Chapter 1 • Theory & Derivations

Coordinates, Index Notation, Summation Convention & Affine Spaces

Foundations of tensor calculus: $n$-dimensional Euclidean and affine spaces, Einstein summation convention, dummy and free index discipline, the generalized Kronecker delta $\delta^i_j$, Levi-Civita permutation pseudo-tensors $\epsilon_{ijk}$ and $\epsilon^{ijk}$, coordinate transformations, and transformation laws for contravariant vectors $A^i$ and covariant vectors $B_i$.

§1.1Curvilinear Coordinates, Scale Factors & Coordinate Transformations

1. Curvilinear Coordinate Systems in $\mathbb{R}^n$

Let $\mathbb{R}^n$ denote $n$-dimensional Euclidean space equipped with standard rectangular Cartesian coordinates $(x^1, x^2, \dots, x^n)$. A system of curvilinear coordinates $(\bar{x}^1, \bar{x}^2, \dots, \bar{x}^n)$ is introduced by establishing a set of $n$ invertible, single-valued, continuously differentiable functions:

x^i = x^i(\bar{x}^1, \bar{x}^2, \dots, \bar{x}^n), \qquad (i = 1, 2, \dots, n)

Definition 1.1 (Regular Coordinate Transformation & Jacobian): A coordinate transformation is said to be regular (admissible) in an open domain $\mathcal{U} \subseteq \mathbb{R}^n$ if:

• The functions $x^i(\bar{x}^1, \dots, \bar{x}^n)$ possess continuous partial derivatives of class at least $C^2$.

• The Jacobian determinant $J$ does not vanish at any point of $\mathcal{U}$:

J = \frac{\partial(x^1, x^2, \dots, x^n)}{\partial(\bar{x}^1, \bar{x}^2, \dots, \bar{x}^n)} = \det\left( \frac{\partial x^i}{\partial \bar{x}^j} \right) \ne 0

By the Inverse Function Theorem, non-vanishing of $J$ guarantees the existence of a unique, continuously differentiable local inverse transformation:

\bar{x}^j = \bar{x}^j(x^1, x^2, \dots, x^n), \qquad (j = 1, 2, \dots, n)

satisfying the inverse Jacobian relationship:

\det\left( \frac{\partial \bar{x}^j}{\partial x^i} \right) = \frac{1}{J} \ne 0

2. Coordinate Curves and Coordinate Surfaces

Fixing all new coordinates except one defines a coordinate curve:

$i$-th Coordinate Curve: The locus of points where $\bar{x}^j = c^j$ (constant) for all $j \ne i$, parameterized solely by $\bar{x}^i$.
$i$-th Coordinate Surface: The locus of points where $\bar{x}^i = c^i$ (constant), while the remaining $n-1$ coordinates vary freely.

The intersection of $n-1$ coordinate surfaces forms a single coordinate curve.

3. Tangent Basis Vectors and Lamé Scale Factors

Let $\mathbf{r} = \mathbf{r}(x^1, \dots, x^n)$ denote the position vector of an arbitrary point in $\mathbb{R}^n$. Under the transformation to curvilinear coordinates $\bar{x}^i$, the tangent vector to the $i$-th coordinate curve is:

\mathbf{e}_i = \frac{\partial \mathbf{r}}{\partial \bar{x}^i} = \sum_{k=1}^n \frac{\partial x^k}{\partial \bar{x}^i} \hat{\mathbf{i}}_k

Definition 1.2 (Scale Factors / Lamé Coefficients): The scale factor $h_i$ associated with coordinate $\bar{x}^i$ is the magnitude of the natural tangent vector $\mathbf{e}_i$:

h_i = \|\mathbf{e}_i\| = \sqrt{\mathbf{e}_i \cdot \mathbf{e}_i} = \sqrt{\sum_{k=1}^n \left( \frac{\partial x^k}{\partial \bar{x}^i} \right)^2}

The corresponding unit tangent vector is $\hat{\mathbf{e}}_i = \frac{\mathbf{e}_i}{h_i}$.

A curvilinear coordinate system is orthogonal if the coordinate curves intersect at right angles at every point, which occurs if and only if:

\mathbf{e}_i \cdot \mathbf{e}_j = 0 \qquad \text{for all } i \ne j

§1.2Einstein Summation Convention & The Kronecker Delta

1. The Einstein Summation Convention

In 1916, Albert Einstein introduced a concise notational convention to streamline tensor calculations.

