Grant Number MCS-790251 (num6m de subvention Ma-790251) French translation: Traduction franpise :... more Grant Number MCS-790251 (num6m de subvention Ma-790251) French translation: Traduction franpise : Jean-Luc Raymond D Globally Rigid Symmetric Tensegrities Abstr;ld f one builds a tensegrity structure with cables and struts, when will it be globally rigid in the sense that there is no I other non-congruent configuration of the points satisfying the cable and strut constraints? We investigate a family of such structures that have dihedral symmetry and we completely characterize those that are globally rigid. This uses some stress-energy techniques that in turn require showing that a certain symmetric matrix is positive semi-definite with the right rank.
In this chapter we introduce the concept of the multiple integral-the precise mathematical expres... more In this chapter we introduce the concept of the multiple integral-the precise mathematical expression for finding the total amount of a quantity in a region in the plane or in space. Examples include area, volume, the total mass of a body, the total electrical charge in a region, or total population of a country. 6.1 Introduction to area, volume, and integral Examples of integrals We introduce the concept of the integral of a function of two or more variables through two problems. Mass. Let D be a set in space. Let f (x, y, z) [mass/volume] be the density at the point (x, y, z) of some material distributed in D. How can we find the total mass M(f, D) contained in D? If the density is between lower and upper bounds and u then the total mass is between the bounds Vol (D) ≤ M(f, D) ≤ u Vol (D) where Vol (D) denotes the volume of D. This estimate for the mass in D is a good start. But perhaps we can do better. Split D into two subsets D 1 and D 2. On each one f has a lower and upper bound, 1 ≤ f (x, y, z) ≤ u 1 in D 1 , 2 ≤ f (x, y, z) ≤ u 2 in D 2. Thus we have
The mathematical description of aspects of the natural world requires a collection of numbers. Fo... more The mathematical description of aspects of the natural world requires a collection of numbers. For example, a position on the surface of the earth is described by two numbers, latitude and longitude. To specify a position above the earth requires a third number, the altitude. To describe the state of a gas we have to specify its density and temperature; if it is a mixture of gases like oxygen and nitrogen, we have to specify their proportion. Such situations are abstracted in the concept of a vector.
In this chapter we introduce the concept of the multiple integral-the precise mathematical expres... more In this chapter we introduce the concept of the multiple integral-the precise mathematical expression for finding the total amount of a quantity in a region in the plane or in space. Examples include area, volume, the total mass of a body, the total electrical charge in a region, or total population of a country. 6.1 Introduction to area, volume, and integral Examples of integrals We introduce the concept of the integral of a function of two or more variables through two problems. Mass. Let D be a set in space. Let f (x, y, z) [mass/volume] be the density at the point (x, y, z) of some material distributed in D. How can we find the total mass M(f, D) contained in D? If the density is between lower and upper bounds and u then the total mass is between the bounds Vol (D) ≤ M(f, D) ≤ u Vol (D) where Vol (D) denotes the volume of D. This estimate for the mass in D is a good start. But perhaps we can do better. Split D into two subsets D 1 and D 2. On each one f has a lower and upper bound, 1 ≤ f (x, y, z) ≤ u 1 in D 1 , 2 ≤ f (x, y, z) ≤ u 2 in D 2. Thus we have
Preliminary report of the results of a project 1 to introduce peer instruction into a multi-secti... more Preliminary report of the results of a project 1 to introduce peer instruction into a multi-section first semester calculus course taught largely by novice instructors. This paper summarizes the instructional approaches instructors chose to use, and the subsequent results of student performance on common exams throughout the course of the term.
ABSTRACT: Preliminary report of the results of a project1 to intro-duce peer instruction into a m... more ABSTRACT: Preliminary report of the results of a project1 to intro-duce peer instruction into a multi-section first semester calculus course taught largely by novice instructors. This paper summarizes the in-structional approaches instructors chose to use, and the subsequent re-sults of student performance on common exams throughout the course of the term.
In this chapter we derive the laws governing the vibration of a stretched string and a stretched ... more In this chapter we derive the laws governing the vibration of a stretched string and a stretched membrane, and the equations governing the propagation of heat. Like the laws of conservation of mass, momentum and energy studied in the previous chapter, and like the electromagnetism laws, these laws are expressed as partial differential equations. We derive some properties and some solutions of these equations. We also state the Schrodinger equation of quantum mechanics, derive a property of the solutions and explain the physical meaning of this property.
