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3.7. ALIGNMENT BUILDING BLOCKS
7
\text inside an align environment:
x= y
1
y
2
+y
3
y
5
+y
8
: : :
by (3.21)
(3.15)
=y
0
y
by (4.1)
(3.16)
=y(0)y
0
by Axiom 1.
(3.17)
\begin{align}
x& = y_1-y_2+y_3-y_5+y_8-\dots
&& \text{by \eqref{eq:C}}\\
& = y’\circ y^*
&& \text{by \eqref{eq:D}}\\
& = y(0) y’
&& \text {by Axiom 1.}
\end{align}
Avariant environment alignat allows the horizontal space between equations
to be explicitly specied. This environment takes one argument, the number of
\equation columns": count the maximum number of &s in any row, add 1 and
divide by 2.
x= y
1
y
2
+y
3
y
5
+y
8
: : : by (3.21)
(3.18)
=y
0
y
by (4.1)
(3.19)
=y(0)y
0
by Axiom 1.
(3.20)
\begin{alignat}{2}
x& = y_1-y_2+y_3-y_5+y_8-\dots
& = y’\circ y^* && \text{by \eqref{eq:D}}\\
& = y(0) y’
&& \text {by Axiom 1.}
\end{alignat}
3.7 Alignment building blocks
Like equation, the multi-equation environments gather, align, and alignat
are designed to produce a structure whose width is the full line width. This
means, for example, that one cannot readily add parentheses around the entire
structure. But variants gathered, aligned, and alignedat are provided whose
total width is the actual width of the contents; thus they can be used as a
component in a containing expression. E.g.,
B
0
= @  E;
E
0
=@  B   4j;
)
Maxwell’s equations
\begin{equation*}
\left.\begin{aligned}
B’&=-\partial\times E,\\
E’&=\partial\times B - 4\pi j,
\end{aligned}
\right\}
\end{equation*}
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8
3. DISPLAYED EQUATIONS
Like the array environment, these -ed variants also take an optional [t] or
[b] argument to specify vertical positioning.
\Cases" constructions like the following are common in mathematics:
(3.21)
P
r j
=
(
0
if r   j is odd;
r! ( 1)
(r j)=2
if r   j is even:
and in the amsmath package there is a cases environment to make them easy
to write:
P_{r-j}=\begin{cases}
0& \text{if $r-j$ is odd},\\
r!\,(-1)^{(r-j)/2}& \text{if $r-j$ is even}.
\end{cases}
Notice the use of \text (cf. x6) and the nested math formulas.
Placing equation numbers can be a rather complex problem in multiline displays.
The environments of the amsmath package try hard to avoid overprinting an
equation number on the equation contents, if necessary moving the number
down or up to a separate line. Diculties in accurately calculating the prole of
an equation can occasionally result in number movement that doesn’t look right.
There is a \raisetag command provided to adjust the vertical position of the
current equation number, if it has been shifted away from its normal position.
To move a particular number up by six points, write \raisetag{6pt}. This
kind of adjustment is ne tuning like line breaks and page breaks, and should
therefore be left undone until your document is nearly nalized, or you may end
up redoing the ne tuning several times to keep up with changing document
contents.