Axiom 1.1 (Einstein Summation Convention): Whenever an index appears twice in a single monomial term—once as a superscript (upper index) and once as a subscript (lower index)—summation over that repeated index is automatically implied over its complete allowable range (typically $1, 2, \dots, n$), without writing the summation symbol $\sum$:

A^i B_i \equiv \sum_{i=1}^n A^i B_i = A^1 B_1 + A^2 B_2 + \dots + A^n B_n

Rigorous Rules Governing Index Balancing:

Dummy (Umbral) Indices:

An index that appears once as a superscript and once as a subscript in a term is a dummy index. A dummy index can be freely replaced by any unused letter without altering the mathematical meaning:

A^i B_i = A^k B_k = A^\alpha B_\alpha

Free Indices:

An index that appears only once in a term is a free index.

Every term in a valid tensor equation must contain the exact same free indices at the exact same height (upper or lower).
If an equation has $p$ free indices, it represents a system of $n^p$ separate scalar equations.

Index Multiplicity Prohibition:

An index must never appear more than twice in any single term. An expression such as $A^i B_i C^i$ is mathematically meaningless and strictly forbidden in tensor calculus.

2. The Kronecker Delta Symbol

The mixed Kronecker delta $\delta_j^i$ is defined by:

\delta_j^i = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \ne j \end{cases}

Algebraic Properties of the Kronecker Delta:

Substitution / Sifting Property:

Contraction with the Kronecker delta replaces the contracted index:

\delta_j^i A^j = A^i, \qquad \delta_j^i B_i = B_j, \qquad \delta_j^i T^j_{\; k} = T^i_{\; k}

Trace / Dimensionality Property:

Summing over both indices yields the dimension $n$ of the underlying space:

\delta_i^i = \delta_1^1 + \delta_2^2 + \dots + \delta_n^n = 1 + 1 + \dots + 1 = n

Chain Rule Inversion:

For any regular coordinate transformation $x^i \leftrightarrow \bar{x}^j$:

\frac{\partial x^i}{\partial \bar{x}^k} \frac{\partial \bar{x}^k}{\partial x^j} = \frac{\partial x^i}{\partial x^j} = \delta_j^i, \qquad \frac{\partial \bar{x}^i}{\partial x^k} \frac{\partial x^k}{\partial \bar{x}^j} = \delta_j^i

§1.3Space of n-Dimensions, Affine Spaces & Jacobian Transitions

1. Spaces of $n$-Dimensions ($V_n, E_n, R_n$)

In differential geometry and physics, we distinguish between three fundamental mathematical structures:

Affine Space ($A_n$): A geometric space consisting of points where parallel displacement and linear combinations of displacement vectors are defined, but no intrinsic concept of angle or metric length exists.
Euclidean Space ($E_n$): An affine space equipped with a constant, positive-definite metric tensor (in Cartesian coordinates, $g_{ij} = \delta_{ij}$).
Riemannian Space ($R_n$ or $V_n$): A differentiable manifold equipped with a position-dependent metric tensor $g_{ij}(x)$, where distance along curves is defined by $ds^2 = g_{ij}(x) dx^i dx^j$.

2. Jacobian Matrix of Coordinate Transitions

Consider two overlapping coordinate charts $x^i$ and $\bar{x}^j$ covering an open region $\mathcal{U} \subseteq V_n$. The transition between charts is governed by the Jacobian transition matrix:

\Lambda^i_{\; j} = \frac{\partial x^i}{\partial \bar{x}^j}, \qquad (\Lambda^{-1})^j_{\; k} = \frac{\partial \bar{x}^j}{\partial x^k}

Their matrix product satisfies identity:

\Lambda^i_{\; j} (\Lambda^{-1})^j_{\; k} = \frac{\partial x^i}{\partial \bar{x}^j} \frac{\partial \bar{x}^j}{\partial x^k} = \delta^i_k

Theorem 1.1 (Transformation of Volume Elements): Let $d^n x = dx^1 dx^2 \cdots dx^n$ denote the differential coordinate volume element in the $x^i$ frame, and $d^n \bar{x} = d\bar{x}^1 \cdots d\bar{x}^n$ in the $\bar{x}^j$ frame. Then:

d^n x = |J| \, d^n \bar{x} = \left| \det\left( \frac{\partial x^i}{\partial \bar{x}^j} \right) \right| d^n \bar{x}

Volume elements do not transform as scalar invariants, but rather as scalar densities of weight $-1$.

