Vector

Calculus

Calculus

Analysis have been adopted from 'the simplest physical ideas which immediately identify them. Thus the idea of divergence is taken directly from Hydrodynamics, and keeping before our minds the picture of ﬂuid leaving (or entering) a small closed space, we define the divergence of a vector function at a point as the limit of the ratio, if one exists, of the surface-integral of the function over a small closed space surrounding the point to the volume enclosed by the surface, a unique limit being supposed to be reached by the closed surface shrinking up' to a point in any manner.

Again, it is found that some vector fields can be specified completely by the gradient of a scalar function, so that the line integral T of the vector function along any closed curve in (simply connected) space would Vanish. Thus the work done is nil along any closed path in a conservative field of force. But in case the vector function cannot be so specified, an expression of this negative quality of the function at a point is naturally sought in its now non-evanescent line integral along a small closed (plane) path surrounding the point. The ratio of this line integral to the area enclosed by our path generally approa ches a limit as the path shrinks up to a point, independently of its original form and of the manner of its shrinking, but dep ending ou the orientation of its plane. The limit moreover has usually a maximum value, subject to the Variation of this orientation, and a vector of magnitude equal to this maximum value and drawn perpendicular to that aspect of the plane which gives us the maximum value is called the curl of the original vector function.

PIBN | 10019407 |

ISBN | 978-1-4400-8038-8 |

ISBN (Hardcover) | 978-0-428-49701-9 |

Language | English |

Category | Calculus |

Pages | 96 |

Words | 19753 |

Vocabulary | 780 |

Hyperbolic

Functions

Functions

Elements of

Vector AnalysisArranged for the Use

of Students in Physics

Vector AnalysisArranged for the Use

of Students in Physics

An Elementary

Course in

Differential

Equations

Course in

Differential

Equations

Advanced

Course in

Algebra

Course in

Algebra

Differential

and Integral

Calculus

and Integral

Calculus

Differential

and Integral

CalculusFor Technical Schools and Colleges

and Integral

CalculusFor Technical Schools and Colleges

Calculus

Theory of

Functions of a

Complex Variable

Functions of a

Complex Variable

Electromagnetic

Radiation and the

Mechanical Reactions,

Arising From It, Being an

Adams Prize Essay in the

University of Cambridge

Radiation and the

Mechanical Reactions,

Arising From It, Being an

Adams Prize Essay in the

University of Cambridge

The Scientific

Papers of J.

Willard GibbsDynamics, Vector Analysis and Multiple

Algebra, Electromagnetic Theory of Light, Etc

Vol. 2 of 2

Papers of J.

