Summary
''Vectorial Mechanics'' has 18 chapters grouped into 3 parts. Part I is on ''vector algebra'' including chapters on a definition of a vector, products of vectors, elementary tensor analysis, and integral theorems. Part II is on ''systems of line vectors'' including chapters on line co-ordinates, systems of line vectors, statics of rigid bodies, the displacement of a rigid body, and the work of a system of line vectors. Part III is on ''dynamics'' includingSummary of reviews
There were significant reviews given near the time of original publication. G.J.Whitrow:Although many books have been published in recent years in which vector andDaniel C. Lewis:tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...methods are used for solving problems in geometry andmathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ..., there has been a lack of first-class treatises which explain the methods in full detail and are nevertheless suitable for the undergraduate student. In applied mathematics no book has appeared till now which is comparable with Hardy's ''Pure Mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...''. ... Just as in Hardy's classic, a new note is struck at the very start: a precise definition is given of the concept "free vector", analogous to the Frege-Russell definition of "cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...." According to Milne, a free vector is the class of all its representations, a typical representation being defined in the customary manner. From a pedagogic point of view, however, the reviewer wonders whether it might have been better to draw attention at this early stage to a concrete instance of a ''free'' vector. The student familiar with physical concepts which have magnitude and position, but not direction, should be made to realise from the very beginning that the free vector is not merely "fundamental in discussing systems of position vectors and systems of line-vectors", but occurs naturally in its own right, as there are physical concepts which have magnitude and direction but not position, e.g. thecouple Couple or couples may refer to : Basic meaning *Couple (app), a mobile app which provides a mobile messaging service for two people *Couple (mechanics), a system of forces with a resultant moment but no resultant force *Couple (relationship), tw ...in statics, and theangular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...of arigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force .... Although the necessary existence theorems must be established at a later stage, and Milne's rigorous proofs are particularly welcome, there is no reason why some instances of free vectors should not be mentioned at this point."
The reviewer has long felt that the role of vector analysis inmechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...has been much overemphasized. It is true that the fundamental equations of motion in their various forms, especially in the case ofrigid bodies In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ..., can be derived with greatest economy of thought by use of vectors (assuming that the requisite technique has already been developed); but once the equations have been set up, the usual procedure is to drop vector methods in their solution. If this position can be successfully refuted, this has been done in the present work, the most novel feature of which is to solve the vectordifferential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...by vector methods without ever writing down the corresponding scalar differential equations obtained by taking components. The author has certainly been successful in showing that this can be done in fairly simple, though nontrivial, cases. To give an example of a definitely nontrivial problem solved in this way, one might mention thenonholonomic A nonholonomic system in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints and non-linear constraints, s ...problem afforded by the motion of a sphere rolling on a roughinclined plane An inclined plane, also known as a ramp, is a flat supporting surface tilted at an angle from the vertical direction, with one end higher than the other, used as an aid for raising or lowering a load. The inclined plane is one of the six clas ...or on a rough spherical surface. The author's methods are interesting and aesthetically satisfying and therefore deserve the widest publication even if they partake of the nature of a tour de force.
References
* E.A.Milne ''Vectorial Mechanics'' (New York: Interscience Publishers INC., 1948). PP. xiii, 382 ASIN: B0000EGLGX * G.J.Whttrowbr>Review of ''Vectorial Mechanics''Notes
{{reflist, 2 1948 non-fiction books Mathematics education in the United Kingdom Mathematics textbooks Vector calculus