Vector and tensor calculus for engineers - ebook

INTRODUCTION

I will write about someone, whom you may not know, but who is absolutely noteworthy. It is a mathematician, Hermann Günter Grassman, born in Szczecin on 15 April 1809 (on the birthday of the Swiss mathematician Leonhard Euler, as noted by German historians). Grassmann first attended a private school and then the integrated Royal and Municipal Gymnasium in Szczecin (at present: Liceum no. IX), in which his father, Justus Günter (1779–1852), taught mathematics, physics and drawing. In 1827, he completed an interesting paper on number theory. He also wrote several papers on geometry, which were later used by physicists and astronomers in the construction of the astronomical and meteorological clock. Hermann’s mother, Johanne Luise Friederike Madewald, was the daughter of a pastor.

Hermann was the third child, after which the Grassmanns had another five girls and four boys. His brothers included Gustaw, Robert and Justus, with whom Hermann would work in the fields of linguistics, physics, mathematics and philosophy. Influenced by pastor Zybbel, he decided to commit to theology and become a pastor. At school, he would not show a particular interest in specific subjects, he mainly learned playing the piano and the basics of music. He passed his secondary school graduation exam on 17 September 1827 and on 15 October 1827, on his graduation day, he delivered a lecture in Latin, On the impact of the reign of Augustus on Roman literature. He achieved the second place in the school’s overall ranking. He was graded Good in mathematics. This shows that his talent and aptitude in mathematics had not awoken during his secondary school period.

After his secondary school graduation exam, he and his older brother Gustaw started their studies in theology at the University in Berlin. At home, he studied classical Greek authors. He attended lectures by different theologians and particularly appreciated Neander and Schleismacher. He read the works of Plato. In 1829, he travelled to Vienna, Tirol, Munich and Switzerland, to the Alps. During his studies and later on, while studying the works of his father, he identified the need for a thorough healing and purification of the mathematical language.

At the end of 1830, Hermann returned to Szczecin and started working as an assistant teacher (Hillfslehrer) at the Royal Seminar integrated with the gymnasium in Szczecin. In the same year, he passed an exam in ancient languages and mathematics. He then resumed his theological studies and in May 1834 he passed his first exam. Over the period of the next two years, he worked as mathematics teacher at the Berlin Industrial School, where he met Jakob Steiner. However, the Berlin environment did not suit him. He returned to Szczecin and started working at the Otto von Bamberg Municipal School (Ottoschule; equivalent of a Realschule, but without Latin), where he taught mathematics, chemistry, German and religion. He worked there until 1842. In the meantime, his father also pursued scientific work and in 1827 published his dissertation on the concepts of pure number theory. The dissertation is interesting in that it is similar to the introduction to the first work by Hermann, Ausdehnugslehre, published in 1844.

In 1839, Hermann G. Grassmann passed his exam in theology with a Very Good grade. One year later, he submitted his first scientific paper to the commission in Berlin, thus obtaining an independent specialisation and the right to teach lectures in mathematics, physics, mineralogy and chemistry. In 1841, he wrote two books: Zarys gramatyki niemieckiej (Outline of German Grammar) and Przewodnik nauki niemieckiej (Guide to German Science) (with his brother, Gustaw). In 1842, he abandoned his career in theology and began to develop and create new scientific disciplines.

In the same year, he started a family. According to historians, the Grassmanns had eleven children (seven of whom lived to adulthood): Justus (born in 1851) was a mathematician and a director of the Friedrich Wilhelm Real Gymnasium, at which his father worked; Max (born in 1852) took a road to mathematics through theology, was a professor at Mary’s Gymnasium; Hermann Ernst (born in 1857) was a Latin teacher in Halle and then became a professor of mathematics at the University in Giessen; Ludolf (born in 1861) was the chief physician in Flensburg, and Richard (born in 1864) was a full professor of mechanical engineering at the School of Technology in Karlsruhe. The other children were: Emma (born in 1850), Agnes (born in 1855), Helmuth (born in 1856), Luisa (born in 1858), Klara (born in 1866) and Konrad (born in 1867).

In 1842, Grassmann published Theorie der Zentralen, in which he presented general theorems on diameters, asymptotes, tangents, curves and areas obtained in special cases, on the basis of the works of Poncelet and Möbius (creator of the barycentric calculus). In this paper, he also presented all of Poncelot’s results. This work demonstrated the creative power of the author, however without any material contribution to the new analytical methods.

