MATH 344: Tensor Analysis - Lecture 1: Vectors

Differentiable Curves in N-Dimensional Space

• Definition of a Differentiable Curve:

  • A differentiable curve $\lambda(t)$ in $N$-dimensional space is defined as a mapping of an interval of the one-dimensional real line into an $N$-dimensional space.
  • Points along the curve are parameterized by the continuous variable $t$.

• Coordinate Representation:

  • Each point $P$ located on the curve $\lambda(t)$ is specified by a set of coordinates $\{x^a(t)\}$, where the index $a$ ranges over $1, 2, \dots, N$.
  • In explicit vector coordinate notation, the point is represented as $P(x(t)) = (x^1(t), x^2(t), \dots, x^N(t))$.
  • The parametric equations defining a given curve in $N$-dimensional space are expressed as $x^a = x^a(t)$.

Parametric Representations and Curve Examples

• Straight Line in Two-Dimensional Space:

  • Parametric equations:
  • $x^1(t) = a^1 t + b^1$
  • $x^2(t) = a^2 t + b^2$
  • Parameter range: $t \in (-\infty, \infty)$
  • Relation obtained by eliminating parameter $t$:
  • $t = \frac{x^1 - b^1}{a^1} = \frac{x^2 - b^2}{a^2}$
  • Geometric path: Represents a straight line in two dimensions.

• Parabola in Two-Dimensional Space:

  • Parametric equations:
  • $x^1(t) = t$
  • $x^2(t) = t^2$
  • Parameter range: $t \in (-\infty, \infty)$
  • Relation obtained by eliminating parameter $t$:
  • $x^2 = (x^1)^2$
  • Geometric path: Represents a parabola.

• Circle in Two-Dimensional Space:

  • Parametric equations:
  • $x^1(t) = a \cos(t)$
  • $x^2(t) = a \sin(t)$
  • Parameter range: $t \in [0, 2\pi)$
  • Relation obtained by eliminating parameter $t$:
  • $(x^1)^2 + (x^2)^2 = a^2$
  • Geometric path: Represents a circle of radius $a$.

• Ellipse in Two-Dimensional Space:

  • Parametric equations:
  • $x^1(t) = a \cos(t)$
  • $x^2(t) = b \sin(t)$
  • Parameter range: $t \in [0, 2\pi)$
  • Relation obtained by eliminating parameter $t$:
  • $\frac{(x^1)^2}{a^2} + \frac{(x^2)^2}{b^2} = 1$
  • Geometric path: Represents an ellipse with semi-major/minor axes $a$ and $b$.

• Hyperbola in Two-Dimensional Space:

  • Parametric equations:
  • $x^1(t) = a \tan(t)$
  • $x^2(t) = b \sec(t)$
  • Relation obtained by eliminating parameter $t$:
  • $\frac{(x^2)^2}{b^2} - \frac{(x^1)^2}{a^2} = 1$
  • Geometric path: Represents a hyperbola.

• Helix (or Spring) in Three-Dimensional Space:

  • Parametric equations:
  • $x^1(t) = a \cos(t)$
  • $x^2(t) = a \sin(t)$
  • $x^3(t) = b t$
  • Parameter range: $t \in [0, \infty)$
  • Geometric path: Represents a 3D helix or spring coiled along the $x^3$-axis.

Alternative Parametric Representations

• Uniqueness Properties:

  • A given set of parametric equations $x^a = x^a(t)$ in $N$-dimensional space uniquely determines a curve $\lambda(t)$.
  • The converse is not true: a given curve does not possess a unique parametric representation and can be described by different parametric equations.

• Rational Parameterization Example for a Circle:

  • Alternative parametric equations for a circle:
  • $x^1(t) = a \left( \frac{1 - t^2}{1 + t^2} \right)$
  • $x^2(t) = a \left( \frac{2t}{1 + t^2} \right)$
  • Parameter range: $t \in (-\infty, \infty)$
  • Proof of circular geometry:
  • $(x^1)^2 + (x^2)^2 = a^2 \left( \frac{1 - t^2}{1 + t^2} \right)^2 + a^2 \left( \frac{2t}{1 + t^2} \right)^2$
  • $(x^1)^2 + (x^2)^2 = a^2 \left( \frac{(1 - t^2)^2 + (2t)^2}{(1 + t^2)^2} \right)$
  • $(x^1)^2 + (x^2)^2 = a^2 \left( \frac{1 - 2t^2 + t^4 + 4t^2}{1 + 2t^2 + t^4} \right)$
  • $(x^1)^2 + (x^2)^2 = a^2 \left( \frac{1 + 2t^2 + t^4}{1 + 2t^2 + t^4} \right) = a^2$
  • Conclusion: The rational parameterization generates the identical circle $(x^1)^2 + (x^2)^2 = a^2$.