This project stems from a collaborative effort by engineering and mathematics faculty at a resear... more This project stems from a collaborative effort by engineering and mathematics faculty at a research university to enhance engineering students’ abilities to transfer and apply mathematics to solve problems in engineering contexts. A recent curriculum innovation resulting from these efforts involves the integration of collaborative, applied, problem-solving workshops into the first-semester engineering mathematics course. This paper will summarize the project team's work to develop two instruments one to gauge students’ abilities in using mathematics in engineering contexts; and the other to gauge students' self-efficacy perceptions related to studying engineering and to learning and applying mathematics – that can be used to assess the effects of this innovation and others like it. The paper will report on the processes being used to develop and adapt the two instruments, the Mathematics Applications Inventory (MAI) and the Engineering and Mathematics Perceptions Survey (EMP...
A Mathematics Applications Inventory (MAI) is being developed by engineering and mathematics facu... more A Mathematics Applications Inventory (MAI) is being developed by engineering and mathematics faculty at Cornell University to assess students’ ability to apply the mathematics they learn in freshman calculus to engineering-related problems. This paper reports on three aspects of this work: (1) Development of the first draft of the MAI, (2) Pilot testing the MAI, and (3) Preliminary analysis of the pilot test data. To develop the MAI, faculty of secondand third-year engineering courses were surveyed about how key concepts and techniques from single variable differential and integral calculus are used in intermediate-level engineering courses. Based on their feedback, as well as feedback from advanced undergraduate engineering students, an initial set of test items was developed. The resulting MAI consists of five open-ended questions with eleven sub-questions. The test is designed to be administered during one hour in paper-and-pencil format. The MAI was administered during the first...
This chapter describes applications of the derivative to methods for finding extreme values of fu... more This chapter describes applications of the derivative to methods for finding extreme values of functions of several variables, and to methods for approximating functions of several variables by polynomials.
The mathematical description of aspects of the natural world requires a collection of numbers. Fo... more The mathematical description of aspects of the natural world requires a collection of numbers. For example, a position on the surface of the earth is described by two numbers, latitude and longitude. To specify a position above the earth requires a third number, the altitude. To describe the state of a gas we have to specify its density and temperature; if it is a mixture of gases like oxygen and nitrogen, we have to specify their proportion. Such situations are abstracted in the concept of a vector.
The concepts and techniques of calculus are indispensable for the description and study of dynami... more The concepts and techniques of calculus are indispensable for the description and study of dynamics, the science of motion in space under the action of forces. Both were created by Isaac Newton in the late seventeenth century and they revolutionized both mathematics and physics. In this chapter we describe the basic concepts and laws of the dynamics of point masses and deduce some of their mathematical consequences.
Grant Number MCS-790251 (num6m de subvention Ma-790251) French translation: Traduction franpise :... more Grant Number MCS-790251 (num6m de subvention Ma-790251) French translation: Traduction franpise : Jean-Luc Raymond D Globally Rigid Symmetric Tensegrities Abstr;ld f one builds a tensegrity structure with cables and struts, when will it be globally rigid in the sense that there is no I other non-congruent configuration of the points satisfying the cable and strut constraints? We investigate a family of such structures that have dihedral symmetry and we completely characterize those that are globally rigid. This uses some stress-energy techniques that in turn require showing that a certain symmetric matrix is positive semi-definite with the right rank.
In this chapter we introduce the concept of the multiple integral-the precise mathematical expres... more In this chapter we introduce the concept of the multiple integral-the precise mathematical expression for finding the total amount of a quantity in a region in the plane or in space. Examples include area, volume, the total mass of a body, the total electrical charge in a region, or total population of a country. 6.1 Introduction to area, volume, and integral Examples of integrals We introduce the concept of the integral of a function of two or more variables through two problems. Mass. Let D be a set in space. Let f (x, y, z) [mass/volume] be the density at the point (x, y, z) of some material distributed in D. How can we find the total mass M(f, D) contained in D? If the density is between lower and upper bounds and u then the total mass is between the bounds Vol (D) ≤ M(f, D) ≤ u Vol (D) where Vol (D) denotes the volume of D. This estimate for the mass in D is a good start. But perhaps we can do better. Split D into two subsets D 1 and D 2. On each one f has a lower and upper bound, 1 ≤ f (x, y, z) ≤ u 1 in D 1 , 2 ≤ f (x, y, z) ≤ u 2 in D 2. Thus we have
The mathematical description of aspects of the natural world requires a collection of numbers. Fo... more The mathematical description of aspects of the natural world requires a collection of numbers. For example, a position on the surface of the earth is described by two numbers, latitude and longitude. To specify a position above the earth requires a third number, the altitude. To describe the state of a gas we have to specify its density and temperature; if it is a mixture of gases like oxygen and nitrogen, we have to specify their proportion. Such situations are abstracted in the concept of a vector.
