From: Cliff Y. <sta...@us...> - 2004-09-25 21:40:16
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Update of /cvsroot/maxima/maximabook/share/physics In directory sc8-pr-cvs1.sourceforge.net:/tmp/cvs-serv2367/share/physics Added Files: dimension.tex Log Message: Import main tex files --- NEW FILE: dimension.tex --- %-*-EMaxima-*- %Definitions to help mactex output look better. \beginmaximanoshow texput(%hbar,"\\hbar")$ texput(%me,"m_e")$ texput(%mp,"m_p")$ texput(%c,"c")$ texput(%%e,"e")$ texput(%mue,"\\mu_e")$ texput(%mup,"\\mu_p")$ texput(%g,"G")$ linenum : 0$ \maximaoutput \endmaximanoshow \noindent Author: Barton Willis University of Nebraska at Kearney Kearney Nebraska \vspace{2ex} \noindent Documentation adapted for the Maxima Book by CY \subsection*{Introduction} \noindent This document demonstrates some of the abilities of a Maxima package called dimension. Not surprisingly, its purpose is to perform dimensional analysis. Maxima comes with an older package dimensional analysis (dimen) that is similar to the one that was in the commercial Macsyma system. The software described in this document differs greatly from the older one. \subsubsection*{Usage} To use the package, you must first load it. From a Maxima prompt, this is done using the command \beginmaximasession load("dimension.mac")$ \maximatexsession \C1. load("dimension.mac")$ \\ \endmaximasession \noindent To begin, we need to assign dimensions to the variables we want to use. Use the {\tt qput} function to do this; for example, to declare $x$ a length, $c$ a speed, and $t$ a time, use the commands \beginmaximasession qput(x, "length", dimension)$ qput(c, "length" / "time", dimension)$ qput(t, "time", dimension)$ \maximatexsession \C2. qput(x, "length", dimension)$ \\ \C3. qput(c, "length" / "time", dimension)$ \\ \C4. qput(t, "time", dimension)$ \\ \endmaximasession \noindent We've defined the dimensions length and time to be strings; doing so reduces the chance that they will conflict with other user variables. To declare a dimensionless variable $\sigma$, use $1$ for the dimension. Thus \beginmaximasession qput(sigma,1,dimension)$ \maximatexsession \C5. qput(sigma,1,dimension)$ \\ \endmaximasession \noindent To find the dimension of an expression, use the {\tt dimension} function. For example \beginmaximasession dimension(4 * sqrt(3) /t); dimension(x + c * t); dimension(sin(c * t / x)); dimension(abs(x - c * t)); dimension(sigma * x / c); dimension(x * sqrt(1 - c * t / x)); \maximatexsession \C6. dimension(4 * sqrt(3) /t); \\ \D6. \frac{1}{\mathrm{time}} \\ \C7. dimension(x + c * t); \\ \D7. \mathrm{length} \\ \C8. dimension(sin(c * t / x)); \\ \D8. 1 \\ \C9. dimension(abs(x - c * t)); \\ \D9. \mathrm{length} \\ \C10. dimension(sigma * x / c); \\ \D10. \mathrm{time} \\ \C11. dimension(x * sqrt(1 - c * t / x)); \\ \D11. \mathrm{length} \\ \endmaximasession \noindent {\tt dimension} applies {\tt logcontract} to its argument; thus expressions involving a difference of logarithms with dimensionally equal arguments are dimensionless; thus \beginmaximasession dimension(log(x) - log(c*t)); \maximatexsession \C12. dimension(log(x) - log(c*t)); \\ \D12. 