Derivative, limit, summation and integral __function in latex

derivatives, Limits, sums and integrals

The expressions are obtained in LaTeX by typing \frac{du}{dt} and \frac{d^2 u}{dx^2. The mathematical symbol is produced using \partial. Thus the Heat equation is obtained in LaTeX by typing

\[\frac{\partial u}{\partial T}
   = h^2 \left (\frac{\partial^2 u}{\partial x^2}
      + \frac{\partial^2 u}{\partial y^ 2}
      + \frac{\partial^2 u}{\partial z^2} \right) \]

To obtain mathematical expressions such as in displayed equations we type \lim_{x \to +\infty}, \inf_{x > s} and \ Sup_k respectively. Thus to obtain (in LaTeX) we type

\[\lim_{x \to 0} \frac{3x^2 +7x^3}{x^2 +5x^4} = 3.\] 

To obtain a summation sign such as we type \sum_{i=1}^{2n}. Thus is obtained by typing

\[\sum_{k=1}^n k^2 = \frac{1}{2} n (n+1). \] 

We are discuss to obtain integrals in mathematical documents. A Typical integral is the following:this is typeset using

\[\int_a^b f (x) \,dx.\] 

The integral sign is typeset using the control sequence \int, and the limits of integration of are treated as a subscript and a superscript on the integral sign.

Most integrals occurring in mathematical documents begin with a integral sign and contain one or more instances of D Foll Owed by another (Latin or Greek) letter, as in DX, Dyand dt. To obtain the correct appearance one should put extra spaces before the D, using \,. Thus and are obtained by typing

\[\int_0^{+\infty} x^n e^{-x} \,dx = n!. \] 

\[\int \cos \theta \,d\theta = \sin \theta.\] 

\[\int_{x^2 + y^2 \leq r^2} f (x,y) \,dx\,dy
   = \int_{\theta=0}^{2\pi} \int_{r=0}^r
      f (r\cos\theta,r\sin\theta) r\ , dr\,d\theta.\] 

and

\[\int_0^r \frac{2x\,dx}{1+x^2} = \log (1+r^2). \] 

respectively.

In some multiple integrals (i.e., integrals containing the more than one integral sign) one finds that LaTeX puts the much spa Ce between the integral signs. The way to improve the appearance of the integral are to use the control sequence \! To remove a thin strip of unwanted spaces. Thus, for example, the multiple integral are obtained by typing

\[\int_0^1 \! \int_0^1 x^2 y^2\,dx\,dy.\] 

Had We typed

\[\int_0^1 \int_0^1 x^2 y^2\,dx\,dy.\] 

We would have obtained

A particularly noteworthy example comes when we are typesetting a multiple integral such as here we use \! Three to obtain suitable spacing between the integral signs. We typeset this integral using

\[\int \!\!\! \int_d F (x,y) \,dx\,dy.\] 

Had We typed

\[\int \int_d F (x,y) \,dx\,dy.\] 

We would have obtained

The following (reasonably complicated) passage exhibits a number of the features which we have been discussing:one would Typeset this in LaTeX by typing

In non-relativistic wave mechanics, the wave function $\psi (\mathbf{r},t) $ of a particle satisfies the \emph{schr\ "{o}ding Er Wave equation} \[i\hbar\frac{\partial \psi}{\partial t} = \frac{-\hbar^2}{2m} \left (\frac{\partial^2}{\partial X^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right) \psi + V \psi.\] It is Customa Ry to normalize the waves equation by demanding that \[\int \!\!\! \int \!\!\! \int_{\textbf{r}^3} \left| \psi (\math bf{r},0) \right|^2\,dx\,dy\,dz = 1.\] A simple calculation using the schr\ ' {O}dinger wave equation shows that \[\frac{d} {DT} \int \!\!\! \int \!\!\! \INT_{\TEXTBF{R}^3} \left| \psi (\mathbf{r},t) \right|^2\,dx\,dy\,dz = 0,\] and hence \[\int \!\!\! \int \!\!\! \int_{\textbf{r}^3} \left| \ps I (\mathbf{r},t) \right|^2\,dx\,dy\,dz = 1\] for all times~ $t $. If We normalize the wave function in this way then, for any (measurable) subset~ $V $ of $\textbf{r}^3$ and time~ $t $, \[\in T \!\!\! \int \!\!\! \int_v \left| \psi (\mathbf{r},t) \right|^2\,dx\,dy\,dz\] represents the probability that the particle are to be found within the region~ $V $ at time~ $t $.