Lecture on Vectors and Tensors by Dan Flesch

Introduction

Goal: Explain what a tensor is in about 12 minutes without heavy use of mathematical equations.
Tools used: Household objects like children's blocks, small arrows, cardboard, and a pointed stick.

Definition: A vector represents a quantity with both magnitude and direction.
Examples:
- Force of gravity
- Earth's magnetic field
- Velocity of a particle in a fluid
Vectors representing other quantities:
- Area: Length of the vector proportional to the area, direction perpendicular to the surface.

Coordinate System: Example of a Cartesian coordinate system (x, y, z axes).
Basis Vectors (Unit Vectors):
- Length of one unit
- Direction aligned with the coordinate axes
- Notations: x-hat (i-hat), y-hat (j-hat), z-hat (k-hat)
Finding Vector Components:
- Project vector onto the x and y axes using a light.
- Shadow on the axis represents the component.

Using Basis Vectors:
- Decompose vector into x-hat and y-hat components.
- Representation as an array of components (e.g., column vectors).

Definition: Vectors are part of a broader class called tensors.
Scalars: Tensors of rank 0 (no directional indicators).
Vectors: Tensors of rank 1 (one index, one basis vector per component).
Rank 2 Tensors:
- 9 components in 3D space, each with two indices.
- Example: Forces inside a solid object.
- Components like a sub xx (x-directed force on x-direction surface).
Rank 3 Tensors:
- 27 components in 3D space, each with three indices.
- Example: a sub xxx (pertaining to three x basis vectors).
- Components organized in slabs (x, y, z as third indices).

Universal Agreement: All observers agree on the combination of components and basis vectors, not individually.
Transformation: Basis vectors and components transform to keep the tensor invariant across reference frames.
Lillian Lieber's Quote: "Tensors are the facts of the universe."