1 \\ \endmaximasession \noindent {\tt dimension} is automatically maps over lists. Thus \beginmaximasession dimension([42, min(x,c*t), max(x,c*t), x^^4, x . c]); \maximatexsession \C13. dimension([42, min(x,c*t), max(x,c*t), x^^4, x . c]); \\ \D13. \left[ 1,\linebreak[0]\mathrm{length},\linebreak[0]\mathrm{length},\linebreak[0]\mathrm{length}^{4},\linebreak[0]\frac{\mathrm{length}^{2}}{\mathrm{time}} \right] \\ \endmaximasession \noindent When an expression is dimensionally inconsistent, {\tt dimension} should signal an error \beginmaximasession dimension(x + c); dimension(sin(x)); \maximatexsession \C14. dimension(x + c); \\ \p Expression is dimensionally inconsistent. #0: dimension(e=x+c)(dimension.mac line 154) -- an error. Quitting. To debug this try DEBUGMODE(TRUE);) \\ \C15. dimension(sin(x)); \\ \p Expression is dimensionally inconsistent. #0: dimension(e=SIN(x))(dimension.mac line 229) -- an error. Quitting. To debug this try DEBUGMODE(TRUE);) \\ \endmaximasession \noindent An {\em equation\/} is dimensionally correct when either the dimensions of both sides match or when one side of the equation vanishes. For example \beginmaximasession dimension(x = c * t); dimension(x * t = 0); \maximatexsession \C16. dimension(x = c * t); \\ \D16. \mathrm{length} \\ \C17. dimension(x * t = 0); \\ \D17. \mathrm{length}\*\mathrm{time} \\ \endmaximasession \noindent When the two sides of an equation have different dimensions and neither side vanishes, {\tt dimension} signals an error \beginmaximasession dimension(x = c); \maximatexsession \C18. dimension(x = c); \\ \p Expression is dimensionally inconsistent. #0: dimension(e=x = c)(dimension.mac line 175) -- an error. Quitting. To debug this try DEBUGMODE(TRUE);) \\ \endmaximasession \noindent The function {\tt dimension} works with derivatives and integrals \beginmaximasession dimension('diff(x,t)); dimension('diff(x,t,2)); dimension('diff(x,c,2,t,1)); dimension('integrate (x,t)); \maximatexsession \C19. dimension('diff(x,t)); \\ \D19. \frac{\mathrm{length}}{\mathrm{time}} \\ \C20. dimension('diff(x,t,2)); \\ \D20. \frac{\mathrm{length}}{\mathrm{time}^{2}} \\ \C21. dimension('diff(x,c,2,t,1)); \\ \D21. \frac{\mathrm{time}}{\mathrm{length}} \\ \C22. dimension('integrate (x,t)); \\ \D22. \mathrm{length}\*\mathrm{time} \\ \endmaximasession Thus far, any string may be used as a dimension; the other three functions in this package, \begin{verb} dimension_as_list \end{verb}, \begin{verb} dimensionless \end{verb}, and \begin{verb} natural_unit \end{verb} all require that each dimension is a member of the list \begin{verb} fundamental_dimensions \end{verb}. The default value is of this list is \beginmaximasession fundamental_dimensions; \maximatexsession \C23. fundamental_dimensions; \\ \D23. \left[ \mathrm{mass},\linebreak[0]\mathrm{length},\linebreak[0]\mathrm{time} \right] \\ \endmaximasession \noindent A user may insert or delete elements from this list. The function \begin{verb} dimension_as_list \end{verb} returns the dimension of an expression as a list of the exponents of the fundamental dimensions. Thus \beginmaximasession dimension_as_list(x); dimension_as_list(t); dimension_as_list(c); dimension_as_list(x/t); dimension_as_list("temp"); \maximatexsession \C24. dimension_as_list(x); \\ \D24. \left[ 0,\linebreak[0]1,\linebreak[0]0 \right] \\ \C25. dimension_as_list(t); \\ \D25. \left[ 0,\linebreak[0]0,\linebreak[0]1 \right] \\ \C26. dimension_as_list(c); \\ \D26. \left[ 