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Covariant transformation

In mathematics and physics, a covariant transformation describes the rule by which the components of a covariant tensor, such as a vector or higher-rank tensor, change under a coordinate transformation in a manifold, preserving the tensor's geometric and physical meaning across different bases.^[1] For a rank-1 covariant vector V_i, the components in the new coordinate system V'_{i'} are given by V'_{i'} = \sum_j \frac{\partial x^j}{\partial x^{i'}} V_j, where x^j are the old coordinates and x^{i'} the new ones.^[1] This law extends to higher-rank covariant tensors by including a factor of \frac{\partial x^k}{\partial x^{l'}} for each lower index, ensuring the tensor's multilinearity and coordinate independence.^[2] Covariant transformations are distinguished from contravariant ones, which use the inverse partial derivatives \frac{\partial x^{i'}}{\partial x^j} for upper indices, reflecting the dual nature of basis vectors and covectors in vector spaces.^[1] These properties satisfy the transitive nature of tensor transformations: applying successive coordinate changes yields the same result as a direct transformation between the initial and final systems.^[2] In non-orthogonal bases, covariant components are defined with respect to the reciprocal basis, where the dot product between basis and reciprocal basis vectors yields the Kronecker delta, facilitating consistent projections.^[1] In theoretical physics, particularly general relativity, covariant transformations underpin the principle of general covariance, ensuring that the fundamental equations—such as Einstein's field equations—retain their form under arbitrary diffeomorphisms of spacetime coordinates.^[3] This invariance allows physical laws to be expressed in a tensorial manner, independent of the observer's coordinate choice, and is essential for describing phenomena like gravity as curvature without privileging any frame.^[3] Applications extend to electromagnetism, where the electromagnetic field tensor transforms covariantly to maintain Maxwell's equations across inertial and non-inertial frames.^[4]

Fundamental Concepts

Covariant Transformation

In tensor analysis, a quantity is said to transform covariantly if, under a change of basis or coordinate system, its components in the new system are obtained by multiplying the original components by the inverse of the transformation matrix.^[5] For a rank-1 covariant tensor T_i, this transformation law is expressed as

T'_i = \frac{\partial x^j}{\partial x'^i} T_j,

where the Einstein summation convention is used over the repeated index j, and x^j and x'^i denote the old and new coordinates, respectively.^[1] This rule ensures that the tensor's geometric interpretation remains consistent across different frames. Covariant objects, such as covectors or the components of a gradient, adhere to this scaling with the inverse Jacobian matrix \left( \frac{\partial x^j}{\partial x'^i} \right), which preserves the tensorial nature of the quantity.^[1] The transformation arises from the requirement that contractions, like the scalar product between a covariant tensor and a contravariant vector, yield invariants under basis changes.^[5] This property distinguishes covariant transformation from contravariant transformation, which applies to vectors and uses the direct Jacobian. The covariant transformation law serves as the foundational rule for defining lower-index tensors in multilinear algebra, enabling the formulation of physical laws that are independent of coordinate choices.^[1] For higher-rank covariant tensors, the law generalizes by applying the inverse Jacobian factor to each covariant index separately.^[5]

Contravariant Transformation

A contravariant transformation refers to the rule by which the components of a tensor with upper indices, such as a vector, change under a coordinate or basis transformation. Specifically, a quantity is said to transform contravariantly if its components in the new system are obtained by multiplying the original components by the Jacobian matrix of the transformation, which describes the partial derivatives of the new coordinates with respect to the old ones. This ensures that the tensor's representation adapts correctly to preserve the underlying geometric or physical object across different frames.^[1] For a rank-1 contravariant tensor \mathbf{V}, the transformation law is given by

V'^i = \frac{\partial x'^i}{\partial x^j} V^j,

where the Einstein summation convention is used over the repeated index j, and x'^i denote the new coordinates. This law applies generally to higher-rank contravariant tensors, where each upper index follows the same multiplication by the corresponding partial derivative. In contrast to covariant transformations, which involve the inverse Jacobian for lower-index components, the contravariant rule aligns with how displacement vectors or tangent space elements shift under passive coordinate changes.^[1] The contravariant transformation plays a crucial role in maintaining the invariance of scalar quantities formed by contractions, such as the dot product between a contravariant vector and a covariant vector via the metric tensor. Under a coordinate change, the opposing transformation behaviors of contravariant and covariant components cancel out in the contraction, yielding a scalar that remains unchanged, thus ensuring the physical or geometric meaning is coordinate-independent.^[6] The origins of contravariant transformations trace back to the foundational work in tensor calculus by Gregorio Ricci-Curbastro and Tullio Levi-Civita, particularly in their 1900 paper on absolute differential calculus, where they systematized these rules to handle coordinate-independent differential operations on manifolds. This framework emphasized passive transformations—changes in the coordinate description rather than active deformations of the space itself—to develop invariant formulations of geometry and physics.^[7]^[8]

