#### We’re pleased to share a post from the Wild On Collective, which comprises Ethan Kleinberg, Joan Wallach Scott, and Gary Wilder, about their new project “Theses on Theory and History.” Scott and Wilder are both Duke University Press authors; Scott is editor of *Women’s Studies on the Edge* and author of *The Fantasy of Feminist History*, and Wilder is author of *Freedom Time* and co-editor of *The Struggle for Life is the Matter*, forthcoming in 2019.

Linked here is a programmatic intervention entitled “Theses on Theory and History” co-written by the three academic historians who currently compose the Wild On Collective. This first publication, which is freely available for web viewing or as a downloadable PDF, emerged from a series of conversations among the three of us that began in fall 2018. Despite our different theoretical investments and analytic orientations, we were each struck by how deeply entrenched realist epistemology and empiricist methodology remains in the field of disciplinary history. This, notwithstanding repeated attempts by successive generations of critics to free historical thinking and knowledge from the fetishes of archival evidence, chronological narrative, and reified boundaries between past and present. We discussed the perverse mechanism whereby the epistemological challenges to conventional history that developed between the 1970s and 1990s were superficially embraced, only in order to be domesticated as new themes or topics to be explored in familiar ways. We concurred that circumscribed assumptions about what counts as historical evidence, argument, and truth are systemically produced by the disciplinary guild.

“Theses on Theory and History” is divided into three sections: one on the assumptions of disciplinary history, another on the strategies through which the field resists “theory” as somehow foreign to real history, and a third which calls programmatically for a theoretically informed practice of critical history. Our aim is to provoke a debate among and beyond professional historians about the intellectual implications of this unstated but regularly enforced disciplinary commonsense concerning descriptive realism and archival empiricism. Specifically, we hope to challenge any artificial separation of empirical research and theoretical reflection; to invite historians to be more conceptually self-aware and critically self-reflexive; to push the field to recognize non-realist and non-empiricist modes of analysis as legitimate ways to know the past; and to remind scholars in other fields that professional history does not possess a monopoly on modes of historical thinking or means of historical insight.

Because these domesticating and disciplining processes are systemic, our theses address all aspects of professional history—training, research, writing, publishing, hiring. Likewise, we believe that any attempt to redress such problems must do so holistically. This intervention is not in any way meant to be a comprehensive inventory of all that is wrong, even theoretically, with the field and the guild (e.g., the persistent Eurocentrism of its frameworks). Even less is it meant to be theoretically prescriptive; we make no claims about which theories historians should engage, how they might be employed, how they might go about theorizing their own work, or to what end. But we do believe that any attempt to change a specific aspect of the field that brackets questions about what counts as evidence and how we produce knowledge is likely to be limited at best. We hope that this initial intervention is only a first step in opening a broader debate about these issues within the field of history. We also hope to create a community of like-minded scholars, within and beyond the field of history, to share concerns about and strategies for doing history otherwise.

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LAGRANGIAN AND JOURNAL and the follows

ABSTRACT

To generalize the adage–and along the way to explain why planes travel this way–we will introduce a special class of curves on surfaces, called geodesics. Geodesics have the useful property that the shortest curve segment connecting two points on a surface is a segment of a geodesic. As we shall see, great circles are geodesics on the sphere, and they therefore have the property that they are the “shortest” curves on the sphere. To examine geodesics, we will develop connections between differential geometry, differential equations, and vector calculus. In order to see geodesics, even when they cannot be found explicitly, the computer algebra system Mathematica will be used.

1 INTRODUCTION

A surface in three-dimensional Euclidean space (R³) is a set of points in R³ that locally look like a plane-that is, given any point on the surface, there is a small neighborhood of that point which appears to be planar. Again, the earth’s surface taken as a sphere is a good example. The earth’s surface curves, yet by looking around, one cannot see this curvature. This is because the area of the earth one can see is a small enough neighborhood of the point where he/she is standing that this neighborhood appears flat. So the sphere is a surface in R³. More technically,

Definition: M R³ is a surface if for any x M, there exists an open neighborhood U R³ containing x, an open neighborhood W R², and a map x: W → U ∩ M that is differentiable with differentiable inverse. Such a function is called a parameterization or a coordinate patch since it allows us to assign coordinates to the surface corresponding to the coordinates of R².

Surfaces do not need to be in R3; replacing R3 with a general space X yields a valid definition. In fact, some of the surfaces we will examine cannot be placed in R3.

