Tensor Calculus: Quick Reference Cheat Sheet

This cheat sheet provides a quick reference for key concepts, notations, and formulas in Tensor Calculus, primarily drawing from the “Crash Course” posts (Parts 1, 2, and 3) on this topic. For detailed explanations and derivations, please refer to the individual course posts.

1. Fundamental Concepts & Notation

Concept	Notation (Components)	Type (Rank)	Description
Scalar	\(s\)	(0,0)	An invariant quantity, a single number from \(\mathbb{R}\).
Vector (Contravariant)	\(V^i\)	(1,0)	Element of vector space \(V\). Components transform contravariantly.
Covector (Covariant)	\(W_j\)	(0,1)	Element of dual space \(V^\ast\) (linear functional). Components transform covariantly.
Tensor (General)	\(T^{i_1 \dots i_p}_{j_1 \dots j_q}\)	(p,q)	Multilinear map: \((V^\ast)^q \times V^p \to \mathbb{R}\).
Kronecker Delta	\(\delta^i_j\)	(1,1)	Identity tensor; 1 if \(i=j\), 0 if \(i \neq j\). Acts as substitution operator.

2. Einstein Summation Convention

Aspect	Description	Example
Rule	Summation implied over any index appearing once up and once down in a single term.	\(\omega_i v^i = \sum_i \omega_i v^i\)
Dummy Index	The index being summed over. Can be relabeled (e.g., \(A^k_k = A^l_l\)).	\(j\) in \(A^i_j B^j_k\)
Free Index	An index appearing only once. Must match on both sides of an equation.	\(i, k\) in \(C^i_k = A^i_j B^j_k\)

3. Basic Tensor Algebra

Operation	Formula (Components)	Resulting Type	Notes
Addition	\((C)^{i_1 \dots}_{j_1 \dots} = (A)^{i_1 \dots}_{j_1 \dots} + (B)^{i_1 \dots}_{j_1 \dots}\)	Same as A, B	Tensors must be of the same type.
Scalar Mult.	\((\alpha T)^{i_1 \dots}_{j_1 \dots} = \alpha (T^{i_1 \dots}_{j_1 \dots})\)	Same as T	\(\alpha\) is a scalar.
Outer Product	\((A \otimes B)^{i_1 \dots k_1 \dots}_{j_1 \dots l_1 \dots} = A^{i_1 \dots}_{j_1 \dots} B^{k_1 \dots}_{l_1 \dots}\)	\((p_A+p_B, q_A+q_B)\)	Product of components, ranks add.
Contraction	E.g., \(S^i = T^{ik}_k\) (contracts 2nd upper with 1st lower)	\((p-1, q-1)\) per pair contracted	Sum over one upper and one lower index. Trace of \(A^i_j\) is \(A^k_k\).
Inner Product	Often outer product + contraction(s). E.g., \(A^i_j B^j_k\) (matrix mult.)	Varies	E.g., \(g_{ij}U^iV^j\) (vector dot product).

4. Coordinate Transformations & Jacobians

Item	Notation	Definition / Relation
Old Coordinates	\(x^k\) (or \(x^1, \dots, x^n\))	Original coordinate system.
New Coordinates	\(x'^{i'}\) (or \(x'^{1}, \dots, x'^n\))	New coordinate system, \(x'^{i'} = x'^{i'}(x^1, \dots, x^n)\).
Forward Jacobian Matrix	\(J^{i'}_k\)	\(\frac{\partial x'^{i'}}{\partial x^k}\)
Inverse Jacobian Matrix	\((J^{-1})^l_{j'}\)	\(\frac{\partial x^l}{\partial x'^{j'}}\)
Jacobian Relationship	-	\(\frac{\partial x'^{i'}}{\partial x^k} \frac{\partial x^k}{\partial x'^{j'}} = \delta^{i'}_{j'}\) (Chain rule)

5. Transformation Laws for Components

Quantity Type & Components	Transformation Rule (\(\text{New components} = \dots \text{Old components}\))	Transformation Nature
Scalar \(\phi\)	\(\phi' = \phi\)	Invariant
Vector Components \(V^k\)	\(V'^{i'} = \frac{\partial x'^{i'}}{\partial x^k} V^k\)	Contravariant
Covector Components \(W_k\)	\(W'_{i'} = \frac{\partial x^k}{\partial x'^{i'}} W_k\)	Covariant
General Tensor \(T^{k_1 \dots k_p}_{l_1 \dots l_q}\)	\(T'^{i'_1 \dots i'_p}_{j'_1 \dots j'_q} = \left( \prod_{a=1}^{p} \frac{\partial x'^{i'_a}}{\partial x^{k_a}} \right) \left( \prod_{b=1}^{q} \frac{\partial x^{l_b}}{\partial x'^{j'_b}} \right) T^{k_1 \dots k_p}_{l_1 \dots l_q}\)	Mixed (p contra, q co)