§1.4Contravariant Vectors vs Covariant Vectors (Covectors / 1-Forms)

1. Contravariant Vectors (Tangent Vectors)

Consider the differential displacement vector between two neighboring points $P(x^i)$ and $Q(x^i + dx^i)$. By the multi-variable chain rule:

d\bar{x}^i = \frac{\partial \bar{x}^i}{\partial x^j} dx^j

Definition 1.3 (Contravariant Vector): A set of $n$ quantities $A^1, A^2, \dots, A^n$ defined at a point $P$ constitutes the components of a contravariant vector (or tensor of rank $(1, 0)$) if, under an arbitrary regular coordinate transformation $x^i \to \bar{x}^i$, its components transform according to the direct Jacobian law:

\bar{A}^i = \frac{\partial \bar{x}^i}{\partial x^j} A^j

The superscript indicates that the components transform with the partial derivatives of the new coordinates with respect to the old coordinates (in the same manner as coordinate differentials $dx^i$).

Physical Examples: Velocity vector $v^i = \frac{dx^i}{dt}$, acceleration vector $a^i = \frac{d^2 x^i}{dt^2}$, displacement $dx^i$, current density 4-vector $J^\mu$.

2. Covariant Vectors (Covectors / Differential 1-Forms)

Consider a scalar field $\phi(x^1, \dots, x^n)$ that is invariant under coordinate changes ($\bar{\phi}(\bar{x}) = \phi(x)$). Computing the gradient components via the chain rule:

\frac{\partial \bar{\phi}}{\partial \bar{x}^i} = \frac{\partial \phi}{\partial x^j} \frac{\partial x^j}{\partial \bar{x}^i}

Definition 1.4 (Covariant Vector / Covector): A set of $n$ quantities $B_1, B_2, \dots, B_n$ defined at a point $P$ constitutes the components of a covariant vector (or covector, 1-form, tensor of rank $(0, 1)$) if, under an arbitrary regular coordinate transformation $x^i \to \bar{x}^i$, its components transform according to the inverse Jacobian law:

\bar{B}_i = \frac{\partial x^j}{\partial \bar{x}^i} B_j

The subscript indicates that the components transform with the partial derivatives of the old coordinates with respect to the new coordinates (in the inverse manner to coordinate differentials).