0,\linebreak[0]1,\linebreak[0]-1 \right] \\ \C27. dimension_as_list(x/t); \\ \D27. \left[ 0,\linebreak[0]1,\linebreak[0]-1 \right] \\ \C28. dimension_as_list("temp"); \\ \D28. \left[ 0,\linebreak[0]0,\linebreak[0]0 \right] \\ \endmaximasession \noindent In the last example, "temp" isn't an element of \begin{verb} fundamental_dimensions \end{verb}; thus, \begin{verb} dimension_as_list \end{verb} reports that "temp" is dimensionless. To correct this, append "temp" to the list \begin{verb} fundamental_dimensions \end{verb} \beginmaximasession fundamental_dimensions : endcons("temp", fundamental_dimensions); \maximatexsession \C29. fundamental_dimensions : endcons("temp", fundamental_dimensions); \\ \D29. \left[ \mathrm{mass},\linebreak[0]\mathrm{length},\linebreak[0]\mathrm{time},\linebreak[0]\mathrm{temp} \right] \\ \endmaximasession \noindent Now we have \beginmaximasession dimension_as_list(x); dimension_as_list(t); dimension_as_list(c); dimension_as_list(x/t); dimension_as_list("temp"); \maximatexsession \C30. dimension_as_list(x); \\ \D30. \left[ 0,\linebreak[0]1,\linebreak[0]0,\linebreak[0]0 \right] \\ \C31. dimension_as_list(t); \\ \D31. \left[ 0,\linebreak[0]0,\linebreak[0]1,\linebreak[0]0 \right] \\ \C32. dimension_as_list(c); \\ \D32. \left[ 0,\linebreak[0]1,\linebreak[0]-1,\linebreak[0]0 \right] \\ \C33. dimension_as_list(x/t); \\ \D33. \left[ 0,\linebreak[0]1,\linebreak[0]-1,\linebreak[0]0 \right] \\ \C34. dimension_as_list("temp"); \\ \D34. \left[ 0,\linebreak[0]0,\linebreak[0]0,\linebreak[0]1 \right] \\ \endmaximasession \noindent To remove "temp" from \begin{verb} fundamental_dimensions \end{verb}, use the {\tt delete} command \beginmaximasession fundamental_dimensions : delete("temp", fundamental_dimensions)$ \maximatexsession \C35. fundamental_dimensions : delete("temp", fundamental_dimensions)$ \\ \endmaximasession The function {\tt dimensionless} finds a {\em basis\/} for the dimensionless quantities that can be formed from a list of dimensioned quantities. For example \beginmaximasession dimensionless([c,x,t]); dimensionless([x,t]); \maximatexsession \C36. dimensionless([c,x,t]); \\ \p Dependent equations eliminated: (1) \\ \D36. \left[ \frac{c\*t}{x},\linebreak[0]1 \right] \\ \C37. dimensionless([x,t]); \\ \p Dependent equations eliminated: (1) \\ \D37. \left[ 1 \right] \\ \endmaximasession \noindent In the first example, every dimensionless quantity that can be formed as a product of powers of $c,x$, and $t$ is a power of $c t/x$; in the second example, the only dimensionless quantity that can be formed from $x$ and $t$ are the constants. The function \begin{verb} natural_unit(e, [v1,v2,...,vn]) \end{verb} finds powers $p_1,p_2, \dots p_n$ such that \[ \mbox{dimension}(e) = \mbox{dimension} (v_1^{p_1} v_2^{p_2} \dots v_n^{p_n}). \] Simple examples are \beginmaximasession natural_unit(x,[c,t]); natural_unit(x,[x,c,t]); \maximatexsession \C38. natural_unit(x,[c,t]); \\ \p Dependent equations eliminated: (1) \\ \D38. \left[ c\*t \right] \\ \C39. natural_unit(x,[x,c,t]); \\ \p Dependent equations eliminated: (1) \\ \D39. \left[ x \right] \\ \endmaximasession Here is a more complex example; we'll study the Bohr model of the hydrogen atom using dimensional analysis. To make things more interesting, we'll include the magnetic moments of the proton and electron as well