Transformation Examples

Derivatives and Functions

Consider a scalar function f defined on a space with coordinates x^j. The partial derivatives \partial_i f = \frac{\partial f}{\partial x^i} form the components of the gradient, which transform as a covector under a change of coordinates. This behavior arises because the differential df is a linear map from tangent vectors to scalars, preserving the value of the function's change along any direction.^[9] To derive the transformation law, introduce new coordinates x'^i. Since f is a scalar, its value remains unchanged, so f(x') = f(x(x')). Applying the chain rule for multivariable differentiation yields

\frac{\partial f}{\partial x'^i} = \frac{\partial f}{\partial x^j} \frac{\partial x^j}{\partial x'^i}.

Here, the indices follow the Einstein summation convention, with j summed over all dimensions. This equation demonstrates the covariant nature of the partial derivatives, as the new components are obtained by contracting the old components with the inverse Jacobian matrix \frac{\partial x^j}{\partial x'^i}.^[10] Geometrically, the gradient indicates the direction of steepest ascent of f, where the directional derivative df(\mathbf{v}) = \partial_i f \, v^i gives the rate of change along a tangent vector \mathbf{v}. Under coordinate transformations, the basis vectors may stretch or contract, requiring the gradient components to scale inversely to maintain this invariant rate. For instance, if the new basis is elongated, the gradient components diminish to preserve the physical slope.^[9] In one dimension, a linear coordinate shift x' = x + b (with constant b) illustrates this numerically. The transformation Jacobian is \frac{\partial x}{\partial x'} = 1. Take f(x) = x^2, so \frac{df}{dx} = 2x. At x = 1, \frac{df}{dx} = 2. Expressing in new coordinates, f(x') = (x' - b)^2, so \frac{df}{dx'} = 2(x' - b). At the corresponding point x' = 1 + b, \frac{df}{dx'} = 2((1 + b) - b) = 2, consistent with the law since the Jacobian factor is unity.^[10]

Basis Vectors and Covectors

In tensor analysis, the basis vectors of a vector space, often denoted as \mathbf{e}_i, transform under a change of coordinates from x^j to x'^i according to the rule \mathbf{e}'_i = \frac{\partial x^j}{\partial x'^i} \mathbf{e}_j.^[11]^[12] This transformation reflects the covariant nature of the basis vectors themselves, as they adjust inversely to the coordinate differentials via the inverse Jacobian matrix.^[11] The partial derivatives \frac{\partial x^j}{\partial x'^i} ensure that the new basis vectors \mathbf{e}'_i are linear combinations of the old ones, preserving the linear structure of the tangent space.^[12] The dual basis, consisting of covectors or one-forms \mathbf{e}^i, exhibits the opposite transformation behavior to maintain consistency with the vector basis. Under the same coordinate change, these covectors transform contravariantly as \mathbf{e}'^i = \frac{\partial x'^i}{\partial x^j} \mathbf{e}^j.^[11]^[12] This uses the direct Jacobian matrix \frac{\partial x'^i}{\partial x^j}, allowing the dual basis to scale in tandem with the coordinate expansion or contraction.^[11] The paired transformations of basis vectors and covectors thus highlight their complementary roles in representing multilinear maps within the space.^[12] A defining property of this duality is the invariance of the pairing between covectors and basis vectors, given by \mathbf{e}^i (\mathbf{e}_j) = \delta^i_j, where \delta^i_j is the Kronecker delta.^[10]^[12] This condition, which equals 1 if i = j and 0 otherwise, remains unchanged under coordinate transformations because the covariant adjustment of \mathbf{e}_i precisely counters the contravariant shift in \mathbf{e}^i.^[10] The preservation ensures that the dual pairing acts as a coordinate-independent selector, fundamental to tensor definitions.^[12] To illustrate in two dimensions, consider an affine transformation that stretches the coordinate axes; the basis vectors \mathbf{e}_1 and \mathbf{e}_2 (represented as arrows) shorten to compensate, while the corresponding covectors \mathbf{e}^1 and \mathbf{e}^2 lengthen inversely, maintaining the invariant area-preserving duality.^[10] This inverse scaling visually underscores how the paired transformations keep the inner product structure intact across frames.^[12]