Rate of Change of a Function Along a Curve

• Problem Statement:

  • Let $f(x^1, x^2, \dots, x^N)$ be a scalar function defined in an $N$-dimensional space, and let $\lambda(t)$ be a given curve.
  • The objective is to determine how the function $f$ changes as one moves along the curve $\lambda(t)$ at a point $P$ on $\lambda(t)$.

• Mathematical Derivation:

  • The total differential change $df$ of the function $f$ across coordinate displacements $dx^a$ is:
  • $df = \frac{\partial f}{\partial x^1} dx^1 + \frac{\partial f}{\partial x^2} dx^2 + \dots + \frac{\partial f}{\partial x^N} dx^N$
  • Expressed using the Einstein Summation Convention (implied summation over repeated index $a$):
  • $df = \frac{\partial f}{\partial x^a} dx^a$
  • The total rate of change of $f$ with respect to parameter $t$ along $\lambda(t)$ is:
  • $\frac{df}{dt} = \frac{\partial f}{\partial x^a} \frac{dx^a}{dt} = \frac{dx^a}{dt} \frac{\partial f}{\partial x^a}$
  • Evaluating this expression specifically at point $P$:
  • $\left( \frac{df}{dt} \right)\Big|_P = \left( \frac{dx^a}{dt} \frac{\partial}{\partial x^a} \right)\Big|_P f = \mathbf{V} f$
  • The linear differential operator $\mathbf{V} = \frac{dx^a}{dt} \frac{\partial}{\partial x^a}$ computes the directional derivative of $f$ along $\lambda(t)$.

Definition and Structure of Vectors in Tensor Analysis

• Operator Definition of a Vector:

  • A vector is defined as a differential operator that acts on a scalar 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.

• Coordinate Basis and Vector Components:

  • An $N$-dimensional vector $\mathbf{V}$ with components $V^a$ ($a = 1, \dots, N$) is expressed in linear combination form as:
  • $\mathbf{V} = V^a \mathbf{E}_a = V^1 \mathbf{E}_1 + V^2 \mathbf{E}_2 + \dots + V^N \mathbf{E}_N$
  • The basis element $\mathbf{E}_a$ is defined as the coordinate partial differential operator:
  • $\mathbf{E}_a = \frac{\partial}{\partial x^a}$
  • $\mathbf{E}_a$ is referred to as the coordinate basis vector.
  • The coefficients $V^a = \frac{dx^a}{dt}$ are the components of the vector $\mathbf{V}$ with respect to the coordinate basis differential operators $\mathbf{E}_a$.

General Parametric Equation of a Straight Line in N-Dimensional Space

• Condition for a Straight Line:

  • In an $N$-dimensional space, a straight line is defined as a curve whose tangent vector $\mathbf{V}$ remains constant at all points $P$ along the curve.

• Derivation:

  • Set the tangent vector components equal to constant values $C^a$:
  • $V^a = \frac{dx^a(t)}{dt} = C^a = \text{Constant} \quad \text{for } a = 1, \dots, N$
  • Integrating both sides with respect to parameter $t$ yields:
  • $x^a(t) = C^a t + K^a$
  • Here, $K^a$ represents integration constants for each component $a = 1, \dots, N$.

• General Parametric Form:

  • The equation $x^a(t) = C^a t + K^a$ represents the general parametric equation for a straight line in an $N$-dimensional space.

Tangent Vector Field of a Circle

• Derivation of Tangent Vector Components:

  • Consider a circle defined in 2D space by $(x^1)^2 + (x^2)^2 = a^2$ with standard parametric equations:
  • $x^1(t) = a \cos(t)$
  • $x^2(t) = a \sin(t)$
  • Differentiating each coordinate with respect to $t$ gives the vector components:
  • $V^1 = \frac{dx^1}{dt} = -a \sin(t) = -x^2$
  • $V^2 = \frac{dx^2}{dt} = a \cos(t) = x^1$

• Operator Expression:

  • The tangent vector $\mathbf{V}$ expressed via coordinate basis operators is:
  • $\mathbf{V} = V^1 \mathbf{E}_1 + V^2 \mathbf{E}_2 = -x^2 \mathbf{E}_1 + x^1 \mathbf{E}_2 = -x^2 \frac{\partial}{\partial x^1} + x^1 \frac{\partial}{\partial x^2}$

• Physical and Directional Behavior:

  • When $x^1 = 0$ (intersections on the $x^2$ axis), $V^1 = -x^2$ and $V^2 = 0$, meaning the rate of change occurs exclusively along the $x^1$ direction.
  • When $x^2 = 0$ (intersections on the $x^1$ axis), $V^1 = 0$ and $V^2 = x^1$, meaning the rate of change occurs exclusively along the $x^2$ direction.