6. The Metric Tensor

Feature	Description / Formula
Definition (\(g_{ij}\))	Symmetric (0,2)-tensor defining geometry (distances, angles). Components depend on coordinate system.
Infinitesimal distance (\(ds^2\))	\(ds^2 = g_{ij} dx^i dx^j\) (Invariant scalar)
Inner Product (\(U \cdot V\))	\(U \cdot V = g_{ij} U^i V^j\) (Invariant scalar)
Transformation of \(g_{ij}\)	\(g'_{i'j'} = \frac{\partial x^k}{\partial x'^{i'}} \frac{\partial x^l}{\partial x'^{j'}} g_{kl}\) (Transforms as a (0,2)-tensor)
Inverse Metric (\(g^{ij}\))	(2,0)-tensor. \(g^{ik} g_{kj} = \delta^i_j\). Transforms as a (2,0)-tensor.
Lowering Index	\(V_i = g_{ij} V^j\) (Maps vector \(V^j\) to associated covector \(V_i\))
Raising Index	\(V^i = g^{ij} V_j\) (Maps covector \(V_j\) to associated vector \(V^i\))
Cartesian Metric (Euclidean)	\(g_{ij} = \delta_{ij}\) (Components: 1 on diagonal, 0 off-diagonal if orthonormal basis)
Polar Metric (2D Euclidean)	\(g_{rr}=1, g_{\theta\theta}=r^2, g_{r\theta}=0\). (Matrix for \((r,\theta)\) coords: diag(1, \(r^2\)))

7. Tensor Differentiation

Concept	Formula / Definition	Notes
Partial Derivative Issue	\(\partial_j V^i\) is NOT generally a tensor if coord. transform is non-linear or basis vectors vary.	Fails to account for changing basis vectors. Exception: \(\partial_j \phi\) for scalar \(\phi\).
Christoffel Symbols (\(\Gamma^k_{ij}\))	\(\Gamma^k_{ij} = \frac{1}{2} g^{kl} (\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij})\)	NOT tensor components. Describe change in basis vectors. Symmetric: \(\Gamma^k_{ij} = \Gamma^k_{ji}\) for Levi-Civita connection. Zero in Cartesian coords for flat space.
Covariant Derivative of Scalar	\(\nabla_j \phi = \partial_j \phi\)	Result: (0,1)-tensor.
Covariant Deriv. of Vector (\(V^i\))	\(\nabla_j V^i \equiv V^i_{;j} = \partial_j V^i + \Gamma^i_{jk} V^k\)	Result: (1,1)-tensor. “+\(\Gamma\)” for each upper index.
Covariant Deriv. of Covector (\(W_i\))	\(\nabla_j W_i \equiv W_{i;j} = \partial_j W_i - \Gamma^k_{ji} W_k\)	Result: (0,2)-tensor. “-\(\Gamma\)” for each lower index.
Covariant Deriv. of \(g_{ij}\)	\(\nabla_k g_{ij} = 0\) and \(\nabla_k g^{ij} = 0\)	Metric compatibility: metric tensor is “constant” w.r.t. \(\nabla\). Raising/lowering commutes w/ \(\nabla\).
Leibniz Rule	Holds, e.g., \(\nabla_k(A^i B_j) = (\nabla_k A^i)B_j + A^i(\nabla_k B_j)\)	Behaves like ordinary derivative for products.
Gradient of \(L\) (Covariant comp.)	\((\text{grad } L)_i = \nabla_i L = \partial_i L\)	Components of grad \(L\) are a (0,1)-tensor.
Gradient of \(L\) (Contrav. comp.)	\((\text{grad } L)^j = g^{ji} \partial_i L\)	“Direction” of ascent/descent, obtained by raising index.
Hessian of \(L\) (Covariant comp.)	\(H_{ij} = \nabla_i (\nabla_j L) = \nabla_i (\partial_j L) = \partial_i \partial_j L - \Gamma^k_{ij} (\partial_k L)\)	Symmetric (0,2)-tensor. For flat space/Cartesian coords, \(H_{ij} = \partial_i \partial_j L\).
Riemann Curvature Tensor (\(R^k_{lji}\))	Measures non-commutativity: \((\nabla_i \nabla_j - \nabla_j \nabla_i) V^k = R^k_{lji} V^l\)	Captures intrinsic curvature of the space/manifold. Zero for flat spaces.