Physical Examples: Gradient of a scalar potential $\nabla_i \phi = \frac{\partial \phi}{\partial x^i}$, electromagnetic 4-potential $A_\mu$, normal vector to a hypersurface $n_i = \partial_i f$.

``` Geometric Duality of Vectors and Covectors

Contravariant Vector Aⁱ Covariant Vector Bᵢ (Tangent Arrow) (Stack of Hyperplanes) ▲ │ │ │ │ / │ │ │ │ / vⁱ = dxⁱ/dt │ │ │ │ • ▼ ▼ ▼ ▼ Direction of displacement Rate of variation / gradient Transforms via ∂x̄ⁱ/∂xʲ Transforms via ∂xʲ/∂x̄ⁱ ```

§1.5Invariance of the Inner Product & Interactive Transformation Engine

1. Invariance of the Inner Product (Contraction of Vector and Covector)

A central postulate of tensor calculus is that physical quantities must not depend on the arbitrary choice of coordinates.

Theorem 1.2 (Scalar Invariance of Contraction): Let $A^i$ be a contravariant vector and $B_i$ be a covariant vector at point $P$. Then the contraction:

\Phi = A^i B_i = A^1 B_1 + A^2 B_2 + \dots + A^n B_n

is an absolute scalar invariant under all regular coordinate transformations:

\bar{\Phi} = \bar{A}^i \bar{B}_i = A^j B_j = \Phi

Proof: Transform each vector according to its definition:

\bar{A}^i = \frac{\partial \bar{x}^i}{\partial x^j} A^j, \qquad \bar{B}_i = \frac{\partial x^k}{\partial \bar{x}^i} B_k

Form the contracted product in the new coordinate system:

\bar{A}^i \bar{B}_i = \left( \frac{\partial \bar{x}^i}{\partial x^j} A^j \right) \left( \frac{\partial x^k}{\partial \bar{x}^i} B_k \right) = \left( \frac{\partial \bar{x}^i}{\partial x^j} \frac{\partial x^k}{\partial \bar{x}^i} \right) A^j B_k

By the chain rule identity:

\frac{\partial x^k}{\partial \bar{x}^i} \frac{\partial \bar{x}^i}{\partial x^j} = \frac{\partial x^k}{\partial x^j} = \delta^k_j

Substituting this Kronecker delta:

\bar{A}^i \bar{B}_i = \delta^k_j A^j B_k = A^j (\delta^k_j B_k) = A^j B_j

The value of $\Phi$ is identical in all coordinate systems. $\blacksquare$

2. Interactive Coordinate Transformation & Dual-Frame Engine

The simulation below provides a real-time laboratory for coordinate geometry:

Continuous Transformation Control: Switch seamlessly between Cartesian, Polar $(r, \theta)$, and Oblique Shear $(u, v)$ frames.
Dual Basis Inspector: Observe the natural tangent vectors $\mathbf{e}_1, \mathbf{e}_2$ alongside the dual reciprocal covectors $\mathbf{e}^1, \mathbf{e}^2$.
Metric Verification: Real-time calculation of $g_{ij} = \mathbf{e}_i \cdot \mathbf{e}_j$ and verification that $\mathbf{e}_i \cdot \mathbf{e}^j = \delta_i^j$.

2D Curvilinear Coordinate Transformations & Basis Vectors Engine

60 FPS Real-Time Canvas Engine

Interact with curvilinear coordinate transformations across Cartesian, Polar, Elliptic, and Hyperbolic coordinate systems. Drag points to dynamically observe the contravariant tangent basis vectors $\mathbf{e}_i = rac{\partial \mathbf{r}}{\partial x^i}$ and covariant normal dual covector basis $\mathbf{e}^i = abla x^i$ with live metric component calculations.

Rigorous Tiered Solved Examination Problems

Step-by-step unskipped derivations, complete proofs, and verification across Foundational, Advanced, and Honors tiers.

Foundational Level

Problem 1.1: Problem 1.1: Vector Component Transformations under 2D Rotation and Shear

Consider the two-dimensional Cartesian plane with coordinates $(x^1, x^2) = (x, y)$.

A new coordinate system $(\bar{x}^1, \bar{x}^2)$ is obtained by rotating the axes counterclockwise by an angle $\theta$:

\bar{x}^1 = x^1 \cos\theta + x^2 \sin\theta, \qquad \bar{x}^2 = -x^1 \sin\theta + x^2 \cos\theta

Compute the Jacobian transformation matrices $\frac{\partial \bar{x}^i}{\partial x^j}$ and $\frac{\partial x^j}{\partial \bar{x}^i}$, and verify $\frac{\partial \bar{x}^i}{\partial x^k} \frac{\partial x^k}{\partial \bar{x}^j} = \delta^i_j$.

Given a contravariant vector with components $A^1 = 3, A^2 = 4$ at the origin, find its components $(\bar{A}^1, \bar{A}^2)$ in the rotated system for $\theta = \pi/4$.
Given a covariant vector with components $B_1 = 3, B_2 = 4$, compute its components $(\bar{B}_1, \bar{B}_2)$ and verify that the scalar contraction $A^i B_i = \bar{A}^i \bar{B}_i$ is strictly conserved.

1. Calculation of the Jacobian Transformation Matrices

The forward coordinate relations are:

\bar{x}^1 = x^1 \cos\theta + x^2 \sin\theta, \qquad \bar{x}^2 = -x^1 \sin\theta + x^2 \cos\theta

The Jacobian matrix $\Lambda^i_{\; j} = \frac{\partial \bar{x}^i}{\partial x^j}$ is:

\left( \frac{\partial \bar{x}^i}{\partial x^j} \right) = \begin{pmatrix} \frac{\partial \bar{x}^1}{\partial x^1} & \frac{\partial \bar{x}^1}{\partial x^2} \\ \frac{\partial \bar{x}^2}{\partial x^1} & \frac{\partial \bar{x}^2}{\partial x^2} \end{pmatrix} = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}

The inverse relations (expressing old coordinates in terms of new) are obtained by inverting the orthogonal rotation matrix:

x^1 = \bar{x}^1 \cos\theta - \bar{x}^2 \sin\theta, \qquad x^2 = \bar{x}^1 \sin\theta + \bar{x}^2 \cos\theta

The inverse Jacobian matrix is:

\left( \frac{\partial x^j}{\partial \bar{x}^i} \right) = \begin{pmatrix} \frac{\partial x^1}{\partial \bar{x}^1} & \frac{\partial x^1}{\partial \bar{x}^2} \\ \frac{\partial x^2}{\partial \bar{x}^1} & \frac{\partial x^2}{\partial \bar{x}^2} \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

Now verify the contraction:

\left( \frac{\partial \bar{x}^i}{\partial x^k} \right) \left( \frac{\partial x^k}{\partial \bar{x}^j} \right) = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} = \begin{pmatrix} \cos^2\theta + \sin^2\theta & -\cos\theta\sin\theta + \sin\theta\cos\theta \\ -\sin\theta\cos\theta + \cos\theta\sin\theta & \sin^2\theta + \cos^2\theta \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \delta^i_j

2. Transformation of Contravariant Vector $A^i = (3, 4)$

For $\theta = \pi/4$, $\cos(\pi/4) = \sin(\pi/4) = \frac{\sqrt{2}}{2}$.

\bar{A}^1 = \frac{\partial \bar{x}^1}{\partial x^1} A^1 + \frac{\partial \bar{x}^1}{\partial x^2} A^2 = 3\left(\frac{\sqrt{2}}{2}\right) + 4\left(\frac{\sqrt{2}}{2}\right) = \frac{7\sqrt{2}}{2}

\bar{A}^2 = \frac{\partial \bar{x}^2}{\partial x^1} A^1 + \frac{\partial \bar{x}^2}{\partial x^2} A^2 = -3\left(\frac{\sqrt{2}}{2}\right) + 4\left(\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{2}

Thus:

\bar{A}^i = \left( \frac{7\sqrt{2}}{2}, \; \frac{\sqrt{2}}{2} \right)

3. Transformation of Covariant Vector $B_i = (3, 4)$

For covariant vectors:

\bar{B}_1 = \frac{\partial x^1}{\partial \bar{x}^1} B_1 + \frac{\partial x^2}{\partial \bar{x}^1} B_2 = 3(\cos\theta) + 4(\sin\theta) = \frac{7\sqrt{2}}{2}

\bar{B}_2 = \frac{\partial x^1}{\partial \bar{x}^2} B_1 + \frac{\partial x^2}{\partial \bar{x}^2} B_2 = 3(-\sin\theta) + 4(\cos\theta) = \frac{\sqrt{2}}{2}

Thus:

\bar{B}_i = \left( \frac{7\sqrt{2}}{2}, \; \frac{\sqrt{2}}{2} \right)

4. Verification of Invariant Contraction

In the original $(x^1, x^2)$ coordinates:

$$A^i B_i = A^1 B_1 + A^2 B_2 = (3)(3) + (4)(4) = 9 + 16 = 25$$

In the rotated $(\bar{x}^1, \bar{x}^2)$ coordinates:

\bar{A}^i \bar{B}_i = \bar{A}^1 \bar{B}_1 + \bar{A}^2 \bar{B}_2 = \left(\frac{7\sqrt{2}}{2}\right)\left(\frac{7\sqrt{2}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) = \frac{49 \times 2}{4} + \frac{1 \times 2}{4} = \frac{98 + 2}{4} = \frac{100}{4} = 25

The scalar product $\bar{A}^i \bar{B}_i = A^i B_i = 25$ is strictly conserved. $\blacksquare$

Advanced Level

Problem 1.2: Problem 1.2: General Curvilinear Coordinate System in $\mathbb{R}^3$, Jacobian & Scale Factors

Consider spherical polar coordinates in $\mathbb{R}^3$ defined by:

x^1 = r \sin\theta \cos\phi, \qquad x^2 = r \sin\theta \sin\phi, \qquad x^3 = r \cos\theta

where $\bar{x}^1 = r, \bar{x}^2 = \theta, \bar{x}^3 = \phi$ with $r > 0, 0 < \theta < \pi, 0 \le \phi < 2\pi$.

Compute the Jacobian matrix $\frac{\partial x^i}{\partial \bar{x}^j}$ and evaluate its determinant $J = \det\left(\frac{\partial x^i}{\partial \bar{x}^j}\right)$.
Calculate the natural basis vectors $\mathbf{e}_1 = \frac{\partial \mathbf{r}}{\partial r}, \mathbf{e}_2 = \frac{\partial \mathbf{r}}{\partial \theta}, \mathbf{e}_3 = \frac{\partial \mathbf{r}}{\partial \phi}$.
Determine the Lamé scale factors $h_r, h_\theta, h_\phi$ and prove that the coordinate system is strictly orthogonal everywhere.
Express the differential volume element $dV$ in terms of $r, \theta, \phi$.

1. Jacobian Matrix and Determinant

The Jacobian matrix $\Lambda^i_{\; j} = \frac{\partial x^i}{\partial \bar{x}^j}$ is:

\left( \frac{\partial x^i}{\partial \bar{x}^j} \right) = \begin{pmatrix} \sin\theta\cos\phi & r\cos\theta\cos\phi & -r\sin\theta\sin\phi \\ \sin\theta\sin\phi & r\cos\theta\sin\phi & r\sin\theta\cos\phi \\ \cos\theta & -r\sin\theta & 0 \end{pmatrix}

Expanding the determinant along the third row:

\begin{aligned} J &= \cos\theta \begin{vmatrix} r\cos\theta\cos\phi & -r\sin\theta\sin\phi \\ r\cos\theta\sin\phi & r\sin\theta\cos\phi \end{vmatrix} - (-r\sin\theta) \begin{vmatrix} \sin\theta\cos\phi & -r\sin\theta\sin\phi \\ \sin\theta\sin\phi & r\sin\theta\cos\phi \end{vmatrix} \\ &= \cos\theta \left( r^2\sin\theta\cos\theta\cos^2\phi + r^2\sin\theta\cos\theta\sin^2\phi \right) + r\sin\theta \left( r\sin^2\theta\cos^2\phi + r\sin^2\theta\sin^2\phi \right) \\ &= \cos\theta (r^2\sin\theta\cos\theta) + r\sin\theta (r\sin^2\theta) \\ &= r^2\sin\theta\cos^2\theta + r^2\sin^3\theta = r^2\sin\theta(\cos^2\theta + \sin^2\theta) = r^2\sin\theta \end{aligned}

Thus, $J = r^2 \sin\theta$, which is strictly positive for $r > 0$ and $0 < \theta < \pi$.

2. Natural Tangent Basis Vectors

Differentiating $\mathbf{r}(r, \theta, \phi)$:

$\mathbf{e}_1 = \mathbf{e}_r = \frac{\partial \mathbf{r}}{\partial r} = (\sin\theta\cos\phi)\hat{\mathbf{i}} + (\sin\theta\sin\phi)\hat{\mathbf{j}} + (\cos\theta)\hat{\mathbf{k}}$
$\mathbf{e}_2 = \mathbf{e}_\theta = \frac{\partial \mathbf{r}}{\partial \theta} = (r\cos\theta\cos\phi)\hat{\mathbf{i}} + (r\cos\theta\sin\phi)\hat{\mathbf{j}} - (r\sin\theta)\hat{\mathbf{k}}$
$\mathbf{e}_3 = \mathbf{e}_\phi = \frac{\partial \mathbf{r}}{\partial \phi} = (-r\sin\theta\sin\phi)\hat{\mathbf{i}} + (r\sin\theta\cos\phi)\hat{\mathbf{j}}$

3. Lamé Scale Factors and Orthogonality Proof

Compute the magnitudes $h_i = \|\mathbf{e}_i\| = \sqrt{\mathbf{e}_i \cdot \mathbf{e}_i}$:

$h_r^2 = \mathbf{e}_r \cdot \mathbf{e}_r = \sin^2\theta\cos^2\phi + \sin^2\theta\sin^2\phi + \cos^2\theta = \sin^2\theta + \cos^2\theta = 1 \implies h_r = 1$