3.9 Vertical spacing and page breaks in multiline displays
You can use the \hdimensioni] command to get extra vertical space be- tween lines in all the amsmath displayed equation environments, as is usual in LAT E X. When the amsmath package is in use page breaks between equa- tion lines are normally disallowed; the philosophy is that page breaks in such material should receive individual attention from the author. To get an in- dividual page break inside a particular displayed equation, a \displaybreak command is provided. \displaybreak is best placed immediately before the \\ where it is to take eect. Like LAT E X’s \pagebreak, \displaybreak takes an optional argument between 0 and 4 denoting the desirability of the page- break. \displaybreak means \it is permissible to break here" without encouraging a break; \displaybreak with no optional argument is the same as \displaybreak and forces a break. If you prefer a strategy of letting page breaks fall where they may, even in the middle of a multi-line equation, then you might put \allowdisplaybreaks in the preamble of your document. An optional argument 1{4 can be used for SDK control service:VB.NET PDF copy, paste image library: copy, paste, cut PDF images Copy, paste and cut PDF image while preview without adobe reader component installed. Image resize function allows VB.NET users to zoom and crop image. www.rasteredge.com SDK control service:C# PDF copy, paste image Library: copy, paste, cut PDF images in C#.NET PDF SDK - Copy, Paste, Cut PDF Image in C#.NET. C#.NET Demo Code: Cut Image in PDF Page in C#.NET. PDF image cutting is similar to image deleting. www.rasteredge.com 3.11. EQUATION NUMBERING 9 ner control:  means allow page breaks, but avoid them as much as possi- ble; values of 2,3,4 mean increasing permissiveness. When display breaks are enabled with \allowdisplaybreaks, the \\* command can be used to prohibit apagebreak after a given line, as usual. Note: Certain equation environments wrap their contents in an unbreakable box, with the consequence that neither \displaybreak nor \allowdisplaybreaks will have any eect on them. These include split, aligned, gathered, and alignedat. 3.10 Interrupting a display The command \intertext is used for a short interjection of one or two lines of text in the middle of a multiple-line display structure (see also the \text command in x6). Its salient feature is preservation of the alignment, which would not happen if you simply ended the display and then started it up again afterwards. \intertext may only appear right after a \\ or \\* command. Notice the position of the word \and" in this example. A 1 =N 0 (; 0 ) (; 0 ); (3.22) A 2 =(; 0 ) (; ); (3.23) and A 3 =N(; !): (3.24) \begin{align} A_1&=N_0(\lambda;\Omega’)-\phi(\lambda;\Omega’),\\ A_2&=\phi(\lambda;\Omega’)-\phi(\lambda;\Omega),\\ \intertext{and} A_3&=\mathcal{N}(\lambda;\omega). \end{align} 3.11 Equation numbering 3.11.1 Numbering hierarchy In LAT E X if you wanted to have equations numbered within sections|that is, have equation numbers (1.1), (1.2), . . . , (2.1), (2.2), . . . , in sections 1, 2, and so forth|you could redene \theequation as suggested in the LAT E Xmanual [7, x6.3, xC.8.4]: \renewcommand{\theequation}{\thesection.\arabic{equation}} This works pretty well, except that the equation counter won’t be reset to zero at the beginning of a new section or chapter, unless you do it yourself using \setcounter. To make this a little more convenient, the amsmath package pro- vides a command \numberwithin. To have equation numbering tied to section numbering, with automatic reset of the equation counter, write \numberwithin{equation}{section} As its name implies, the \numberwithin command can be applied to any counter, not just the equation counter. SDK control service:VB.NET PDF Page Delete Library: remove PDF pages in vb.net, ASP. doc.Save(outPutFilePath). How to VB.NET: Delete Consecutive Pages from PDF. doc.Save(outPutFilePath). How to VB.NET: Delete Specified Pages from PDF. www.rasteredge.com SDK control service:VB.NET PDF Page Insert Library: insert pages into PDF file in vb. Page: Insert PDF Pages. |. Home ›› XDoc.PDF ›› VB.NET PDF: Insert PDF Page. Add and Insert Multiple PDF Pages to PDF Document Using VB. www.rasteredge.com 10 4. MISCELLANEOUS MATHEMATICAL FEATURES 3.11.2 Cross references to equation numbers To make cross-references to equations easier, an \eqref command is provided. This automatically supplies the parentheses around the equation number. I.e., if \ref{abc} produces 3.2 then \eqref{abc} produces (3.2). 