as the universal gravitational constant in with our list of physical quantities. Let $\hbar$ be Planck's constant, $e$ the electron charge, $\mu_e$ the magnetic moment of the electron, $\mu_p$ the magnetic moment of the proton, $m_e$ the mass of the electron, $m_p$ the mass of the proton, $G$ the universal gravitational constant, and $c$ the speed of light in a vacuum. For this problem, we might like to display the square root as an exponent instead of as a radical; to do this, set {\tt sqrtdispflag} to false \beginmaximasession SQRTDISPFLAG : false$ \maximatexsession \C40. SQRTDISPFLAG : false$ \\ \endmaximasession \noindent Assuming a system of units where Coulomb's law is \[ \mbox{force} = \frac{\mbox{product of charges}}{\mbox{distance}^2}, \] we have \beginmaximasession qput(%hbar, "mass" * "length"^2 / "time",dimension)$ qput(%%e, "mass"^(1/2) * "length"^(3/2) / "time",dimension)$ qput(%mue, "mass"^(1/2) * "length"^(5/2) / "time",dimension)$ qput(%mup, "mass"^(1/2) * "length"^(5/2) / "time",dimension)$ qput(%me, "mass",dimension)$ qput(%mp, "mass",dimension)$ qput(%g, "length"^3 / ("time"^2 * "mass"), dimension)$ qput(%c, "length" / "time", dimension)$ \maximatexsession \C41. qput(%hbar, "mass" * "length"^2 / "time",dimension)$ \\ \C42. qput(%%e, "mass"^(1/2) * "length"^(3/2) / "time",dimension)$ \\ \C43. qput(%mue, "mass"^(1/2) * "length"^(5/2) / "time",dimension)$ \\ \C44. qput(%mup, "mass"^(1/2) * "length"^(5/2) / "time",dimension)$ \\ \C45. qput(%me, "mass",dimension)$ \\ \C46. qput(%mp, "mass",dimension)$ \\ \C47. qput(%g, "length"^3 / ("time"^2 * "mass"), dimension)$ \\ \C48. qput(%c, "length" / "time", dimension)$ \\ \endmaximasession \noindent The numerical values of these quantities may defined using {\tt numerval}. We have \beginmaximasession numerval(%%e, 1.5189073558044265d-14*sqrt(kg)*meter^(3/2)/sec)$ numerval(%hbar, 1.0545726691251061d-34*kg*meter^2/sec)$ numerval(%c, 2.99792458d8*meter/sec)$ numerval(%me, 9.1093897d-31*kg)$ numerval(%mp, 1.6726231d-27*kg)$ \maximatexsession \C49. numerval(%%e, 1.5189073558044265d-14*sqrt(kg)*meter^(3/2)/sec)$ \\ \C50. numerval(%hbar, 1.0545726691251061d-34*kg*meter^2/sec)$ \\ \C51. numerval(%c, 2.99792458d8*meter/sec)$ \\ \C52. numerval(%me, 9.1093897d-31*kg)$ \\ \C53. numerval(%mp, 1.6726231d-27*kg)$ \\ \endmaximasession \noindent To begin, let's use only the variables $e, c, \hbar, m_e$, and $m_p$ to find the dimensionless quantities. We have \beginmaximasession dimensionless([%hbar, %me, %mp, %%e, %c]); \maximatexsession \C54. dimensionless([%hbar, %me, %mp, %%e, %c]); \\ \D54. \left[ \frac{m_e}{m_p},\linebreak[0]\frac{c\*\hbar}{e^{2}},\linebreak[0]1 \right] \\ \endmaximasession \noindent The second element of this list is the reciprocal of the fine structure constant. To find numerical values, use {\tt float} \beginmaximasession float(%); \maximatexsession \C55. float(%); \\ \D55. \left[ 5.446169970987487 \times 10^{-4},\linebreak[0]137.03599074450503,\linebreak[0]1.0 \right] \\ \endmaximasession The natural units of energy are given by \beginmaximasession natural_unit("mass" * "length"^2 / "time"^2, [%hbar, %me, %mp, %%e, %c]); \maximatexsession \C56. natural_unit("mass" * "length"^2 / "time"^2, [%hbar, %me, %mp, %%e, %c]); \\ \D56. \left[ c^{2}\*m_e,\linebreak[0]\frac{c^{3}\*\hbar\*m_p}{e^{2}} \right] \\ \endmaximasession \noindent Let's see what happens when we include $\mu_e, \mu_p$, and $G$. We have \beginmaximasession dimensionless([%hbar, %%e, %mue, %mup, %me, %mp, %g, %c]); \maximatexsession \C57. dimensionless([%hbar, %%e, %mue, %mup, %me, %mp, %g, %c]); \\ \D57. \left[ \frac{\mu_p}{\mu_e},\linebreak[0]\frac{c^{2}\*m_e\*\mu_e}{e^{3}},\linebreak[0]\frac{c^{2}\*m_p\*\mu_e}{e^{3}},\linebreak[0]\frac{e^{4}\*G}{c^{4}\*\mu_e^{2}},\linebreak[0]\frac{c\*\hbar}{e^{2}},\linebreak[0]1 \right] \\ \endmaximasession To find the natural units of mass, length, time, speed, force, and energy, use the commands \beginmaximasession natural_unit("mass", [%hbar, %%e, %me, %mp, %mue, %mup, %g, %c]); natural_unit("length", [%hbar, %%e, %me, %mp, %mue, %mup, %g, %c]); natural_unit("time", [%hbar, %%e, %me, %mp, %mue, %mup, %g, %c]); natural_unit("mass" * "length" / "time"^2, [%hbar, %%e, %me, %mp, %mue, %mup, %g, %c]); natural_unit("mass" * "length"^2 / "time"^2, [%hbar, %%e, %me, %mp, %mue, %mup, %g, %c]); \maximatexsession \C58. natural_unit("mass", [%hbar, %%e, %me, %mp, %mue, %mup, %g, %c]); \\ \D58. \left[ m_p,\linebreak[0]\frac{c^{2}\*m_e^{2}\*\mu_e}{e^{3}},\linebreak[0]\frac{c^{2}\*m_e^{2}\*\mu_p}{e^{3}},\linebreak[0]\frac{G\*m_e^{3}}{e^{2}},\linebreak[0]\frac{c\*\hbar\*m_e}{e^{2}} \right] \\ \C59. natural_unit("length", [%hbar, %%e, %me, %mp, %mue, %mup, %g, %c]); \\ \D59. \left[ \frac{e^{2}\*m_p}{c^{2}\*m_e^{2}},\linebreak[0]\frac{\mu_e}{e},\linebreak[0]\frac{\mu_p}{e},\linebreak[0]\frac{G\*m_e}{c^{2}},\linebreak[0]\frac{\hbar}{c\*m_e} \right] \\ \C60. natural_unit("time", [%hbar, %%e, %me, %mp, %mue, %mup, %g, %c]); \\ \D60. \left[ \frac{e^{2}\*m_p}{c^{3}\*m_e^{2}},\linebreak[0]\frac{\mu_e}{e\*c},\linebreak[0]\frac{\mu_p}{e\*c},\linebreak[0]\frac{G\*m_e}{c^{3}},\linebreak[0]\frac{\hbar}{c^{2}\*m_e} \right] \\ \C61. natural_unit("mass" * "length" / "time"^2, [%hbar, %%e, %me, %mp, %mue, %mup, %g, %c]); \\ \D61. \left[ \frac{c^{4}\*m_e\*m_p}{e^{2}},\linebreak[0]\frac{c^{6}\*m_e^{3}\*\mu_e}{e^{5}},\linebreak[0]\frac{c^{6}\*m_e^{3}\*\mu_p}{e^{5}},\linebreak[0]\frac{c^{4}\*G\*m_e^{4}}{e^{4}},\linebreak[0]\frac{c^{5}\*\hbar\*m_e^{2}}{e^{4}} \right] \\ \C62. natural_unit("mass" * "length"^2 / "time"^2, [%hbar, %%e, %me, %mp, %mue, %mup, %g, %c]); \\ \D62. \left[ c^{2}\*m_p,\linebreak[0]\frac{c^{4}\*m_e^{2}\*\mu_e}{e^{3}},\linebreak[0]\frac{c^{4}\*m_e^{2}\*\mu_p}{e^{3}},\linebreak[0]\frac{c^{2}\*G\*m_e^{3}}{e^{2}},\linebreak[0]\frac{c^{3}\*\hbar\*m_e}{e^{2}} \right] \\ \endmaximasession \noindent The first element of this list is the rest mass energy of the proton. The dimension package can handle vector operators such as dot and cross products, and the vector operators div, grad, and curl. To use the vector operators, we'll first declare them \beginmaximasession prefix(div)$ prefix(curl)$ infix("~")$ \maximatexsession \C63. prefix(div)$ \\ \C64. prefix(curl)$ \\ \C65. infix("~")$ \\ \endmaximasession \noindent Let's work with the electric and magnetic fields; again assuming a system of units where Coulomb's law is \[ \mbox{force} = \frac{\mbox{product of charges}}{\mbox{distance}^2} \] the dimensions of the electric and magnetic field are \beginmaximasession qput(e, sqrt("mass") / (sqrt("length") * "time"), dimension)$ qput(b, sqrt("mass") / (sqrt("length") * "time"),dimension)$ qput(rho, sqrt("mass")/("time" * "length"^(3/2)), dimension)$ qput(j, sqrt("mass") / ("time"^2 * sqrt("length")), dimension)$ \maximatexsession \C66. qput(e, sqrt("mass") / (sqrt("length") * "time"), dimension)$ \\ \C67. qput(b, sqrt("mass") / (sqrt("length") * "time"),dimension)$ \\ \C68. qput(rho, sqrt("mass")/("time" * "length"^(3/2)), dimension)$ \\ \C69. qput(j, sqrt("mass") / ("time"^2 * sqrt("length")), dimension)$ \\ \endmaximasession Finally, declare the speed of light $c$ as \beginmaximasession qput(c, "length" / "time", dimension); \maximatexsession \C70. qput(c, "length" / "time", dimension); \\ \D70. \frac{\mathrm{length}}{\mathrm{time}} \\ \endmaximasession \noindent Let's find the dimensions of $\| \mathbf{E} \|^2, \mathbf{E} \cdot \mathbf{B}, \| \mathbf{B} \|^2$, and $\mathbf{E} \times \mathbf{B} / c$. We have \beginmaximasession dimension(e.e); dimension(e.b); dimension(b.b); dimension((e ~ b) / c); \maximatexsession \C71. dimension(e.e); \\ \D71. \frac{\mathrm{mass}}{\mathrm{length}\*\mathrm{time}^{2}} \\ \C72. dimension(e.b); \\ \D72. \frac{\mathrm{mass}}{\mathrm{length}\*\mathrm{time}^{2}} \\ \C73. dimension(b.b); \\ \D73. \frac{\mathrm{mass}}{\mathrm{length}\*\mathrm{time}^{2}} \\ \C74. dimension((e ~ b) / c); \\ \D74. \frac{\mathrm{mass}}{\mathrm{length}^{2}\*\mathrm{time}} \\ \endmaximasession \noindent The physical significance of these quantities becomes more apparent if they are integrated over $\mathbf{R^3}$. Defining \beginmaximasession qput(v, "length"^3, dimension); \maximatexsession \C75. qput(v, "length"^3, dimension); \\ \D75. \mathrm{length}^{3} \\ \endmaximasession \noindent We now have \beginmaximasession dimension('integrate(e.e, v)); dimension('integrate(e.b, v)); dimension('integrate(b.b, v)); dimension('integrate((e ~ b) / c,v)); \maximatexsession \C76. dimension('integrate(e.e, v)); \\ \D76. \frac{\mathrm{length}^{2}\*\mathrm{mass}}{\mathrm{time}^{2}} \\ \C77. dimension('integrate(e.b, v)); \\ \D77. \frac{\mathrm{length}^{2}\*\mathrm{mass}}{\mathrm{time}^{2}} \\ \C78. dimension('integrate(b.b, v)); \\ \D78. \frac{\mathrm{length}^{2}\*\mathrm{mass}}{\mathrm{time}^{2}} \\ \C79. dimension('integrate((e ~ b) / c,v)); \\ \D79. \frac{\mathrm{length}\*\mathrm{mass}}{\mathrm{time}} \\ \endmaximasession \noindent It's clear that $\| \mathbf{E} \|^2, \mathbf{E} \cdot \mathbf{B}$ and $\| \mathbf{B} \|^2$ are energy densities while $\mathbf{E} \times \mathbf{B} / c$ is a momentum density. Let's also check that the Maxwell equations are dimensionally consistent. \beginmaximasession dimension(DIV(e)= 4*%pi*rho); dimension(CURL(b) - 'diff(e,t) / c = 4 * %pi * j / c); dimension(CURL(e) + 'diff(b,t) / c = 0); dimension(DIV(b) = 0); \maximatexsession \C80. dimension(DIV(e)= 4*%pi*rho); \\ \D80. \frac{\iexpt{\mathrm{mass}}{\frac{1}{2}}}{\iexpt{\mathrm{length}}{\frac{3}{2}}\*\mathrm{time}} \\ \C81. dimension(CURL(b) - 'diff(e,t) / c = 4 * %pi * j / c); \\ \D81. \frac{\iexpt{\mathrm{mass}}{\frac{1}{2}}}{\iexpt{\mathrm{length}}{\frac{3}{2}}\*\mathrm{time}} \\ \C82. dimension(CURL(e) + 'diff(b,t) / c = 0); \\ \D82. \frac{\iexpt{\mathrm{mass}}{\frac{1}{2}}}{\iexpt{\mathrm{length}}{\frac{3}{2}}\*\mathrm{time}} \\ \C83. dimension(DIV(b) = 0); \\ \D83. \frac{\iexpt{\mathrm{mass}}{\frac{1}{2}}}{\iexpt{\mathrm{length}}{\frac{3}{2}}\*\mathrm{time}} \\ \endmaximasession |