In 1839, Grassmann published Theorie der Ebbe und Flut (The Low and High Tide Theory), in which he used the vector calculus for the first time. In 1844, he published his foundational work, Die lineale Ausdehnungslehre, ein nuer Zweig der Mathematik, written in the difficult Old German language. Already the initial part of the title poses difficulties in translation to Modern German. In English literature, it reads as follows: The Theory of Extension, a New Branch of Mathematics. Into Polish, the title could be translated as Wykłady o liniowej teorii rozszerzenia, nowa gałąź matematyki (as it applies not only to vector calculus). In Polish literature, this paper was first referred to by M.T. Huber in his book Mechanika ogólna i techniczna, in an interesting footnote on p. 25.

In Ausdehnungslehre, Grassmann uses small calculi, but at an unprecedented level of abstraction. Due to the philosophical style of the work, it was not appreciated by the contemporary mathematicians, however – as confirmed by Erdmann – the three axioms presented in the Riemann–Helmholtz theory correspond to the theses expressed by Grassmann and considered as adequate in this theory. The book was highly appreciated by Möbius, however he would not review it and recommended Drobisch, whom he regarded as a more competent mathematician. However, he offered Grassmann to write an abstract of Ausdehnungslehre as an article for the Archiv der Mathematik journal. The second edition of this work was published in 1862. Its revised edition with an extensive introduction by the author was printed in 1878 in Leipzig, under the following title: Die Ausdehnungslehre von 1844 oder Die lineale Ausdehnunglehre: ein nuer Zweig der Mathematik dargestellt und durch Andendungen auf die ubrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert (Verlag von Otto Wiegand). The work consists of 302 pages, including one with 17 figures at the end.

As noted by the author himself, he was fully convinced about the novelty of his work. In the introduction, he emphasised its influence on the works of other authors, including Rudolf Clebsch. Grassmann defined the concepts of vector independence, vector space dimension, exterior and interior multivector multiplication. He was the first to introduce the concept of exterior product (ein ausseres Produkt) as vector product (cf. p. 60–64, ed. of 1877; symbols according to the original notation). In contemporary literature, antisymmetric tensor product, exterior product or wedge product is recorded differently, as while – this notation was probably first used by Cartan in 1922. It should be noted, however, that the author did not use the word “vector” (German: Vektor) directly. He uses the term of ausseres Produkt zweier Strecken (in German die Strecke is a line). The concept of vector space was introduced between 1918 and 1820 in a series of papers by the eminent German scientist Hermann Weyl (1885-1955, PhD at the University of Göttingen in 1908, supervisor David Hilbert). Grassmann also introduced the concept of n-dimensional space, expanded its geometry, first affine and then metric geometry. He provided the concepts of exterior algebra, nowadays referred to as Grassmann algebra.

Hermann G. Grassmann was also interested in physics. In 1845, he published a paper on electrodynamics. His results were confirmed by Clausius in 1876. In 1853, he formulated the laws of active colour mixing (nowadays also referred to as Grassmann’s laws), which became the basics of colorimetry (confirmed in the work by prof. Prayer of 1876). In 1860–1862, he was active in linguistics and developed a Sanskrit dictionary (the literary language of Ancient, Medieval and Early Modern India) for Rigveda (the oldest Indo-Aryan piece of literature, worked on by Grassmann in 1872–1875). He loved music and collected Pomeranian folk songs. He became a correspondent-member of the Göttingen Scientific Society in 1871 and a member of the American Oriental Society in 1876. In 1876, the University of Tübingen (the Faculty of Philosophy) conferred the title of doctor honoris causa upon him.

The final years of Grassmann’s life were a time of physical suffering (he used a wheelchair), but also recognition of his works. Despite his leg condition, he was transported to lectures in the Gymnasium. He worked until the end of his days. He did not live to see the second edition of his work published. He died in the early morning hours of 26 August 1877, at his company apartment in Szczecin, Königsplatz 9 (at present: Professors’ Houses, pl. Żołnierza Polskiego).

So, it was neither Adhémar Jean Claude Bareé de Saint-Venant, nor William Rowan Hamilton, nor even Josiah Willard Gibbs who invented vector calculus (the latter discovered Grassmann’s work as late as 1877). In any case, Gibbs wrote: “(…) in mechanics, kinematics, astronomy, physics or crystallography, Grassmann’s point analysis will rarely be wanted”.

One of the first mathematicians to appreciate Grassmann’s works was Hermann Hankel in 1867 (Theorie der komplexen Zahlensysteme). In 1878, William Kingdon Clifford of the Johns Hopkins University wrote in the paper “Application of Grassmann’s Algebra”, published in the first issue of American Journal of Mathematics of 1878: “(…) I may, perhaps, therefore be permitted to express my profound admiration of that extraordinary work, and my conviction that its principles will exercise a vast influence upon the future of mathematical science”.