3.11.3 Subordinate numbering sequences The amsmath package provides also a subequations environment to make it easy to number equations in a particular group with a subordinate numbering scheme. For example \begin{subequations} ... \end{subequations} causes all numbered equations within that part of the document to be numbered (4.9a) (4.9b) (4.9c) . . . , if the preceding numbered equation was (4.8). A \label command immediately after \begin{subequations} will produce a \ref of the parent number 4.9, not 4.9a. The counters used by the subequations environ- ment are parentequation and equation and \addtocounter, \setcounter, \value, etc., can be applied as usual to those counter names. To get anything other than lowercase letters for the subordinate numbers, use standard LAT E X methods for changing numbering style [7, x6.3, xC.8.4]. For example, redening \theequation as follows will produce roman numerals. \begin{subequations} \renewcommand{\theequation}{\theparentequation \roman{equation}} ... |4| Miscellaneous mathematical features 4.1 Matrices The amsmath package provides some environments for matrices beyond the basic array environment of LAT E X. The pmatrix, bmatrix, Bmatrix, vmatrix and Vmatrix have (respectively) ( ), [ ], f g, j j, and k k delimiters built in. For naming consistency there is a matrix environment sans delimiters. This is not entirely redundant with the array environment; the matrix environments all use more economical horizontal spacing than the rather prodigal spacing of the array environment. Also, unlike the array environment, you don’t have to give column specications for any of the matrix environments; by default you can have up to 10 centered columns.2 (If you need left or right alignment in a column or other special formats you must resort to array.) 2 More precisely: The maximum number of columns in a matrix is determined by the counter MaxMatrixCols (normal value = 10), which you can change if necessary using LAT E X’s \setcounter or \addtocounter commands. SDK control service:C# PDF Page Insert Library: insert pages into PDF file in C#.net doc2.Save(outPutFilePath); Add and Insert Multiple PDF Pages to PDF Document Using C#. Add and Insert Blank Pages to PDF File in C#.NET. www.rasteredge.com SDK control service:C# PDF Page Delete Library: remove PDF pages in C#.net, ASP.NET doc.Save(outPutFilePath); Demo Code: How to Delete Consecutive Pages from PDF in C#.NET. Demo Code: How to Delete Specified Pages from PDF in C#.NET. www.rasteredge.com 4.3. DOTS 11 To produce a small matrix suitable for use in text, there is a smallmatrix environment (e.g., a b cd ) that comes closer to tting within a single text line than a normal matrix. Delimiters must be provided; there are no p,b,B,v,V versions of smallmatrix. The above example was produced by \bigl( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr) \hdotsfor{hnumberi} produces a row of dots in a matrix spanning the given number of columns. For example, a b c d e : : : : : : : \begin{matrix} a&b&c&d\\ e&\hdotsfor{3} \end{matrix} The spacing of the dots can be varied through use of a square-bracket option, for example, \hdotsfor[1.5]{3}. The number in square brackets will be used as a multiplier (i.e., the normal value is 1.0). (4.1) 0 B B @ D 1 t a 12 t 2 :: : a 1n t n a 21 t 1 D 2 t :: : a 2n t n :: : : : : : : : : : : : : : : : : : : : : a n1 t 1 a n2 t 2 :: : D n t 1 C C A ; \begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\ -a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\ \hdotsfor{4}\\ -a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix} 4.2 Math spacing commands The amsmath package slightly extends the set of math spacing commands, as shown below. Both the spelled-out and abbreviated forms of these commands are robust, and they can also be used outside of math. Abbrev. Spelled out Example Abbrev. Spelled out Example no space )( no space )( \, \thinspace )( \! \negthinspace )( \: \medspace )( \negmedspace )( \; \thickspace )( \negthickspace )( \quad ) ( \qquad ) ( For the greatest possible control over math spacing, use \mspace and ‘math units’. One math unit, or mu, is equal to 1/18 em. Thus to get a negative \quad you could write \mspace{-18.0mu}. 4.3 Dots For preferred placement of ellipsis dots (raised or on-line) in various contexts there is no general consensus. It may therefore be considered a matter of taste. By using the semantically oriented commands SDK control service:C# PDF remove image library: remove, delete images from PDF in C#. NET comment annotate PDF, VB.NET delete PDF pages, VB.NET Able to cut and paste image into another PDF PDF image in preview without adobe PDF reader component. www.rasteredge.com SDK control service:How to C#: Basic SDK Concept of XDoc.PDF for .NET example, you may easily create, load, combine, and split PDF file(s), and add, create, insert, delete, re-order, copy, paste, cut, rotate, and save PDF page(s www.rasteredge.com 12 4. MISCELLANEOUS MATHEMATICAL FEATURES  \dotsc for \dots with commas"  \dotsb for \dots with binary operators/relations"  \dotsm for \multiplication dots"  \dotsi for \dots with integrals"  \dotso for \other dots" (none of the above) instead of \ldots and \cdots, you make it possible for your document to be adapted to dierent conventions on the y, in case (for example) you have to submit it to a publisher who insists on following house tradition in this respect. The default treatment for the various kinds follows American Mathematical Society conventions: Then we have the series A_1, A_2, \dotsc, the regional sum A_1 +A_2 +\dotsb , the orthogonal product A_1 A_2 \dotsm , and the infinite integral \[\int_{A_1}\int_{A_2}\dotsi.
