MATH 344: Tensor Analysis - Lecture 1: Vectors

Vocabulary flashcards covering key definitions and formulas from MATH 344 Lecture 1 on vectors, parametric curves, coordinate bases, and differential operators.

Differentiable curve in $n$-dimensional space

A mapping of an interval of a one-dimensional real line into $n$-dimensional space, denoted as $\lambda(t)$, where each point $P$ on the curve is specified by the coordinates $\{x^a(t)\}$ for $a = 1, \dots, N$.

Helix (or spring)

A curve in 3-dimensional space specified by the parametric equations $x^1 = a \cos(t)$, $x^2 = a \sin(t)$, and $x^3 = b t$, where $t \in [0, \infty)$.

Uniqueness property of parametric representations

The principle that parametric equations $x^a = x^a(t)$ uniquely determine a curve $\lambda(t)$ in $n$-dimensional space, but the converse is not true as a given curve can have different parametric representations.

Rate of change of a function along a curve

The change of a function $f(x^1, x^2, \dots, x^n)$ along a curve $\lambda(t)$ at point $P$, given by $\left(\frac{df}{dt}\right)_{\lambda(t)} = \frac{\partial f}{\partial x^a} \frac{dx^a}{dt} = \mathbf{V} f$ using the Einstein summation convention.

Vector (Differential Operator)

A differential operator that acts on a function at a given point on a given curve to determine the change of the function along the tangent of the curve at that point.

Co-ordinate basis vector

The differential operator $\mathbb{E}_a = \frac{\partial}{\partial x^a}$, used to expand an $n$-dimensional vector as $\mathbf{V} = V^a \mathbb{E}_a = V^1 \mathbb{E}_1 + V^2 \mathbb{E}_2 + \dots + V^N \mathbb{E}_N$.

Vector components ($V^a$)

The coefficients $V^a$ (for $a = 1, \dots, N$) of a vector with respect to the coordinate basis differential operators $\mathbb{E}_a$.

General parametric equation for a straight line

The equation $x^a(t) = C^a t + K^a$ for $a = 1, \dots, N$ in $n$-dimensional space, representing a curve whose tangent vector $V^a = \frac{dx^a(t)}{dt} = C^a$ remains constant at all points.

Tangent vector of a circle

For a circle parametrized by $x^1 = a \cos(t)$ and $x^2 = a \sin(t)$, the tangent vector is $\mathbf{V} = V^1 \mathbb{E}_1 + V^2 \mathbb{E}_2 = -x^2 \frac{\partial}{\partial x^1} + x^1 \frac{\partial}{\partial x^2}$, where $V^1 = -x^2$ and $V^2 = x^1$.