Differential Forms

Differential forms provide a natural framework for understanding covariant and contravariant transformations on manifolds, particularly through the behavior of basis 1-forms and their role in integration. The basis 1-forms dx^i, which span the cotangent space at each point in a coordinate chart, transform contravariantly under a change of coordinates from x^j to x'^i. This means dx'^i = \frac{\partial x'^i}{\partial x^j} dx^j, ensuring that the dual relationship with the contravariant basis vectors \frac{\partial}{\partial x^i} is preserved across coordinate systems. A concrete example arises in curvilinear coordinates on Euclidean space, such as the transition from Cartesian coordinates (x, y) to polar coordinates (r, \theta), where x = r \cos \theta and y = r \sin \theta. Here, the basis 1-forms transform as dr = \cos \theta \, dx + \sin \theta \, dy and d\theta = -\frac{\sin \theta}{r} dx + \frac{\cos \theta}{r} dy, reflecting the partial derivatives of the new coordinates with respect to the old ones and adjusting the infinitesimal line elements under reparametrization to maintain geometric consistency. In contrast to this contravariant behavior of the basis, differential forms as a whole exhibit covariant transformation properties when considered in the context of integration, achieved through the pullback operation. For a diffeomorphism \phi: M [\to](/page/TO) N and a k-form \omega on N, the pullback \phi^* \omega on M ensures that the integral \int_M \phi^* \omega = \int_N \omega (up to orientation) is coordinate-independent, as the transformation compensates for changes in the basis forms. This covariance is particularly evident in the volume form on an n-dimensional manifold, where the standard volume element dV = dx^1 \wedge \cdots \wedge dx^n transforms under a coordinate change such that dV = \left| \det \frac{\partial x}{\partial x'} \right| dV' , where dV' = dx'^1 \wedge \cdots \wedge dx'^n , incorporating the absolute value of the Jacobian determinant to preserve the positive measure for integration regardless of the coordinate system.^[13]

Tensor Components

Without Coordinates

In abstract vector spaces, tensors are defined as multilinear maps that generalize the notions of scalars, vectors, and linear functionals without reference to any specific basis or coordinate system. A tensor of type (p, q) over a vector space V is a multilinear function T: (V^)^p \times V^q \to \mathbb{R}, where V^ denotes the dual space of V consisting of all linear functionals on V, p represents the number of contravariant slots (which accept elements from V^*), and q represents the number of covariant slots (which accept elements from V).^[14]^[15] This definition captures the intrinsic, coordinate-free nature of tensors as higher-order generalizations of vectors (p=1, q=0) and covectors (p=0, q=1), emphasizing their multilinearity—linearity in each argument while holding the others fixed.^[16] The space of all such (p, q)-tensors, denoted T^{p,q}(V), forms a vector space itself and can be constructed algebraically as the tensor product T^{p,q}(V) = V^{\otimes p} \otimes (V^)^{\otimes q}, where \otimes denotes the tensor product of vector spaces.^[15] In this component-free perspective, the transformation properties of tensors are implicit in the structure of the tensor product: changing bases in V induces corresponding changes in V^ via the dual map, ensuring that the overall multilinear action remains consistent across different choices of bases.^[14] Elements of T^{p,q}(V) are finite sums of pure tensors of the form v_1 \otimes \cdots \otimes v_p \otimes \alpha_1 \otimes \cdots \otimes \alpha_q, where v_i \in V and \alpha_j \in V^*, and the multilinearity follows from the universal property of the tensor product, which guarantees a unique linear extension of the map (v_1, \dots, v_p, \alpha_1, \dots, \alpha_q) \mapsto T(v_1, \dots, v_p, \alpha_1, \dots, \alpha_q).^[16] This algebraic construction underscores the basis-independent essence of tensors, as the tensor product is defined intrinsically without coordinates.^[15] A representative example is the metric tensor, which serves as a (0,2) covariant tensor in T^{0,2}(V) = (V^*)^{\otimes 2}. In the coordinate-free view, the metric g is a symmetric bilinear form g: V \times V \to \mathbb{R} that pairs two vectors to produce an inner product, such as g(u, v) for u, v \in V, without invoking components or bases.^[14] This pairing induces a norm and angle structure on V, and its bilinearity ensures it is linear in each vector argument separately.^[15] The invariance of tensor operations, particularly scalar contractions, highlights their basis-independent character. A full contraction of a tensor in T^{p,q}(V) with p = q pairs all contravariant and covariant slots to yield a scalar in \mathbb{R}, such as tracing over matching indices in a (1,1)-tensor to obtain a number; this result is unchanged under any change of basis because the contraction is defined via the natural duality pairing between V and V^*, which is intrinsic to the vector space structure.^[16] Partial contractions similarly reduce the tensor type while preserving multilinearity and independence from coordinates.^[14]

