L A TEX Lab 3: advanced concepts

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1 L A TEX Lab 3: advanced concepts Simon Shaw November 20, 205 Lab 3: macros Suppose that you wanted to write many instances of dy dx. In code this is \frac{dy}{dx} But you might still think it is a lot to type many, many times. You can get around this by defining a macro: \newcommand{\dydx}{\frac{dy}{dx}} And then: typing $\dydx$ will produce dy. dx dy But what if you wanted to write several variants: dx, df dz, ds dt etc? You can abbreviate this by defining a macro with arguments: \newcommand{\deriv}[2]{\frac{d#}{d#2}} And then: typing \[\deriv{s}{t}\deriv{t}{s}=.\] will produce dsdt dtds =. While \[\deriv{y}{x}\deriv{x}{y}=.\] will produce dy dx dxdy =. How about these? dy dx, d2 f dm+ s dz 2, etc? dtm+ You can deal with these by defining a macro with an optional argument: \newcommand{\derivn}[3][]{\frac{d^{#}#2}{d#3^{#}}}

2 And then: typing \[\derivn{y}{x}+\derivn[2]{f}{z}+\derivn[m+]{s}{t}=?\] will produce dy dx + d2 f dz + dm+ s 2 dt m+ =? You can have only one optional argument, and up to nine arguments in total. The \newcommand{...} definitions should go in the document preamble. One last thing for this section: Which do you prefer: x y z or x y z? If the latter then you can re-new the \le and \ge commands with \renewcommand{\le}{\leqslant} \renewcommand{\ge}{\geqslant} 2 Lab 3: environments Follow the lecture up to here and make sure all necessary preamble material has been added. Lemma 2.. For all 2 2 matrices ( a b c d) such that ad bc 0 we have, ( ) ( ) a b d b = c d ad bc c a. Proof. Just multiply to get ( a b c d )( c d b a ) = ( ad bc 0 0 ad bc ). Theorem 2.2. A 2 2 matrix A = ( a b c d) is invertible if and only if ad bc 0. Proof. From Lemma 2. we see that if ad bc 0 the inverse of A exists. On the other hand, since the inverse must be unique if it exists, if ad bc = 0 Lemma 2. shows that A can have no inverse. Proposition 2.3. For all real numbers x and y and for all ǫ > 0 we have xy x2 2ǫ 2 + ǫ2 y 2 2. Proof. Simply note that 0 (x/ǫ ǫy) 2 and expand the bracket. Corollary 2.4 ((to Prop. 2.3) The Arithmetic-Geometric Mean Inequality). For all real numbers x and y we have Proof. Take ǫ = in Prop xy x2 2 + y2 2. 2

3 Left centre centre right L C C R L2 C2 C2 R2 Table : A simple table showing left, centre and right alignment, and horizontal and vertical lines. 3 Lab 3: tables and floats Table has been created by using the tabular environment inside the table environment. If we had just used the tabular environment like this: Left centre centre right L C C R L2 C2 C2 R2 Then the table would appear in the same place it was typed. The table environment allows us to place the table inside a floating environment with a caption and a label. The floating environment is then positioned in the document automatically by L A TEX, and the label can be used to cross reference the Table number using the \label and \ref commands that we have seen before. 4 Lab 3: tikz Again, follow the lecture and make sure that all necessary preamble material has been added. The tikz plot is shown in Figure and another example is shown in Figure 2. Figure 3 shows the illustration of the integral test from lectures. It is clear that this is sophisticated material. There is a lot to take in. 0 y = 3sin(2x ) z = 2 x 2 / Figure : Plot of y = 3sin(2x ) and z = 2 x 2 /2 for x [ 3,5] 3

4 0 y = ln(x ) z = 2 x 2 / Figure 2: Plot of y = ln(x ) and z = 2 x 2 /2 for x [ 3,5] Figure 3: Integral test illustration from lectures. 4

5 These examples will reward study and you can ask in the labs if you get stuck. Or you can look online there is a huge amount of information. Let s just note that the tikz plot is contained in a tikzpicture environment. This itself is within a figure environment which, like the table environment also floats and allows for a labelled and numbered caption. You need to process this with DVI->PS and then PS->PDF to get the tikz graphics. 5 Lab session 3 Download lab3.tex and examine the source code that creates the examples in this lecture. Feel free to experiment and alter. Also, for the next lab session you should download the handwritten material (from BBL or the web page) in the file lab3-proofs-to-create.pdf Spend some time understanding these results they are important and then create your own versions in L A TEX. References [] Nicholas J. Higham. Handbook of writing for the mathematical sciences. SIAM, 998. [2] Leslie Lamport. L A TEX Users Guide & Reference Manual. Addison-Wesley Publishing Company, 986. [3] Helmut Kopka and Patrick W. Daly. A Guide to L A TEX2ε. Document preparation for beginners and advanced users. Addison Wesley, 995, second edition. 5