$h_\theta^2 = \mathbf{e}_\theta \cdot \mathbf{e}_\theta = r^2\cos^2\theta\cos^2\phi + r^2\cos^2\theta\sin^2\phi + r^2\sin^2\theta = r^2(\cos^2\theta + \sin^2\theta) = r^2 \implies h_\theta = r$
$h_\phi^2 = \mathbf{e}_\phi \cdot \mathbf{e}_\phi = r^2\sin^2\theta\sin^2\phi + r^2\sin^2\theta\cos^2\phi = r^2\sin^2\theta \implies h_\phi = r\sin\theta$

Now compute the mutual dot products for $i \ne j$:

$\mathbf{e}_r \cdot \mathbf{e}_\theta = r\sin\theta\cos\theta\cos^2\phi + r\sin\theta\cos\theta\sin^2\phi - r\sin\theta\cos\theta = r\sin\theta\cos\theta - r\sin\theta\cos\theta = 0$
$\mathbf{e}_r \cdot \mathbf{e}_\phi = -r\sin^2\theta\cos\phi\sin\phi + r\sin^2\theta\sin\phi\cos\phi + 0 = 0$
$\mathbf{e}_\theta \cdot \mathbf{e}_\phi = -r^2\sin\theta\cos\theta\cos\phi\sin\phi + r^2\sin\theta\cos\theta\sin\phi\cos\phi + 0 = 0$

Since $\mathbf{e}_i \cdot \mathbf{e}_j = 0$ for all $i \ne j$, spherical coordinates form a strictly orthogonal curvilinear coordinate system.

4. Differential Volume Element

For an orthogonal coordinate system, the volume element is the product of the differential arc lengths:

dV = (h_r dr)(h_\theta d\theta)(h_\phi d\phi) = (1 \, dr)(r \, d\theta)(r\sin\theta \, d\phi) = r^2 \sin\theta \, dr \, d\theta \, d\phi

Notice that this matches $|J| \, dr \, d\theta \, d\phi = r^2 \sin\theta \, dr \, d\theta \, d\phi$ identically! $\blacksquare$

Honors / Proof Challenge

Problem 1.3: Problem 1.3: Rigorous Tensor Criteria & The Failure of the Hessian Matrix

Let $\phi(x^1, \dots, x^n)$ be an arbitrary scalar field of class $C^2$ on an $n$-dimensional differentiable manifold $V_n$.

Prove rigorously that the gradient vector components $G_i = \frac{\partial \phi}{\partial x^i}$ transform as a covariant vector under any arbitrary regular coordinate transformation $x^i \to \bar{x}^i$.
Prove that the tangent velocity components $V^i = \frac{dx^i}{dt}$ along a parameterized curve $x^i(t)$ transform as a contravariant vector.
Examine the Hessian matrix of second partial derivatives:

H_{ij} = \frac{\partial^2 \phi}{\partial x^i \partial x^j}

Derive its exact transformation law under $x^i \to \bar{x}^i$.

Prove that $H_{ij}$ fails to transform as a tensor, identify the exact mathematical term responsible for this failure, and state under what restricted class of coordinate transformations $H_{ij}$ behaves as a tensor.

1. Proof that the Gradient $G_i$ is a Covariant Vector

Let $\phi(x)$ be a scalar field. Since it is an invariant scalar:

\bar{\phi}(\bar{x}^1, \dots, \bar{x}^n) = \phi(x^1(\bar{x}), \dots, x^n(\bar{x}))

Differentiating with respect to the new coordinate $\bar{x}^i$ using the chain rule:

\bar{G}_i = \frac{\partial \bar{\phi}}{\partial \bar{x}^i} = \frac{\partial \phi}{\partial x^k} \frac{\partial x^k}{\partial \bar{x}^i} = \frac{\partial x^k}{\partial \bar{x}^i} G_k

This is precisely the defining transformation law of a covariant vector (Definition 1.4). Thus, $G_i = \partial_i \phi$ is a true covariant vector. $\blacksquare$

2. Proof that Tangent Velocity $V^i$ is a Contravariant Vector

Let $x^i(t)$ be a curve parameterized by parameter $t$ (an invariant scalar). The coordinates in the new frame are $\bar{x}^i(t) = \bar{x}^i(x^1(t), \dots, x^n(t))$. Differentiating with respect to $t$ using the chain rule:

\bar{V}^i = \frac{d\bar{x}^i}{dt} = \frac{\partial \bar{x}^i}{\partial x^k} \frac{dx^k}{dt} = \frac{\partial \bar{x}^i}{\partial x^k} V^k