Then we have the series A
1
;A
2
;: : : ,
the regional sum A
1
+A
2
+   , the
orthogonal product A
1
A
2
 , and
the innite integral
Z
A
1
Z
A
2
 :
4.4 Nonbreaking dashes
Acommand \nobreakdash is provided to suppress the possibility of a linebreak
after the following hyphen or dash. For example, if you write ‘pages 1{9’ as
pages 1\nobreakdash--9 then a linebreak will never occur between the dash
and the 9. You can also use \nobreakdash to prevent undesirable hyphen-
ations in combinations like $p$-adic. For frequent use, it’s advisable to make
abbreviations, e.g.,
\newcommand{\p}{$p$\nobreakdash}% for "\p-adic"
\newcommand{\Ndash}{\nobreakdash--}% for "pages 1\Ndash 9"
%
For "\n dimensional" ("n-dimensional"):
\newcommand{\n}{$n$\nobreakdash-\hspace{0pt}}
The last example shows how to prohibit a linebreak after the hyphen but allow
normal hyphenation in the following word. (It suces to add a zero-width space
after the hyphen.)
4.5 Accents in math
In ordinary LAT
E
Xthe placement of the second accent in doubled math accents
is often poor. With the amsmath package you will get improved placement of
the second accent:
^
^
A(\hat{\hat{A}}).
The commands \dddot and \ddddot are available to produce triple and
able in LAT
E
X.
To get a superscripted hat or tilde character, load the amsxtra package and
use \sphat or \sptilde. Usage is A\sphat (note the absence of the ^ character).
To place an arbitrary symbol in math accent position, or to get under accents,
see the accents package by Javier Bezos.
4.10. AFFIXING SYMBOLS TO OTHER SYMBOLS
13
4.6 Roots
In ordinary L
A
T
E
Xthe placement of root indices is sometimes not so good:
p
k
(\sqrt[\beta]{k}). In the amsmath package \leftroot and \uproot allow
you to adjust the position of the root:
\sqrt[\leftroot{-2}\uproot{2}\beta]{k}
will move the beta up and to the right:
p
k. The negative argument used with
\leftroot moves the  to the right. The units are a small amount that is a
4.7 Boxed formulas
The command \boxed puts a box around its argument, like \fbox except that
the contents are in math mode:
(4.2)
 C(() + 
M
(0; ))
\boxed{\eta \leq C(\delta(\eta) +\Lambda_M(0,\delta))}
4.8 Over and under arrows
Basic LAT
E
Xprovides \overrightarrow and \overleftarrow commands. Some
additional over and under arrow commands are provided by the amsmath package
to extend the set:
\overleftarrow
\underleftarrow
\overrightarrow
\underrightarrow
\overleftrightarrow
\underleftrightarrow
4.9 Extensible arrows
\xleftarrow and \xrightarrow produce arrows that extend automatically to
accommodate unusually wide subscripts or superscripts. These commands take
one optional argument (the subscript) and one mandatory argument (the su-
perscript, possibly empty):
(4.3)
A
n+ 1
B
ni 1
!
T
C
4.10 Axing symbols to other symbols
L
A
T
E
Xprovides \stackrel for placing a superscript above a binary relation. In
the amsmath package there are somewhat more general commands, \overset
and \underset, that can be used to place one symbol above or below another
symbol, whether it’s a relation or something else. The input \overset{*}{X}
will place a superscript-size  above the X:
X; \underset is the analog for
See also the description of \sideset in x7.2.
14
4. MISCELLANEOUS MATHEMATICAL FEATURES
4.11 Fractions and related constructions
4.11.1 The \frac, \dfrac, and \tfrac commands
The \frac command, which is in the basic command set of LAT
E
X, takes two
arguments|numerator and denominator|and typesets them in normal fraction
form. The amsmath package provides also \dfrac and \tfrac as convenient
abbreviations for {\displaystyle\frac ... } and {\textstyle\frac ... }.