2 FORMULATION

For a sphere of radius r centered at the origin, a coordinate patch is x(u, v) = (rcos(u)cos(v), rsin(u)cos(v), rsin(v)), for – < u < , and – < v < A coordinate patch is said to be orthogonal if its first partial derivatives are orthogonal-that is, if xu•xv = 0. Clearly, for an orthogonal patch x, xu x xv is never zero. This means it is possible to construct a unit normal at any point on the surface. Also, because xu and xv vary smoothly on M, so will U. If any two points on a surface can be connected by a curve contained in the surface, the surface is said to be connected.

Using the sphere as an example, the parametrization of the sphere given above is orthogonal. xu = (-r sin(u) cos(v), r cos(u) cos(v), 0), and xv = (-r cos(u) sin(v), -r sin(u) sin(v), r cos(v)). So xu • xv = r2 sin(u) sin(v) cos(u) cos(v) – r2 sin(u) sin(v) cos(u) cos(v) + 0 = 0. The unit normal at x(u, v) is , as the reader can easily verify either by hand or using Mathematica.

A useful and important construct on a surface M is the tangent plane to the surface at a point p. Parameterize M in a neighborhood of p by x(u, v), with x(u0, v0) = p. Then, the tangent plane to M at p-denoted by TpM-is the two dimensional vector space spanned by {xu(u0, v0), xv(u0, v0)}. It is fairly easy to show that this space is equivalent to the space of all vectors v such that v = ’(t0), where

3 SOLUTION The Geodesic Equations

Important curves on surfaces are curves called geodesics. Geodesics are essentially the extensions into M of straight lines in the plane—that is, relative to the surface, there appears to be no acceleration. Formally,

Definition: For a surface M in Euclidean three-space, a geodesic is a curve α:[0, 1]→M where α’’ is always normal to M.

Since α’’ is always normal to M, that means that the dot product of α’’ and a vector in in TpM is always zero. In particular, α’’ • α’ = 0. Let s(t) be the speed of α at t; . Differentiating s with respect to t,

(1)

All correspondence concerning this publication or the reproduction or translation of all or part of the document should be addressed to the Director of School, Out of School and Higher Education of the Council of Europe (F-67075 Strasbourg Cedex).

The reproduction of extracts is authorised, except for commercial purposes, on condition that the source is quoted.

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HISTORY OF LAGRANGIAN AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

ABSTRACT Teaching Scheme. Hours / Week. Semester Examination Scheme of … Note: For audit courses students are given certificate by the institutes based on the … Building Technology and Materials, Second Year Civil Engineering …. 1) Ordinary and Partial differential equations applied to structural analysis and fluid.

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LAGRANGIAN ELLIPTIC HYPERBOLIC EQUATIONS

ABSTRACT

In the real world, it is very difficult to explain behavior as a function of only one variable, and economics is no different. More specific economic interpretations will be discussed in the next section, but for now, we’ll just concentrate on developing the techniques we’ll be using.

First, to define the functions themselves. We want to describe behavior where a variable is dependent on two or more variables. Every rule and notation described from now on is the same for two variables, three variables, four variables, and so on, so we’ll use the simplest case; a function of two independent variables. Conventionally, z is the dependent variable (like y in univariate functions) and x and y are the independent variables (like x in univariate functions):

For example, suppose that the following function describes some behavior:

Differentiating this function still means the same thing–still we are looking for functions that give us the slope, but now we have more than one variable, and more than one slope.

Visualize this by recalling from graphing what a function with two independent variables looks like. Whereas a 2-dimensional picture can represent a univariate function, our z function above can be represented as a 3-dimensional shape. Think of the x and y variables as being measured along the sides of a chessboard. Then every combination of x and y would map onto a square somewhere on the chessboard. For example, suppose x=1 and y=1. Start at one of the corners of the chessboard. Then move one square in on the x side for x=1, and one square up into the board to represent y=1. Now, calculate the value of z.

The function z takes on a value of 4, which we graph as a height of 4 over the square that represents x=1 and y=1. Map out the entire function this way, and the result will be a shape, usually looking like a mountain peak in typical economic analysis problems.

Now back to slope. Imagine standing on the mountain shape, facing parallel to the x side of the chessboard. If you allow x to increase, while holding y constant, then you would move forward in a straight line along the mountain shape. We define the slope in this direction as the change in the z variable, or a change in the height of the shape, in response to a movement along the chessboard in one direction, or a change in the variable x, holding y constant.