Hermann G. Grassmann was forgotten for almost 150 years, until the conference in Lieschow on Rügen Island in 1994, held for the anniversary of the publishing of his first work, Ausdehnungslehre. Nowadays, he is considered as one of the greatest thinkers of the 19th century science. The mathematicians themselves assert that his mathematical achievements are not yet fully known and properly understood.

This book uses the following layout: chapters 1 and 2 are the basics of vector and tensor calculus. The practical application of vector calculus is presented in chapters 3 and 4, using calculations related to selected problems of kinematics (plane motion and spherical motion), and tensor calculus was used to calculate the moments of inertia of lines, planar figures and solids (chapter 5). The problems to solve on your own can be found in chapter 6.

And finally, a pleasant duty. I would like to thank the Rector of the West Pomeranian University of Technology in Szczecin, dr hab. inż. Jacek Wróbel for allocating the funds for the publishing of this book. I would like to thank the reviewer of the book, prof. Paweł Dłużewski of the Institute of Fundamental Technological Research of the Polish Academy of Sciences in Warszawa, for undertaking the review, writing critical comments and the provided consultations. I would like to thank Mr Grzegorz Urawski for his great care in the preparation of the graphic side of this book and my editor, Izabela Jaźwińska (WN PWN), for her very good cooperation. Finally, I would like to thank my wife Barbara and my daughters Kamila and Sabina for their endless support and years of patience.

If you have any questions or comments regarding my book, please send them to my e-mail address: [email protected].

References

Browne J., Grassmann Algebra, Volume 1: Foundations. Exploring extended vector algebra with Mathematica, Barnard Publishing, 2012.

Clifford W.K., Application of Grassmann’s extensive algebra, American Journal of Mathematics, 1(4): 350-358, 1878.

Domaradzki S., Hermann Grassmann (1809-1877). Jego pracowite życie i renesansowe zainteresowania, Matematyka XIX wieku. Materiały z II Ogólnopolskiej Szkoły Historii Matematyki, ed. S. Fudali, University of Szczecin, Szczecin 1988.

Jach D., Hermann Günter Grassmann (1809-1877), Zeszyty Naukowe Uniwersytetu Szczecińskiego, no. 169, Acta Mathematica Pomeranica, no. 3, 1995.

Winitzki S., Linear algebra via exterior products, www.lulu.com.CHAPTER 1 FUNDAMENTALS OF VECTOR AND TENSOR CALCULUS

1.1. Basic concepts. Scalars and vectors

One of the methods of classification of physical quantities is based on the determination of quantity with a pre-determined unit of measurement. If a single number is enough to determine the specific quantity, such quantities are referred to as scalar quantities and the numbers determining them as scalars, as they correspond to points of a specific scale. The scalars include i.a. temperature, density, energy, potential. Vector length is also a scalar. These quantities are also called invariants, meaning that they are unaffected by changes of the axes of the coordinate system, e.g. by rotation of the system.

Other quantities, e.g. velocity, acceleration, force, torque, cannot be clearly determined by their measure. Their effect also depends on the direction and the sense. Such quantities are referred to as vector quantities. Vector quantities can be represented as a segment with a certain length and direction.

Vector quantities representing physical quantities should, in addition to the three characteristics given (vector length, direction and sense), have a specific position in a given space. For this reason, we divide vectors into three groups:

(a) free vectors,

(b) sliding vectors,

(c) bound vectors.

Free vectors represent physical quantities without specifying their position in space. This is exemplified by the vectors of displacement, velocity and acceleration of a rigid body in its translation. All points of the rigid body have the same physical properties (the same displacement, velocity and acceleration), regardless of the point of application.

Sliding vectors determine a physical quantity correctly if they lie in the same straight line. An example is the force vector. The effect of a force vector will not change as long as it is applied to any point along the direction of the vector. Bound vectors have a specific point of application. This is exemplified by the displacement vector of a specific point of a deformable body, the velocity or the displacement vectors of a specific point of a rigid body in its free motion.

This classification of vectors in mechanics is important and assignment of the given vector to the incorrect group, while retaining the three basic vector properties (magnitude, direction and sense), could lead to a wrong physical interpretation of the mathematical operations on vectors. Vectors are equivalent if they represent the same vector quantity. Two free vectors are always equivalent. Sliding vectors are equivalent if they lie in the same straight line. Bound vectors are equivalent if they are applied to the same point.