(4.4)
1
k
log
2
c(f)
1
k
log
2
c(f)
r
1
k
log
2
c(f)
r
1
k
log
2
c(f)
\begin{equation}
\frac{1}{k}\log_2 c(f)\;\tfrac{1}{k}\log_2 c(f)\;
\sqrt{\frac{1}{k}\log_2 c(f)}\;\sqrt{\dfrac{1}{k}\log_2 c(f)}
\end{equation}
4.11.2 The \binom, \dbinom, and \tbinom commands
For binomial expressions such as
n
k
amsmath has \binom, \dbinom and \tbinom:
(4.5)
2
k
k
1
2
k 1
+
k
2
2
k 2
2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}
4.11.3 The \genfrac command
The capabilities of \frac, \binom, and their variants are subsumed by a general-
ized fraction command \genfrac with six arguments. The last two correspond
to \frac’s numerator and denominator; the rst two are optional delimiters
(as seen in \binom); the third is a line thickness override (\binom uses this to
set the fraction line thickness to 0|i.e., invisible); and the fourth argument
is a mathstyle override: integer values 0{3 select respectively \displaystyle,
\textstyle, \scriptstyle, and \scriptscriptstyle. If the third argument
is left empty, the line thickness defaults to ‘normal’.
\genfrac{left-delim}{right-delim}{thickness}{mathstyle}
{numerator}{denominator}
To illustrate, here is how \frac, \tfrac, and \binom might be dened.
\newcommand{\frac}{\genfrac{}{}{}{}{#1}{#2}}
\newcommand{\tfrac}{\genfrac{}{}{}{1}{#1}{#2}}
\newcommand{\binom}{\genfrac{(}{)}{0pt}{}{#1}{#2}}
If you nd yourself repeatedly using \genfrac throughout a document for a
particular notation, you will do yourself a favor (and your publisher) if you
dene a meaningfully-named abbreviation for that notation, along the lines of
\frac and \binom.
The primitive generalized fraction commands \over, \overwithdelims, \atop,
\atopwithdelims, \above, \abovewithdelims produce warning messages if
used with the amsmath package, for reasons discussed in technote.tex.
4.14. DELIMITERS
15
4.12 Continued fractions
The continued fraction
(4.6)
1
p
2+
1
p
2+
1
p
2+   
can be obtained by typing
\cfrac{1}{\sqrt{2}+
\cfrac{1}{\sqrt{2}+
\cfrac{1}{\sqrt{2}+\dotsb
}}}
This produces better-looking results than straightforward use of \frac. Left
or right placement of any of the numerators is accomplished by using \cfrac[l]
4.13 Smash options
The command \smash is used to typeset a subformula with an eective height
and depth of zero, which is sometimes useful in adjusting the subformula’s
position with respect to adjacent symbols. With the amsmath package \smash
has optional arguments t and b, because occasionally it is advantageous to be
able to \smash" only the top or only the bottom of something while retaining
the natural depth or height. For example, when adjacent radical symbols are
unevenly sized or positioned because of dierences in the height and depth of
their contents, \smash can be employed to makethem more consistent. Compare
p
x+
p
y+
p
zand
p
x+
p
y+
p
z, where the latter was produced by $\sqrt{x} +\sqrt{\smash[b]{y}} + \sqrt{z}$.
4.14 Delimiters
4.14.1 Delimiter sizes
The automatic delimiter sizing done by \left and \right has two limitations:
First, it is applied mechanically to produce delimiters large enough to encompass
the largest contained item, and second, the range of sizes is not even approxi-
mately continuous but has fairly large quantum jumps. This means that a math
fragment that is innitesimally too large for a given delimiter size will get the
next larger size, a jump of 3pt or so in normal-sized text. There are two or
three situations where the delimiter size is commonly adjusted, using a set of
commands that have ‘big’ in their names.
Delimiter
text
\left
\bigl
\Bigl
\biggl
\Biggl
size
size
\right
\bigr
\Bigr
\biggr
\Biggr
Result
(b)(
c
d
) (b)
c
d
b

c
d
b

c
d
b

c
d
b
c
d
!