Formally, the definition is: the partial derivative of z with respect to x is the change in z for a given change in x, holding y constant. Notation, like before, can vary. Here a to a movement forward on the chessboard

1 INTRODUCTION

The rules of partial differentiation follow exactly the same logic as univariate differentiation. The only difference is that we have to decide how to treat the other variable. Recall that in the previous section, slope was defined as a change in z for a given change in x or y, holding the other variable constant. There’s our clue as to how to treat the other variable. If we hold it constant, that means that no matter what we call it or what variable name it has, we treat it as a constant. Suppose, for example, we have the following equation:

If we are taking the partial derivative of z with respect to x, then y is treated as a constant. Since it is multiplied by 2 and x and is constant, it is also defined as a coefficient of x. Therefore,

Therefore, once all other variables are held constant, then the partial derivative rules for dealing with coefficients, simple powers of variables, constants, and sums/differences of functions remain the same, and are used to determine the function of the slope for each independent variable. Let’s use the function from the previous section to illustrate.

First, different

understand the concept of a partial derivative as the rate that something is changing, calculating partial derivatives usually isn’t difficult. (Unfortunately, there are special cases where calculating the partial derivatives is hard.) As these examples show, calculating a partial derivatives is usually just like calculating an ordinary derivative of one-variable calculus. You just have to remember with which variable you are taking the derivative.When we first considered what the derivative of a vector function might mean, there was really not much difficulty in understanding either how such a thing might be computed or what it might measure. In the case of functions of two variables, things are a bit harder to understand. If we think of a function of two variables in terms of its graph, a surface, there is a more-or-less obvious derivative-like question we might ask, namely, how “steep” is the surface. But it’s not clear that this has a simple answer, nor how we might proceed. We will start with what seem to be very small steps toward the goal; surprisingly, it turns out that these simple ideas hold the keys to a more general understanding.

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LAGRANGIAN AS NONHOLONOMIC OPERATOR

ABSTRACT

This summary closes with a proof of the length-minimizing properties of geodesics. Examples of surfaces are

given and plotted in Mathematica. We also describe geodesics on these surfaces and plot select examples. The surfaces chosen include some with Clairaut patches, some without, and some surfaces in R3 and some not in R3.For example, assume the earth to be a sphere. New York City and Madrid, Spain are both at latitudes of about 40°N. Yet an airplane taking the shortest distance between the two does not follow the 40th parallel. Rather, it arcTo generalize the adage–and along the way to explain why planes travel this way–we will introduce a special class of curves on surfaces, called geodesics. Geodesics have the useful property that the shortest curve segment connecting two points on a surface is a segment of a geodesic. As we shall see, great circles are geodesics on the sphere, and they therefore have the property that they are the “shortest” curves on the sphere. To examine geodesics, we will develop connections between differential geometry, differential equations, and vector calculus

1 . INTRODUCTION

Differential Geometry

A surface in three-dimensional Euclidean space (R³) is a set of points in R³ that locally look like a plane-that is, given any point on the surface, there is a small neighborhood of that point which appears to be planar. Again, the earth’s surface taken as a sphere is a good example. The earth’s surface curves, yet by looking around, one cannot see this curvature. This is because the area of the earth one can see is a small enough neighborhood of the point where he/she is standing that this neighborhood appears flat. So the sphere is a surface in R³. More technically,

Definition: M R³ is a surface if for any x M, there exists an open neighborhood U R³ containing x, an open neighborhood W R², and a map x: W → U ∩ M that is differentiable with differentiable inverse. Such a function is called a parameterization or a coordinate patch since it allows us to assign coordinates to the surface corresponding to the coordinates of R².

Surfaces do not need to be in R3; replacing R3 with a general space X yields a valid definition. In fact, some of the surfaces we will examine cannot be placed in R3.

For a sphere of radius r centered at the origin, a coordinate patch is x(u, v) = (rcos(u)cos(v), rsin(u)cos(v), rsin(v)), for – < u < , and – < v < A coordinate patch is said to be orthogonal if its first partial derivatives are orthogonal-that is, if xu•xv = 0. Clearly, for an orthogonal patch x, xu x xv is never zero. This means it is possible to construct a unit normal at any point on the surface. Also, because xu and xv vary smoothly on M, so will U. If any two points on a surface can be connected by a curve contained in the surface, the surface is said to be connected.

Using the sphere as an example, the parametrization of the sphere given above is orthogonal. xu = (-r sin(u) cos(v), r cos(u) cos(v), 0), and xv = (-r cos(u) sin(v), -r sin(u) sin(v), r cos(v)). So xu • xv = r2 sin(u) sin(v) cos(u) cos(v) – r2 sin(u) sin(v) cos(u) cos(v) + 0 = 0. The unit normal at x(u, v) is , as the reader can easily verify either by hand or using Mathematica.