1.2. Designation of vectors

1.2.1. Vector addition

A vector (more specifically: a bound vector) beginning at A and ending at B is an organised pair of points A and B. A vector beginning at A and ending at B is designated as . Vectors will be also designated with bold letters, e.g. a, b, c. The distance between points A and B is the length of or magnitude of vector and designated as or AB or, accordingly, the length of vector a by or a. The sense of the vector is the sense of the AB ray. Two vectors have the same direction if the straight lines determined by these vectors have the same direction. Two vectors, a and b, are called equal if they have the same direction, the same sense and the same lengths. Two vectors are called opposite if they have equal lengths, the same direction, but opposite senses. The vector opposite to a is designated as . A zero vector 0 is a vector, the initial and terminal points of which coincide.

The sum of vectors a and b is vector c, which is formed as follows: draw the a vec point of vector is the O point its terminal point is the terminal point of the vector b (Fig. 1.1). The sum of the vectors does not depend on the selection of point O. The difference of vectors a and b is the sum of the vector a and the vector opposite to the vector b. This difference is defined using the formula:

.

The sum and the difference can be represented using a parallelogram, as shown in Fig. 1.1.

Figure 1.1

The associative law is applicable to vector addition.

Figure 1.2

In accordance with the associative law, the resulting vector r (Fig. 1.2), being the sum of three vectors a, b and c, may be calculated as follows:

.

The commutative law is applicable here, i.e. the order of vector addition is free, that is:

.

The properties of the vector addition can be summarised as follows:

(commutative law),

(associative law),

,

.

The sum of any number of vectors may be obtained using the presented method (Fig. 1.3). The closing side of the polygon is a sum of vectors, however the order of vector addition is free. The sum vector is formed by connecting the initial point of the first vector with the terminal point of the last vector.

Figure 1.3

1.2.2. Vector multiplication by a number, a linear combination of vectors

The product of a non-zero vector a and number m is a vector with m-times length of the vector a, i.e.:

.

The resulting vector b is parallel to the vector a, however if m < 0, then the vectors are anti-parallel, if m > 0, then vectors re co-parallel.

Each vector with a length of 1 is referred to as unit vector or versor. If is the versor of the vector a, .

The versor is a vector that is co-parallel to the vector a. The vector a can then be presented as product of the versor and modulus (of the length) of the vector a: .

The properties of vector multiplication by a number result in the following relationships:

,

,

,

.

Vector

, (1.1)

is the linear combination of vectors , , ..., . These vectors are linearly dependent if there exist numbers, not all of which are equal to zero, i.e.

, such that .

Two linearly dependent vectors are termed collinear vectors. Two non-zero and collinear vectors are termed parallel vectors. Three linearly dependent vectors a, b, c are termed coplanar vectors if there exist three numbers α, β, γ, all of which are non-zero and compliant with the relationship:

. (1.2)

Coplanar vectors lie in a single plane. Each of the three vectors in a single plane can be presented as combination that is linearly dependent on the two other vectors, e.g. if , then

. (1.3)

If, in addition, vectors a and b are not collinear vectors, they determine a plane once their initial points at placed at the common point O. Then, the vector c can be decomposed into the directions of vectors a and b (Fig. 1.4), whereas , , , , , .

Figure 1.4

1.3 Einstein summation convention

Use of the Cartesian orthogonal coordinate system is sufficient to describe many physical phenomena.

We are accustomed to designating the axes of the coordinate system as x, y, z, and the corresponding axis versors as i, j, k. Let us introduce other designations. Let us designate the axes of the system as , and versors as. For the convenience of notation, dependencies will be written in the index form, using the so-called Einstein summation convention. In accordance with said convention, the axes of the coordinate system will be designated as , and axis versors as , where: i = 1, 2, 3. If formulae include expressions with repetitive indices, it means the summation of all the expressions, in which the repeated index assumes all possible values. The repeated indices will be referred to as dummy (summation) indices and the other indices as free indices (or current indices). Dummy indices always disappear after the expansion of the formula. Any other letters can be used alternatively to designate the dummy indices, e.g.:

,

where indices i, j, k, m = 1, 2, 3.

In this expression k or m are dummy indices (repeated), while i and j are free indices. For example, if we chose free indices i = 2 and j = 3, then the form of our expression is:

.

The index notation and the summation convention given here are commonly used in tensor calculus.

EXAMPLE 1.1

Expand the expression , assuming that i, j, k = 1, 2, 3.

Solution

This expression will include dummy indices k and l, as well as free indices i and j. The summation follows the dummy indices, while the free indices remain in the formula.

