Mathematical Preliminaries

Overview

This page provides an overview of the basics of vector and tensor theory, touching on the fundamental principles that make up vector and tensor calculus. These ideas will be viewed through the lens of indicial notation, a mathematical notation that significantly simplifies complex expressions involving vectors and tensors. Coordinate transformations will also be explored in this section.

Vector Theory

This section will include an overview of vector basics including vector algebra, vector and scalar fields, as well as fundamental identities involving vectors.

Vector Algebra

A vector is a directed line segment, possessing the qualities of both magnitude and direction. The magnitude being the scale of the vector and the direction being the relative orientation of the vector in a given coordinate system.

(Visual representation of a vector)

In mathematical notation, the vector, , can be written as follows:

where , , and are the unit vectors (magnitude of one) in the , , and directions, respectively. , , and are the scalar values corresponding to each of these three unit vectors. Each of these three terms are called components of vector . That is, each component describes the portion of that is parallel to the respective axis of said component. Together, all three components make up the resultant vector of .

Given the components of , the total magnitude can be found:

Let another vector, be given as:

two vectors can be added such that:

Note: , , and are an alternative and common way to denote the unit vectors , , and , respectively. Vectors can also be multiplied by means of a scalar, vector, or dyadic product. A scalar product of two vectors, also known as a dot product, will yield a scalar value and is given as:

A vector product, also known as a cross product, yields a vector that is orthogonal to both original vectors and is given as

The dyadic product of two vectors yields a 3x3 matrix, or a tensor, and is given as:

Scalar and Vector Fields

A field is any quantity that varies spatially. That is, if a value depends on and .

A scalar field is defined by a scalar quantity expressed as a function of cartesian coordinates. An example of a scalar field could be a map showing the temperature at different locations.

A vector field is formed when the components of a vector depend on the spatial coordinates of the vector. An example of a vector field could be a weather map showing the wind direction at different points in space.

Scalar and vector fields have many useful applications, but to be able to explore these applications, the fundamentals of vector calculus must be understood. The del/nabla operator, can be applied to both scalars and vectors in several ways in order to obtain useful information about a given value or field.

The gradient of a scalar, , yields a vector, defined as follows:

The divergence of a vector, , is given as the dot product between the vector and the del operator and yields a scalar as follows:

The curl of a vector, , also yields a vector and is given as the cross product between the vector and the del operator as follows:

The gradient of a vector, , yields a tensor and is given as the dyadic product between the vector and the del operator as follows:

(A scalar field with lines of constant value plot)

(Scalar field vector diagram)

The figure displays a sample scalar field. The plotted lines represent trajectories where the value of the scalar field is constant. Each vector plotted in the field is the gradient taken at the point of the base of the vector. The gradient vectors are always orthogonal to the lines of constant value. This is because the gradient vector points toward the direction of greatest increase of the field, which is always orthogonal to the direction of no change.

Indicial Notation

Indicial notation is a mathematical shorthand that is useful for concisely representing complex operations. Any number of dimensions can be represented through this notation, but for the sake of this course, we will work with the standard three dimensions, representative of the three spatial dimensions.

Using indicial notation a vector, , consisting of three components (, , ) can be written simply as where . A tensor can be similarly condensed. Given:

the same tensor can be written simply as , with and .

(The number of terms represented by a certain notation)