A useful and important construct on a surface M is the tangent plane to the surface at a point p. Parameterize M in a neighborhood of p by x(u, v), with x(u0, v0) = p. Then, the tangent plane to M at p-denoted by TpM-is the two dimensional vector space spanned by {xu(u0, v0), xv(u0, v0)}. It is fairly easy to show that this space is equivalent to the space of all vectors v such that v = ’(t0), where is a curve on M with (t0)=p (see [3]). Since TpM is a vector space, an inner product can be defined on it. If an inner product is defined consistently on every tangent plane of M, then M is said to be a geometric surface.

2 . FORMULATION)). Then (0)=p gives initial conditions u(0) and v(0). (0)=v gives initial conditions u(0) and v(0). Then, by the fundamental existence and uniqueness theorems of ordinary differential equations [1], exists and is unique.

If every geodesic can be extended infinitely without leaving the surface, then the surface is called a complete surface.

Usually, the geodesic equations cannot be solved by hand. For this reason, it is useful to be able to solve the geodesic equations numerically; we give a Mathematica procedure for this below. On a few surfaces, such as the sphere, explicit solutions can be found. One class of surfaces for which the geodesic equations can often be explicitly solved are surfaces given a Clairaut patch. A Clairaut patch is an orthogonal patch in which |xu| and |xv| are both independent of v. This implies and , or simply xuv•xu = xvv•xv = 0. Also, since xu•xv = 0, . Therefore, the geodesic equations reduce in this case to

(6) |xu|2 u + u2 xuu•xu + v2xvv•xu = 0

(7) |xv|2 v + 2uvxuv•xv = 0

Isolating the v-terms in (7) gives

(8) , where G= xv•xv.

Integrating both sides of (8) with respect to t, and making the substitution w = dv/dt, . Taking the exponential of both sides gives the equation

(9)

for some constant k. Now we use the fact that we may assume the geodesic α(t) = x(u(t), v(t)) to be unit speed. This means that , or

(10)

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ON THE THESES OF LAGRANGIAN AND HISTORY

ABSTRACT

Quantities that have direction as well as magnitude are called as vectors. Examples of vectors are velocity, acceleration, force, momentum etc.

Vectors can be added and subtracted. Let a and b be two vectors. To get the sum of the two vectors, place the tail of b onto the head of a and the distance between the tail of a and b is a+b.

Multiplication of a vector by a positive scalar k multiplies the magnitude but leaves the direction unchanged. If k=2 then the magnitude of a doubles but the direction remains the same.

Dot product of two vectors is the product of a vector to the projection of the other vector on the vector. a. b is called the dot product of the two vectors.

a. b = . If the two vectors are parallel, then a. b = and if the two vectors are perpendicular to each other, then a. b = 0

Cross Product of any two vectors is defined by a b= c = , where is a unit vector (vector of length 1) pointing perpendicular to the plane of a and b. But as there are two directions perpendicular to any plane, the ambiguity is resolved by the right hand rule: let your fingers point in the direction of the first vector and curl around (via the smaller angle) towards the second; then your thumb indicates the direction of .

A Unit vector is a vector whose magnitude is 1 and point is a particular direction. Without loss of generality, we can assume to be three distinct unit vectors along the x, y, and z-axis relatively.

Then,

and

Also,

In cylindrical coordinate systems, a vector , where are the unit

1 INTRODUCTION

m. Imagine that the vector is a force whose units are given in Newtons. Imagine vector is a radius vector through which the force acts in meters. What is the value of the torque , in this case?

n. Now imagine that continues to be a force vector and is a displacement vector whose units are meters. What is the work done in applying force through a displacement ?

o. What is the vector sum of a vector given by 40 m, 30 degrees and a vector given by 12 m, 225 degrees? Use the method of resolving vectors into their components and then adding the components.

3. Consider three vectors:

A. Find .

b. Find .

c. Find .

PROBLEMS OF CALCULUS, Prof. Mircea Orasanu

Here consists that, pentru un sir de descompuneri ale lui de norma tinzand la la zero

Avem limdin norma

Pe de alta parte pentru exista astfel incat sa avem dist de indata, ,iar aceasta implica

Existenta unui morfism intre si aici observatii

.se poate extinde pentru panza cand x= f, y = g, z = h stfel ca matricea || f’u,..| are in

fiecare

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ACADEMIC WORKS

ABSTRACT

Thus, = . Thus,

evaluating the given limit, we have that

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I did not get to meet all of you but I know I’ve benefited from your help this summer! Thanks, and good luck on your future endeavors!

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