This expression can be written in the double sum form, whereas the summation first follows l, and then k:

By replacing the indices i and j with any numbers, e.g. i = 1, j= 3, we obtain the expression for :

1.4. Vectors in the rectangular coordinate system

Let a be any vector in a plane in a rectangular coordinate system Oxyz and be the projections of that vector on the axes Ox, Oy and Oz (Fig. 1.5).

If are versors of axes (or unit vectors with senses coincident with the senses of the axes) and let us assume that the terminal points of vectors and (theseare position vectors) have coordinates and , respectively, then vector is written in the following form (Fig. 1.5):

or .

The numbers are vector coordinates. Coordinates can be positive, negative or equal to zero. In the three-dimensional space, the given vector a will be written as follows (see Fig. 1.5):

.

Figure 1.5

From the definition of the sum of vectors, the formula follows:

.

Axis versors (or basis vectors) can be presented as follows:

, , . (1.4)

Alternatively, the versors of the coordinate system will also be designated as i, j, k.

Let the non-zero vector form angles with the coordinate axes (Fig. 1.5). The cosines of these angles are the directional cosines of the vector a, whereas:

, , . (1.5)

By squaring both sides of the equation (1.5) and adding side-by-side, we obtain:

. (1.6)

The sum or difference of two vectors, and , is formed by adding or subtracting vector coordinates:

,(1.7a)

. (1.7b)

The projection of the vector a on the axis Os is a vector (Fig. 1.6), the initial point of which is the projection of the starting point and the terminal point is the projection of the terminal point of the vector a on this axis. The coordinate of the vector a relative to the axis is the projection measure (or coordinate) of the vector relative to this axis. Assuming that is the versor of the axis , we obtain:

, . (1.8)

We will relate all calculations to the right-handed coordinate system (Fig. 1.7).

Figure 1.6

Figure 1.7

EXAMPLE 1.2

Three vectors are given: Present the vector c as a linear combination of vectors a and b.

Solution

Let us write the vector in the following form: . By multiplying both sides of the vector equation by versors i and j, we obtain two linear equations:

, ,

therefore

, .

By solving the system of equations, we obtain: i .

Answer

.

EXAMPLE 1.3

In the rectangle ABCD, M and N are the centres of sides , . Decompose the vector into the directions of vectors and .

Figure 1.8

Solution

As the points M and N are centres of the sides of the rectangle ABCD (Fig. 1.8), therefore

, , ,

where: i, j, k are the versors of axes.

As assumed, the vector c must be the sum of two vectors, i.e. , therefore:

.

By comparing now the coordinates at the same versors on both sides of the equation, we obtain two equations:

, .

By solving the system of equations, we obtain: .

Answer

.

1.5. Scalar product

The scalar product a × b of two non-zero vectors a and b is the number equal to the product of the lengths of these vectors and the cosine of the angle between them:

, where . (1.9)

Scalar product properties:

– scalar product is commutative,

,

– the law of distributive multiplication over addition.

The definition of scalar product entails the property of orthogonality of two non-zero vectors a and b and the formula for the length of the vector a. Namely, two vectors are orthogonal (the angle between the vectors is right) if their scalar product is equal to zero, i.e. .

Formula for the length of the vector a:

. (1.10)

In a rectangular coordinate system, for the given two vectors and , the scalar product in is expressed in the index notation in the formula:

(1.11)

In the above case of scalar multiplication, the Kronecker delta symbol is used. According to its definition, we have:

Assuming orthogonality of the coordinate system, we obtain .

If a and b are non-zero vectors, then using the definition of scalar product, the angle between the vectors is calculated using the formula:

. (1.12)

EXAMPLE 1.4

Demonstrate the validity of the formula (a+ b)²+ (a – b)²= 2(a²+ b²) and describe its geometrical meaning.

Solution

.

Using this formula (1.9), let us calculate:

and ,

therefore

.

Answer

The sum of the square lengths of the sides of the parallelogram is equal to the sum of the squares of its diagonals.

EXAMPLE 1.5

Find the interior angles of the triangle with the following vertices , , .

Figure 1.9

Solution

Method I

, , ,

, , .

Let us calculate the angles using the formula (1.9). Maintaining the appropriate senses of the vectors C forming the triangle (Fig. 1.9), we obtain:

,

,

.

Method II

Since the vectors a, b and c form a closed triangle (Fig. 1.9), we obtain thus (we are looking for cos α).

We square both sides of the equation and obtain:

.

We use the scalar product properties and calculate:

, ,

and ,

thus

.

Similarly, formulae for the other angles can be obtained:

,

.

The formulae obtained above present the Carnot’s theorem (also referred to as the law of cosines).