In this chapter we introduce the concept of the multiple integral-the precise mathematical expres... more In this chapter we introduce the concept of the multiple integral-the precise mathematical expression for finding the total amount of a quantity in a region in the plane or in space. Examples include area, volume, the total mass of a body, the total electrical charge in a region, or total population of a country. 6.1 Introduction to area, volume, and integral Examples of integrals We introduce the concept of the integral of a function of two or more variables through two problems. Mass. Let D be a set in space. Let f (x, y, z) [mass/volume] be the density at the point (x, y, z) of some material distributed in D. How can we find the total mass M(f, D) contained in D? If the density is between lower and upper bounds and u then the total mass is between the bounds Vol (D) ≤ M(f, D) ≤ u Vol (D) where Vol (D) denotes the volume of D. This estimate for the mass in D is a good start. But perhaps we can do better. Split D into two subsets D 1 and D 2. On each one f has a lower and upper bound, 1 ≤ f (x, y, z) ≤ u 1 in D 1 , 2 ≤ f (x, y, z) ≤ u 2 in D 2. Thus we have
Preliminary report of the results of a project 1 to introduce peer instruction into a multi-secti... more Preliminary report of the results of a project 1 to introduce peer instruction into a multi-section first semester calculus course taught largely by novice instructors. This paper summarizes the instructional approaches instructors chose to use, and the subsequent results of student performance on common exams throughout the course of the term.
ABSTRACT: Preliminary report of the results of a project1 to intro-duce peer instruction into a m... more ABSTRACT: Preliminary report of the results of a project1 to intro-duce peer instruction into a multi-section first semester calculus course taught largely by novice instructors. This paper summarizes the in-structional approaches instructors chose to use, and the subsequent re-sults of student performance on common exams throughout the course of the term.
In this chapter we derive the laws governing the vibration of a stretched string and a stretched ... more In this chapter we derive the laws governing the vibration of a stretched string and a stretched membrane, and the equations governing the propagation of heat. Like the laws of conservation of mass, momentum and energy studied in the previous chapter, and like the electromagnetism laws, these laws are expressed as partial differential equations. We derive some properties and some solutions of these equations. We also state the Schrodinger equation of quantum mechanics, derive a property of the solutions and explain the physical meaning of this property.
This project stems from a collaborative effort by engineering and mathematics faculty at a resear... more This project stems from a collaborative effort by engineering and mathematics faculty at a research university to enhance engineering students’ abilities to transfer and apply mathematics to solve problems in engineering contexts. A recent curriculum innovation resulting from these efforts involves the integration of collaborative, applied, problem-solving workshops into the first-semester engineering mathematics course. This paper will summarize the project team's work to develop two instruments one to gauge students’ abilities in using mathematics in engineering contexts; and the other to gauge students' self-efficacy perceptions related to studying engineering and to learning and applying mathematics – that can be used to assess the effects of this innovation and others like it. The paper will report on the processes being used to develop and adapt the two instruments, the Mathematics Applications Inventory (MAI) and the Engineering and Mathematics Perceptions Survey (EMP...
A Mathematics Applications Inventory (MAI) is being developed by engineering and mathematics facu... more A Mathematics Applications Inventory (MAI) is being developed by engineering and mathematics faculty at Cornell University to assess students’ ability to apply the mathematics they learn in freshman calculus to engineering-related problems. This paper reports on three aspects of this work: (1) Development of the first draft of the MAI, (2) Pilot testing the MAI, and (3) Preliminary analysis of the pilot test data. To develop the MAI, faculty of secondand third-year engineering courses were surveyed about how key concepts and techniques from single variable differential and integral calculus are used in intermediate-level engineering courses. Based on their feedback, as well as feedback from advanced undergraduate engineering students, an initial set of test items was developed. The resulting MAI consists of five open-ended questions with eleven sub-questions. The test is designed to be administered during one hour in paper-and-pencil format. The MAI was administered during the first...
This chapter describes applications of the derivative to methods for finding extreme values of fu... more This chapter describes applications of the derivative to methods for finding extreme values of functions of several variables, and to methods for approximating functions of several variables by polynomials.
The mathematical description of aspects of the natural world requires a collection of numbers. Fo... more The mathematical description of aspects of the natural world requires a collection of numbers. For example, a position on the surface of the earth is described by two numbers, latitude and longitude. To specify a position above the earth requires a third number, the altitude. To describe the state of a gas we have to specify its density and temperature; if it is a mixture of gases like oxygen and nitrogen, we have to specify their proportion. Such situations are abstracted in the concept of a vector.
The concepts and techniques of calculus are indispensable for the description and study of dynami... more The concepts and techniques of calculus are indispensable for the description and study of dynamics, the science of motion in space under the action of forces. Both were created by Isaac Newton in the late seventeenth century and they revolutionized both mathematics and physics. In this chapter we describe the basic concepts and laws of the dynamics of point masses and deduce some of their mathematical consequences.
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