This is precisely the defining transformation law of a contravariant vector (Definition 1.3). Thus, $V^i = \frac{dx^i}{dt}$ is a true contravariant vector. $\blacksquare$

3. Transformation Law of the Hessian Matrix $H_{ij}$

In the original coordinates, $H_{kl} = \frac{\partial^2 \phi}{\partial x^k \partial x^l} = \frac{\partial G_k}{\partial x^l}$. In the new coordinates:

\bar{H}_{ij} = \frac{\partial^2 \bar{\phi}}{\partial \bar{x}^i \partial \bar{x}^j} = \frac{\partial}{\partial \bar{x}^j} \left( \frac{\partial \bar{\phi}}{\partial \bar{x}^i} \right) = \frac{\partial}{\partial \bar{x}^j} \left( \frac{\partial x^k}{\partial \bar{x}^i} \frac{\partial \phi}{\partial x^k} \right)

Applying the product rule of differentiation:

\bar{H}_{ij} = \frac{\partial x^k}{\partial \bar{x}^i} \frac{\partial}{\partial \bar{x}^j}\left(\frac{\partial \phi}{\partial x^k}\right) + \frac{\partial \phi}{\partial x^k} \frac{\partial}{\partial \bar{x}^j}\left(\frac{\partial x^k}{\partial \bar{x}^i}\right)

Using the chain rule on the first term:

\frac{\partial}{\partial \bar{x}^j}\left(\frac{\partial \phi}{\partial x^k}\right) = \frac{\partial x^l}{\partial \bar{x}^j} \frac{\partial}{\partial x^l}\left(\frac{\partial \phi}{\partial x^k}\right) = \frac{\partial x^l}{\partial \bar{x}^j} \frac{\partial^2 \phi}{\partial x^k \partial x^l} = \frac{\partial x^l}{\partial \bar{x}^j} H_{kl}

For the second term:

\frac{\partial}{\partial \bar{x}^j}\left(\frac{\partial x^k}{\partial \bar{x}^i}\right) = \frac{\partial^2 x^k}{\partial \bar{x}^j \partial \bar{x}^i}

Combining both terms yields the exact transformation law:

\bar{H}_{ij} = \frac{\partial x^k}{\partial \bar{x}^i} \frac{\partial x^l}{\partial \bar{x}^j} H_{kl} + \frac{\partial^2 x^k}{\partial \bar{x}^i \partial \bar{x}^j} \frac{\partial \phi}{\partial x^k}

4. Non-Tensor Character and Remediation

For $H_{ij}$ to be a valid rank-$(0, 2)$ covariant tensor, it would have to transform strictly homogeneously as:

\bar{H}_{ij} = \frac{\partial x^k}{\partial \bar{x}^i} \frac{\partial x^l}{\partial \bar{x}^j} H_{kl}

However, the actual transformation law contains an inhomogeneous second-derivative term:

\text{Inhomogeneous Obstruction} = \frac{\partial^2 x^k}{\partial \bar{x}^i \partial \bar{x}^j} \frac{\partial \phi}{\partial x^k}

Due to this extra term, $H_{ij}$ is NOT a tensor under general coordinate transformations!

Special Cases where $H_{ij}$ Behaves as a Tensor:

Affine / Linear Transformations: If the coordinate transformation is purely linear:

x^k = A^k_{\; m} \bar{x}^m + c^k \implies \frac{\partial^2 x^k}{\partial \bar{x}^i \partial \bar{x}^j} = 0

The second partial derivatives vanish identically, so $H_{ij}$ transforms as a tensor under affine transformations (rotations, translations, scaling).

Critical / Stationary Points: At a point where $\nabla \phi = 0$ ($\frac{\partial \phi}{\partial x^k} = 0$), the inhomogeneous term vanishes, meaning the Hessian is a local tensor at critical points!
General Resolution in Curved Spaces: To obtain a true tensor everywhere in curved manifolds, one must replace ordinary partial derivatives with covariant derivatives:

\nabla_j \nabla_i \phi = \frac{\partial^2 \phi}{\partial x^j \partial x^i} - \Gamma_{ji}^k \frac{\partial \phi}{\partial x^k}

The Christoffel symbol $\Gamma_{ji}^k$ cancels the inhomogeneous second-derivative term, yielding a true tensor! $\blacksquare$