Vector and Tensor Analysis: A Comprehensive Guide for Engineers and Scientists

Vector and Tensor Analysis: A Comprehensive Introduction

Vector and tensor analysis is a branch of mathematics that deals with geometric objects that have magnitude and direction (vectors) or magnitude and multiple directions (tensors). Vectors and tensors are useful for describing physical phenomena in a coordinate-independent way, such as forces, displacements, stresses, strains, etc. In this article, we will introduce the basic concepts of vector and tensor analysis, such as their definitions, operations, properties, transformations and applications. We will also use some examples to illustrate how vector and tensor analysis can help us understand the behavior of physical systems.

Vector and Tensor Analysis

What are vectors and tensors?

Vectors and tensors are mathematical objects that can represent physical quantities that depend on direction. For example, a force is a vector because it has a magnitude (how strong it is) and a direction (where it is applied). A stress is a tensor because it has a magnitude (how much pressure it exerts) and two directions (one normal to the surface where it acts, and one parallel to the direction of the force).

Physical vectors and mathematical vectors

A physical vector is a directed line segment with an arrow head that indicates its direction. For example, we can draw a vector to represent the force acting on an object or the displacement of a point. A physical vector has three characteristics: magnitude, direction and sense. The magnitude is the length of the line segment, the direction is the angle between the line segment and a reference axis, and the sense is whether the arrow points towards or away from the origin.

A mathematical vector is an abstract object that satisfies certain rules or axioms of vector addition and scalar multiplication. A mathematical vector does not depend on any specific coordinate system or frame of reference. It can be represented by a set of numbers called components that vary according to the choice of coordinates. A mathematical vector can also be represented by a symbol with an arrow above it, such as $\veca$ or $\vecF$.

Physical vectors are examples of mathematical vectors, but not all mathematical vectors are physical vectors. For instance, we can define a mathematical vector that represents the color of a pixel in an image by using three components: red, green and blue. This vector does not have a physical meaning in terms of magnitude or direction.

Dot product and cross product of vectors

The dot product (or scalar product) of two vectors $\veca$ and $\vecb$ is a scalar quantity that measures how much they are aligned with each other. It is defined as:

$$\veca \cdot \vecb = \veca \vecb \cos \theta$$ where $\veca$ and $\vecb$ are the magnitudes of the vectors, and $\theta$ is the angle between them. The dot product has the following properties:

• It is commutative: $\veca \cdot \vecb = \vecb \cdot \veca$

• It is distributive: $\veca \cdot (\vecb + \vecc) = \veca \cdot \vecb + \veca \cdot \vecc$

• It is associative with scalar multiplication: $(k\veca) \cdot \vecb = k(\veca \cdot \vecb) = \veca \cdot (k\vecb)$

• It is zero if and only if the vectors are perpendicular: $\veca \cdot \vecb = 0$ iff $\theta = 90^\circ$

The dot product can be used to calculate the projection of one vector onto another, the angle between two vectors, the work done by a force along a displacement, etc.

The cross product (or vector product) of two vectors $\veca$ and $\vecb$ is a vector quantity that measures how much they are perpendicular to each other. It is defined as:

$$\veca \times \vecb = \veca \vecb \sin \theta \hatn$$ where $\veca$ and $\vecb$ are the magnitudes of the vectors, $\theta$ is the angle between them, and $\hatn$ is a unit vector that is perpendicular to both $\veca$ and $\vecb$ and follows the right-hand rule. The cross product has the following properties:

• It is anti-commutative: $\veca \times \vecb = -\vecb \times \veca$

• It is distributive: $\veca \times (\vecb + \vecc) = \veca \times \vecb + \veca \times \vecc$

It is associative with scalar multiplication: $(k\veca) \times \vecb = k(\veca \times \endim_end>

• \beginim_startassistant \beginim_end \beginim_startassistant \beginim_end \vecb) = k(\veca \times \vecb) = \veca \times (k\vecb)$

• It is zero if and only if the vectors are parallel: $\veca \times \vecb = 0$ iff $\theta = 0^\circ$ or $180^\circ$

The cross product can be used to calculate the area of a parallelogram spanned by two vectors, the normal vector to a plane or a surface, the torque exerted by a force around a point, etc.

