Mathematical Preliminaries - 1: Vector Representations

Summary

This lecture introduces the mathematical foundations of applied elasticity, focusing on key preliminary concepts. The instructor outlines different vector and tensor representations, emphasizing compact, component, matrix, and index notations. Through these notations, the course aims to simplify complex algebraic expressions, critical for continuum mechanics. The instructor also highlights the importance of understanding index notation, offering essential rules for its use. The lecture serves as a starting point for deeper discussions on tensor operations, laying the groundwork for more advanced topics in elasticity and material behavior.

Takeaways

• 😀 The course on applied elasticity is mathematically focused, and its applications will be explained through mathematical language.
• 😀 The first chapter covers mathematical preliminaries that are essential for understanding the concepts in subsequent chapters of applied elasticity.
• 😀 The course takes a selective approach to learning, focusing only on the necessary mathematical concepts for the course rather than covering all of tensor algebra or intensive calculus.
• 😀 Vectors can be represented in multiple ways, including compact notation (with an arrow or tilde above the vector), component notation (using unit vectors and scalar components), and matrix notation.
• 😀 Component notation breaks a vector into scalar components along chosen coordinate axes, but these components change if the coordinate system is changed.
• 😀 Matrix representation of vectors can be written as either column or row vectors, but for this course, the column matrix form will be used.
• 😀 Index notation simplifies algebra and calculations by omitting summation signs and unit vectors, making it more concise and easier to use in continuum mechanics.
• 😀 In index notation, a repeated index indicates summation, and the index can be replaced by any other letter without affecting the meaning.
• 😀 It is crucial not to repeat an index more than once, as this would make the expression meaningless.
• 😀 The number of free indices in an expression indicates the order of the tensor. A single free index indicates a vector (order 1), and two free indices indicate a second-order tensor (like stress or strain tensors).

Q & A

What is the main focus of the first chapter in applied elasticity?

The first chapter focuses on mathematical preliminaries and utilities that will be used throughout subsequent chapters of applied elasticity. These include various representations of vectors and tensors, which are foundational for understanding the course.

Why does the instructor choose a selective approach to teaching mathematical preliminaries?

The instructor selects a more targeted approach to save time, particularly because of the online format of the course. Instead of covering the full depth of topics like tensor algebra, they focus only on the concepts that are necessary for the course.

What are the different ways of representing a vector as mentioned in the transcript?

A vector can be represented in compact notation (with an arrow or tilde above or below), component notation (expressed as scalar components along coordinate axes), matrix representation (as column or row matrices), and index notation (where the vector components are written as indices).

What is the difference between compact notation and component notation?

In compact notation, the vector is represented simply as a symbol with an arrow or tilde. In component notation, the vector is expressed as a sum of its scalar components along unit vectors in a chosen coordinate system.

What is the significance of the unit vectors (e1 hat, e2 hat, e3 hat) in the component notation?

The unit vectors (e1 hat, e2 hat, e3 hat) define the direction of the vector components along the respective coordinate axes (x1, x2, x3). These unit vectors are a choice, and the same vector can be expressed in different coordinate systems with different unit vectors.

Why is matrix representation of vectors referred to as 'column' or 'row' form, and which one is preferred for this course?

Matrix representation expresses the vector components as either a column or row matrix. For this course, the column matrix form is chosen for consistency and clarity in subsequent discussions.

What is the core idea behind the index notation for vectors?

In index notation, vectors are written as a simple series of components, such as vi, without requiring summation signs or unit vectors. The repeated index implies summation over the components of the vector.

What does a repeated index in the index notation signify?

A repeated index indicates summation. For example, ai bi means the sum of the components a1b1 + a2b2 + a3b3, which is essentially the dot product of two vectors.

What are some key rules to remember when using index notation?

Some key rules include: no summation signs or unit vectors are needed, a repeated index implies summation, repeated indices are like dummy variables and can be replaced by other letters, and an index should never be repeated more than once in the same term.

How does the order of a tensor relate to the number of free indices in index notation?

The order of a tensor is determined by the number of free indices in a term. A vector (first-order tensor) has one free index, a second-order tensor like the stress or strain tensor has two free indices, and so on.