The book publishing program at Duke University Press is growing!

This month we add a new acquisitions editor—Elizabeth Ault—to our team. Elizabeth started at the Press in 2012, and she has been working with our editorial director, Ken Wissoker, on his book projects. In 2014 Elizabeth was promoted to assistant editor as she began to acquire projects of her own, and in 2016 she was promoted to associate editor. She has steadily built a list in African studies and has been regularly attending the African Studies Association conference on behalf of the Press. She has also acquired titles in film and media studies and American studies and has worked with the editors of our journal *Camera Obscura* to restart their book series.

Most recently, Elizabeth launched a new books series “Theory in Forms”—edited by Achille Mbembe, Nancy Hunt, and Juan Obarrio–which will focus on theory from the Global South. The series builds upon Duke’s commitment to innovative, interdisciplinary, and international scholarship and also points to some of the new directions that Elizabeth’s list will take.

Elizabeth plans to acquire titles in African studies, urban studies, Middle East studies, geography, theory from the South, Black and Latinx studies, disability studies, trans studies, and critical prison studies. As is characteristic of our list, these areas overlap and intersect with other editors’ areas of acquisitions. We take pride in the intellectual synergy that comes from the intersections between our editors’ lists (as well as between our book and journal publications), and we hope that adding another editor to our team will allow Duke UP to expand the intellectual breadth of our list even further.

Elizabeth says, “It’s an exciting time for me – and for the Press! I’m looking forward to finding surprising turns in established fields of inquiry as well as supporting emerging conversations, particularly those between activists and academics. I’m so thrilled that I’ll be able to more fully support the authors and series editors I’ve already been working with, and also that I’ll get to learn fields that will be new to me and to DUP, expanding our spirit of interdisciplinary inquiry.”

Prior to joining Duke UP, Elizabeth earned an A.B. in American Studies from Brown University and a Ph.D. in American Studies from the University of Minnesota. She has published her research in *Television & New Media*, among other places. While in graduate school, Elizabeth worked at the Minnesota Historical Society Press, where she helped to write the catalog for The 1968 Exhibit. In addition to her editorial work, Elizabeth is an active participant in Durham community organizations like Southerners on New Ground and the Durham Prison Books Collective.

To submit your book project to Duke University Press, contact Elizabeth or another of our acquisitions editors by email. See the requirements here.

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Mathematica, Matlab, Maple, Mathcad: Why would I want to use one of those?

There are at least 2 reasons you should know how to use one of more of these tools:

• It will allow you to solve equations you can’t easily solve otherwise, including complicated derivatives and integrals (and PDEs of course)

• It will allow you to learn more Mathematics without the drudgery of doing everything longhand, and mathematics, after all, may be the only truth in the worldof the input can be cryptic and takes a while to learn. As with most everything else in life, you work from examples and only learn the parts you need (by trial and error) for the task at hand.

Methods

I’m going to focus on Mathematica here because I wanted to learn it; I had previous experience with all of the others, though not in solving PDEs. I also thought it would be the best for this purpose. Now I see that it appears to be rather limited in what it can do in the PDE realm. You are welcome and encouraged to use any of the programs mentioned above that will solve the PDEs in this assignment.

Preliminaries

Execute commands in Mathematica by holding down ‘shift’ and pressing enter while on the line you wish to execute. For your sanity, I encourage you to begin every notebook with Remove[“Global`*”].

Plotting

The syntax of the ‘plot’ statement is Plot[what to plot, {variable, low end of range, high end of range}].

Plotting functions of 1 and 2 variables is easy:

Bad things

The syntax

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here we can approach since many situations as observed prof dr mircea orasanu and prof drd horia orasanu for the above discussions as followed with A mechanical system is scleronomous if the equations of constraints do not contain the time as an explicit variable and the equation of constraints can be described by generalized coordinates. Such constraints are called scleronomic constraints. The opposite of scleronomous is rheonomous. n 3-D space, a particle with mass m {\displaystyle m\,\!} m\,\!, velocity v {\displaystyle \mathbf {v} \,\!} {\mathbf {v}}\,\! has kinetic energy T {\displaystyle T\,\!} T\,\!

T = 1 2 m v 2 . {\displaystyle T={\frac {1}{2}}mv^{2}\,\!.} T={\frac {1}{2}}mv^{2}\,\!.

Velocity is the derivative of position with respect to time. Use chain rule for several variablMain article: Generalized velocity

In 3-D space, a particle with mass m {\displaystyle m\,\!} m\,\!, velocity v {\displaystyle \mathbf {v} \,\!} {\mathbf {v}}\,\! has kinetic energy T {\displaystyle T\,\!} T\,\!

T = 1 2 m v 2 . {\displaystyle T={\frac {1}{2}}mv^{2}\,\!.} T={\frac {1}{2}}mv^{2}\,\!.

Velocity is the derivative of position with respect to time. Use chain rule for several variables:

v = d r d t = ∑ i ∂ r ∂ q i q ˙ i + ∂ r ∂ t . {\displaystyle \mathbf {v} ={\frac {d\mathbf {r} }{dt}}=\sum _{i}\ {\frac {\partial \mathbf {r} }{\partial q_{i}}}{\dot {q}}_{i}+{\frac {\partial \mathbf {r} }{\partial t}}\,\!.} {\mathbf {v}}={\frac {d{\mathbf {r}}}{dt}}=\sum _{i}\ {\frac {\partial {\mathbf {r}}}{\partial q_{i}}}{\dot {q}}_{i}+{\frac {\partial {\mathbf {r}}}{\partial t}}\,\!.

