Index Notation and Embedded Surfaces
Index notation and differential geometry provide compact ways to describe vectors, tensors, and curved surfaces. This short note collects a few general definitions that are useful across applied mathematics.
Free and dummy indices
Under the Einstein summation convention, an index that appears twice in one term is summed. For example,
\[ a_i b_i=\sum_i a_i b_i. \]
Such a repeated index is called a dummy index. Its name can be changed without changing the expression:
\[ a_i b_i=a_j b_j. \]
An index appearing once is a free index. It identifies a component of the resulting object. Every term in a valid tensor equation must have the same free indices. For instance,
\[ c_i=A_{ij}b_j \]
defines the components of a vector, while an equation that leaves different free indices on its two sides is not well formed.
Vectors, covectors, and the metric
A vector has components \(v^i\), whereas a covector has components \(\alpha_i\). A covector acts linearly on a vector:
\[ \alpha(v)=\alpha_i v^i. \]
A metric \(g_{ij}\) converts vectors to covectors and supplies lengths and angles:
\[ v_i=g_{ij}v^j, \qquad \lVert v\rVert^2=g_{ij}v^i v^j. \]
The inverse metric \(g^{ij}\) raises indices:
\[ v^i=g^{ij}v_j. \]
In Cartesian coordinates with the Euclidean metric, raising or lowering an index does not change the numerical components. In general coordinates, that shortcut is no longer valid.
A parametrized surface
Let a smooth surface in three-dimensional Euclidean space be represented by
\[ X(u^1,u^2). \]
The coordinate tangent vectors are
\[ e_a=\frac{\partial X}{\partial u^a}, \qquad a=1,2. \]
Their inner products form the induced metric:
\[ g_{ab}=e_a\cdot e_b. \]
This metric records the geometry experienced by vectors tangent to the surface. The corresponding area element is
\[ dA=\sqrt{\det(g_{ab})}\,du^1du^2. \]
When the two tangent vectors are linearly independent, a choice of orientation gives the unit normal
\[ n=\frac{e_1\times e_2}{\lVert e_1\times e_2\rVert}. \]
Reversing the parameter order reverses the normal, so orientation-dependent signs must always be stated explicitly.
Tangential projection
For a unit normal \(n\), the orthogonal projector onto the tangent plane is
\[ P=I-n\otimes n. \]
It satisfies
\[ P^2=P, \qquad Pn=0. \]
If a scalar field is defined near the surface, its ambient gradient can be projected onto the tangent plane:
\[ \nabla_{\!S}f=P\nabla f. \]
This separates variation along the surface from variation in the normal direction.
Derivatives in changing coordinates
Ordinary partial derivatives differentiate tensor components, but a curvilinear coordinate basis also changes from point to point. The covariant derivative accounts for both effects. For a vector,
\[ \nabla_j v^i =\partial_j v^i+\Gamma^i{}_{jk}v^k, \]
where \(\Gamma^i{}_{jk}\) are the connection coefficients.
For the metric-compatible, torsion-free connection,
\[ \Gamma^i{}_{jk} =\frac12 g^{i\ell} \left( \partial_j g_{\ell k} +\partial_k g_{\ell j} -\partial_\ell g_{jk} \right). \]
These additional terms are not forces or new physics. They express how the chosen basis varies across the coordinate domain.
A compact checklist
- Repeated indices are summed; free indices must match across an equation.
- Raising and lowering indices requires a metric.
- A parametrization supplies tangent vectors, while their inner products supply the induced metric.
- The determinant of the induced metric determines the surface area measure.
- A unit normal fixes an orientation, so sign conventions should be declared.
- Tangential derivatives can be obtained by projection.
- Covariant derivatives are needed when the coordinate basis varies.
These rules provide a clean starting point for more advanced work with curved coordinates and embedded manifolds.