta,\gamma\right)=\frac{-1}{\sqrt{\delta}}\operatorname{% arctan}\left(\frac{\sqrt{\delta}\sin\theta\sin\phi\sin\psi}{\alpha^{2}-1-% \alpha^{2}\cos\theta\cos\phi\cos\psi}\right),

were added which indicate that care must be taken with the multivalued functions in (19.11.5). See (Cayley, 1961, pp. 103-106).

Suggested by Albert Groenenboom.

… ►

Equation (36.2.18), Subsections §§36.12(i), 36.15(i), 36.15(ii)

The vector at the origin, previously given as $0$ , has been clarified to read $\boldsymbol{{0}}$ .

… ►

Equations (4.45.8), (4.45.9)

These equations have been rewritten to improve the numerical computation of $\operatorname{arctan}x$ .

… ►

Equation (21.3.4)

21.3.4

\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}+\mathbf{m}_{1}}{% \boldsymbol{{\beta}}+\mathbf{m}_{2}}\left(\mathbf{z}\middle|\boldsymbol{{% \Omega}}\right)={\mathrm{e}}^{2\pi\mathrm{i}\boldsymbol{{\alpha}}\cdot\mathbf{% m}_{2}}\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta% }}}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)

Originally the vector $\mathbf{m}_{2}$ on the right-hand side was given incorrectly as $\mathbf{m}_{1}$ .

Reported 2012-08-27 by Klaas Vantournhout.

…