Willard GibbsDynamics, Vector Analysis and Multiple

Algebra, Electromagnetic Theory of Light, Etc

Vol. 2 of 2

The

Calculus for

Engineers

Calculus for

Engineers

Vector

Calculus

Calculus

A Manual of

Quaternions

Quaternions

An Elementary

Treatise on

Quaternions

Treatise on

Quaternions

An Elementary

Treatise on

Quaternions

Treatise on

Quaternions

Vector

AnalysisAn Introduction to Vector-Methods and Their

Various Applications to Physics and Mathematics

AnalysisAn Introduction to Vector-Methods and Their

Various Applications to Physics and Mathematics

The Dynamics of

Particles and of Rigid,

Elastic, and Fluid BodiesBeing Lectures on

Mathematical Physics

Particles and of Rigid,

Elastic, and Fluid BodiesBeing Lectures on

Mathematical Physics

Harmonic

Functions

Functions

Vector

CalculusWith Applications to Physics

CalculusWith Applications to Physics

An

Introduction

to Mathematics

Introduction

to Mathematics

A Treatise on

the Calculus

of Finite

Differences

the Calculus

of Finite

Differences

Graphical

Calculus

Calculus

Elements of

Quaternions

Vol. 1

Quaternions

Vol. 1

A Brief History

of MathematicsAn Authorized Translation of Dr. Karl Fink's

Geschichte Der Elementar-Mathematik

of MathematicsAn Authorized Translation of Dr. Karl Fink's

Geschichte Der Elementar-Mathematik

Elementary

Vector AnalysisWith Application to

Geometry and Physics

Vector AnalysisWith Application to

Geometry and Physics

A History of

Mathematics

Mathematics

Elements

of DynamicAn Introduction to the Study of Motion and

Rest in Solid and Fluid Bodies; Kinematic

Vol. 1

of DynamicAn Introduction to the Study of Motion and

Rest in Solid and Fluid Bodies; Kinematic

Vol. 1

The Philosophy

of MathematicsTranslated From the Cours De

Philosophie Positive of Auguste Comte

of MathematicsTranslated From the Cours De

Philosophie Positive of Auguste Comte

Elements

of DynamicAn Introduction to the Study of Motion and

Rest in Solid and Fluid Bodies; Kinematic

Vol. 1

of DynamicAn Introduction to the Study of Motion and

Rest in Solid and Fluid Bodies; Kinematic

Vol. 1

Exact and

Asymptotic

Solutions of the

Cauchy Problem

Asymptotic

Solutions of the

Cauchy Problem

Vector

AnalysisA Text-Book for the Use of Students

of Mathematics and Physics

AnalysisA Text-Book for the Use of Students

of Mathematics and Physics

The Messenger

of MathematicsMay, 1922 April, 1923

Vol. 52

of MathematicsMay, 1922 April, 1923

Vol. 52

Spherical

TrigonometryFor Colleges and Secondary Schools

TrigonometryFor Colleges and Secondary Schools

The Method of

Fluxions and

Infinite SeriesWith Its Application to the

Geometry of Curve-Lines

Fluxions and

Infinite SeriesWith Its Application to the

Geometry of Curve-Lines

Differential

and Integral

Calculus

and Integral

Calculus

The

Nature of

Mathematics

Nature of

Mathematics

Notes on

Magneto-HydrodynamicsTheory of Maxwell's Equations

Without Displacement Current

Vol. 5

Magneto-HydrodynamicsTheory of Maxwell's Equations

Without Displacement Current

Vol. 5

A Treatise on

Electricity

and Magnetism

Vol. 1

Electricity

and Magnetism

Vol. 1

Lectures on

MathematicsDelivered From Aug. 28 to Sept. 9, 1893 Before Members of the

Congress of Mathematics Held in Connection With the World's

Fair in Chicago at Northwestern University, Evanston, Ill

MathematicsDelivered From Aug. 28 to Sept. 9, 1893 Before Members of the

Congress of Mathematics Held in Connection With the World's

Fair in Chicago at Northwestern University, Evanston, Ill

Advanced

CalculusA Text Upon Select Parts of Differential

Calculus, Differential Equations,

Integral Calculus, Theory of Functions

CalculusA Text Upon Select Parts of Differential

Calculus, Differential Equations,

Integral Calculus, Theory of Functions

Theoretical

PhysicsMechanics and Heat, Newton-Carnot

Vol. 1

PhysicsMechanics and Heat, Newton-Carnot

Vol. 1

General Investigations

of Curved Surfaces

of 1827 and 1825Translated With Notes

and a Bibliography

of Curved Surfaces

of 1827 and 1825Translated With Notes

and a Bibliography

Advanced

CalculusA Text Upon Select Parts of Differential Calculus,

Differential Equations, Integral Calculus,

Theory of Functions, With Numerous Exercises

CalculusA Text Upon Select Parts of Differential Calculus,

Differential Equations, Integral Calculus,

Theory of Functions, With Numerous Exercises

The Wireless

Telegraphist's

Pocket Book of

Notes, Formulae,

and Calculations

Telegraphist's

Pocket Book of

Notes, Formulae,

and Calculations

Geometry and

TrigonometryGeometry, Plane Trigonometry,

Natural Trigonometric Functions,

Logarithmic Trigonometric Functions

TrigonometryGeometry, Plane Trigonometry,

Natural Trigonometric Functions,

Logarithmic Trigonometric Functions

History of

Modern

Mathematics

Modern

Mathematics

Vector

Analysis and

Quaternions

Analysis and

Quaternions

Elements of

Quaternions

Quaternions

A Treatise on

Differential

EquationsSupplementary Volume

Differential

EquationsSupplementary Volume