Scalar triple product and vector triple product

The scalar triple product (or mixed product) of three vectors $\veca$, $\vecb$ and $\vecc$ is a scalar quantity that measures the volume of the parallelepiped formed by them. It is defined as:

$$[\veca, \vecb, \vecc] = \veca \cdot (\vecb \times \vecc)$$ where $\veca \cdot (\vecb \times \vecc)$ is the dot product of $\veca$ with the cross product of $\vecb$ and $\vecc$. The scalar triple product has the following properties:

• It is unchanged under a circular shift of its three operands: $[\veca, \vecb, \vecc] = [\vecb, \vecc, \veca] = [\vecc, \veca, \vecb]$

• It is negated under a swap of any two of its three operands: $[\veca, \vecb, \vecc] = -[\veca, \vecc, \vecb] = -[\vecb, \veca, \vecc] = -[\vecc, \vecb, \veca]$

• It can also be expressed as the determinant of the matrix that has the three vectors as its rows or columns: $[\veca, \vecb, \vecc] = det[\begin pmatrix\mathbf a &\mathbf b &\mathbf c \\end pmatrix]$

• It is zero if and only if the three vectors are coplanar: $[\veca, \vecb, \vecc] = 0$ iff $\theta = 0^\circ$ or $180^\circ$ for any pair of vectors

The scalar triple product can be used to test whether three vectors are linearly independent, to find the volume of a tetrahedron with given vertices, to find the equation of a plane passing through three points, etc.

The vector triple product (or box product) of three vectors $\veca$, $\vecb$ and $\vecc$ is a vector quantity that is perpendicular to the plane spanned by $\veca$ and $\vecb$ and has a magnitude equal to the area of the parallelogram multiplied by the component of $\vec c$ along the direction of $\veca \times \vecb$. It is defined as:

$$[\veca, \vecb, \vecc] \times \vecd = (\veca \times \vecb) \times (\vecc \times \vecd)$$ where $(\veca \times \vecb) \times (\vecc \times \vecd)$ is the cross product of the cross products of the pairs of vectors. The vector triple product has the following properties:

• It is anti-commutative: $[\veca, \vecb, \vecc] \times \vecd = -\vecd \times [\veca, \vecb, \vecc]$

• It satisfies the BAC-CAB rule: $[\veca, \vecb, \vecc] \times \vecd = (\veca \cdot [\vecc, \vecd, \vecb])\vecc - (\veca \cdot [\vecb, \vecd, \vecc])\vecb$

• It can also be expressed as a determinant of a matrix that has the four vectors as its rows or columns: $[\veca, \vecb, \vecc] \times \vecd = det[\begin pmatrix\mathbf a &\mathbf b &\mathbf c &\mathbf d\\end pmatrix]$

• It is zero if and only if the four vectors are coplanar: $[\veca, \vecb, \vecc] \times \vecd = 0$ iff $\theta = 0^\circ$ or $180^\circ$ for any pair of vectors

The vector triple product can be used to find the area of a parallelepiped spanned by three vectors, to find the normal vector to a tetrahedron with given vertices, to find the equation of a line passing through two points and perpendicular to a plane, etc.

Components of a vector and index notation

The components of a vector are the scalar projections of the vector onto the axes of a coordinate system. For example, in a Cartesian coordinate system with unit vectors $\hati$, $\hatj$ and $\hatk$ along the $x$, $y$ and $z$ axes respectively, we can write any vector $\vecv$ as:

$$\vec v = v_x\hati + v_y\hatj + v_z\hatk$$ where $v_x$, $v_y$ and $v_z$ are the components of $\vec v$ along the $x$, $y$ and $z$ axes respectively. We can also write them as a column matrix or a row matrix:

$$\begin align* \begin pmatrix v_x\\ v_y\\ v_z\\ \end pmatrix &= \begin pmatrix v_1\\ v_2\\ v_3\\ \end pmatrix\\ &= (v_x, v_y, v_z)\\ &= (v_1, v_2, v_3) \end align* $$ To simplify the notation further, we can use index notation and write $\vec v$ as:

$$\begin align* \begin pmatrix v_x\\ v_y\\ v_z\\ \end pmatrix &= (v_x, v_y, v_z)\\ &= (v_1, v_2, v_3)\\ &= v_i\hati\\ &= v_i \end align* $$ where $i$ is an index that can take values from 1 to 3 (or from x to z). The index notation allows us to write vector equations in a compact and general form, without specifying the coordinate system or the number of dimensions. For example, we can write the dot product of two vectors $\vec a$ and $\vec b$ as:

$$\vec a \cdot \vec b = a_ib_i$$ where the summation over the repeated index $i$ is implied. This is called the Einstein summation convention. Similarly, we can write the cross product of two vectors $\vec a$ and $\vec b$ as:

$$\vec a \times \vec b = \epsilon_ijka_jb_k\hati$$ where $\epsilon_ijk$ is the Levi-Civita symbol that is defined as:

$$\epsilon_ijk = \begin cases +1 & \text if (i, j, k) \text is an even permutation of (1, 2, 3)\\ -1 & \text if (i, j, k) \text is an odd permutation of (1, 2, 3)\\ 0 & \text if i = j \text or j = k \text or k = i \end cases $$ The index notation is very useful for writing and manipulating tensor equations, as we will see in the next section. 71b2f0854b