Therefore,

T = 1 2 m ( ∑ i ∂ r ∂ q i q ˙ i + ∂ r ∂ t ) 2 . {\displaystyle T={\frac {1}{2}}m\left(\sum _{i}\ {\frac {\partial \mathbf {r} }{\partial q_{i}}}{\dot {q}}_{i}+{\frac {\partial \mathbf {r} }{\partial t}}\right)^{2}\,\!.} T={\frac {1}{2}}m\left(\sum _{i}\ {\frac {\partial {\mathbf {r}}}{\partial q_{i}}}{\dot {q}}_{i}+{\frac {\partial {\mathbf {r}}}{\partial t}}\right)^{2}\,\!.

Rearranging the terms carefully,[1]

T = T 0 + T 1 + T 2 : {\displaystyle T=T_{0}+T_{1}+T_{2}\,\!:} T=T_{0}+T_{1}+T_{2}\,\!:

T 0 = 1 2 m ( ∂ r ∂ t ) 2 , {\displaystyle T_{0}={\frac {1}{2}}m\left({\frac {\partial \mathbf {r} }{\partial t}}\right)^{2}\,\!,} T_{0}={\frac {1}{2}}m\left({\frac {\partial {\mathbf {r}}}{\partial t}}\right)^{2}\,\!,

T 1 = ∑ i m ∂ r ∂ t ⋅ ∂ r ∂ q i q ˙ i , {\displaystyle T_{1}=\sum _{i}\ m{\frac {\partial \mathbf {r} }{\partial t}}\cdot {\frac {\partial \mathbf {r} }{\partial q_{i}}}{\dot {q}}_{i}\,\!,} T_{1}=\sum _{i}\ m{\frac {\partial {\mathbf {r}}}{\partial t}}\cdot {\frac {\partial {\mathbf {r}}}{\partial q_{i}}}{\dot {q}}_{i}\,\!,

T 2 = ∑ i , j 1 2 m ∂ r ∂ q i ⋅ ∂ r ∂ q j q ˙ i q ˙ j , {\displaystyle T_{2}=\sum _{i,j}\ {\frac {1}{2}}m{\frac {\partial \mathbf {r} }{\partial q_{i}}}\cdot {\frac {\partial \mathbf {r} }{\partial q_{j}}}{\dot {q}}_{i}{\dot {q}}_{j}\,\!,} T_{2}=\sum _{{i,j}}\ {\frac {1}{2}}m{\frac {\partial {\mathbf {r}}}{\partial q_{i}}}\cdot {\frac {\partial {\mathbf {r}}}{\partial q_{j}}}{\dot {q}}_{i}{\dot {q}}_{j}\,\!,

where T 0 {\displaystyle T_{0}\,\!} T_{0}\,\!, T 1 {\displaystyle T_{1}\,\!} T_{1}\,\!, T 2 {\displaystyle T_{2}\,\!} T_{2}\,\! are respectively homogeneous functions of degree 0, 1, and 2 in generalized velocities. If this system is scleronomous, then the position does not depend explicitly with time:

∂ r ∂ t = 0 . {\displaystyle {\frac {\partial \mathbf {r} }{\partial t}}=0\,\!.} {\frac {\partial {\mathbf {r}}}{\partial t}}=0\,\!.

Therefore, only term T 2 {\displaystyle T_{2}\,\!} T_{2}\,\! does not vanish:

T = T 2 . {\displaystyle T=T_{2}\,\!.} T=T_{2}\,\!.

Kinetic energy is a homogeneous function of degree 2 in generalized velocities .

Example: pendulum

A simple pendulum

As shown at right, a simple pendulum is a system composed of a weight and a string. The string is attached at the top end to a pivot and at the bottom end to a weight. Being inextensible, the string’s length is a constant. Therefore, this system is scleronomous; it obeys scleronomic constraint

x 2 + y 2 − L = 0 , {\displaystyle {\sqrt {x^{2}+y^{2}}}-L=0\,\!,} {\sqrt {x^{2}+y^{2}}}-L=0\,\!,

where ( x , y ) {\displaystyle (x,y)\,\!} (x,y)\,\! is the position of the weight and L {\displaystyle L\,\!} L\,\! is length of the string.

A simple pendulum with oscillating pivot point

Take a more complicated example. Refer to the next figure at right, Assume the top end of the string is attached to a pivot point undergoing a simple harmonic motion

x t = x 0 cos ω t , {\displaystyle x_{t}=x_{0}\cos \omega t\,\!,} x_{t}=x_{0}\cos \omega t\,\!,

where x 0 {\displaystyle x_{0}\,\!} x_{0}\,\! is amplitude, ω {\displaystyle \omega \,\!} \omega \,\! is angular frequency, and t {\displaystyle t\,\!} t\,\! is time.

Although the top end of the string is not fixed, the length of this inextensible string is still a constant. The distance between the top end and the weight must stay the same. Therefore, this system is rheonomous as it obeys constraint explicitly dependent on time

( x − x 0 cos ω t ) 2 + y 2 − L = 0 . {\displaystyle {\sqrt {(x-x_{0}\cos \omega t)^{2}+y^{2}}}-L=0\,\!.} {\sqrt {(x-x_{0}\cos \omega t)^{2}+y^{2}}}-L=0\,\!.

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There is great value in being able to derive the finite difference expressions; you are in a position to write your own computer code to solve these and similar equations, which arise in many fields of endeavor.

There are also new tools that allow you bypass this step; you provide the PDE and the computer uses (hopefully appropriate) numerical methods to provide a solution

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