With Coordinates

In a chosen coordinate system, the components of a tensor are expressed using indices that distinguish between contravariant (upper) and covariant (lower) positions, enabling explicit calculations of how the tensor behaves under coordinate transformations.^[17] For instance, a contravariant vector V^i has components that scale with the direct partial derivatives of the new coordinates with respect to the old ones, while a covariant vector V_i scales inversely.^[17] The transformation law for a general tensor follows a pattern where each contravariant index acquires a factor of the Jacobian matrix \frac{\partial x'^i}{\partial x^j}, and each covariant index acquires a factor of its inverse \frac{\partial x^k}{\partial x'^l}.^[17] For a (1,1) tensor T^i_j, which has one contravariant and one covariant index, the components in the new coordinate system x' transform as

T'^i_j = \frac{\partial x'^i}{\partial x^k} \frac{\partial x^l}{\partial x'^j} T^k_l,

where summation over repeated indices k and l is implied.^[17] This ensures the tensor's multilinearity is preserved across coordinate changes, with the direct Jacobian acting on the upper index and the inverse on the lower.^[17] A practical example arises in physics with the stress tensor \sigma^i_j, a (1,1) tensor representing force per unit area in mixed components. Under a coordinate rotation defined by an orthogonal rotation matrix R, the transformed stress tensor becomes \sigma' = R \sigma R^T, or in index notation, \sigma'^i_j = R^i_k R^l_j \sigma^k_l, since rotations preserve the metric and the inverse Jacobian equals the transpose for orthogonal transformations.^[18] For a 2D case with initial components \sigma = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix} (in MPa) and a 50° counterclockwise rotation, the new components are approximately \sigma' = \begin{pmatrix} 2.36 & -0.68 \\ -0.68 & 1.64 \end{pmatrix}, illustrating how shear and normal stresses redistribute.^[18] To interconvert between contravariant and covariant components within the same coordinate system, the metric tensor g_{ij} is employed, which defines the inner product and allows index raising or lowering. Specifically, a contravariant vector V^j is lowered to its covariant form via V_i = g_{ij} V^j, with the inverse metric g^{ij} used for raising: V^i = g^{ij} V_j. In Euclidean space with the identity metric g_{ij} = \delta_{ij}, the components coincide, but in curved spaces like general relativity, this operation accounts for the geometry.

Dual Properties

Duality in Vector Spaces

In finite-dimensional vector spaces over a field F, the dual space V^* of a vector space V is defined as the set of all linear functionals from V to F, equipped with the structure of a vector space under pointwise addition and scalar multiplication.^[19] This construction establishes a natural duality, where elements of V^*, often called covectors, act on vectors in V to produce scalars.^[20] The duality is manifested through the natural pairing \langle \omega, v \rangle = \omega(v) for \omega \in V^* and v \in V, which is bilinear and invariant under linear transformations.^[19] Given a basis \{e_j\} for V, the dual basis \{e^i\} for V^* satisfies \langle e^i, e_j \rangle = \delta^i_j, where \delta^i_j is the Kronecker delta (equal to 1 if i = j and 0 otherwise), ensuring the bases are uniquely determined and the dimension of V^* equals that of V.^[19] Under a change of basis, vectors in V transform contravariantly, with components v'^i = \frac{\partial x'^i}{\partial x^j} v^j, while covectors in V^* transform covariantly, with components \omega'_i = \frac{\partial x^j}{\partial x'^i} \omega_j, preserving the invariance of the pairing \langle \omega', v' \rangle = \langle \omega, v \rangle.^[10] This complementary transformation behavior underscores the duality between contravariant and covariant objects. A fundamental result is the natural isomorphism between V and its double dual V^{**} = (V^*)^*, given by the evaluation map \mathrm{ev}: V \to V^{**} defined by \mathrm{ev}(v)(\omega) = \langle \omega, v \rangle for v \in V and \omega \in V^*. This map is linear and bijective for finite-dimensional spaces, as injectivity follows from the fact that if \langle \omega, v \rangle = 0 for all \omega \in V^* then v = 0, and dimensions match to ensure surjectivity.^[19]

Covariant-Contra Variant Pairs

In tensor analysis, practical examples of covariant-contravariant pairs illustrate how these dual objects interact in physical contexts. The position vector, represented by components \mathbf{r}^i, transforms as a contravariant vector under coordinate changes, scaling with the basis vectors to maintain its directional integrity.^[21] In contrast, the gradient of a scalar field, with components \nabla_i \phi, behaves as a covariant vector, transforming inversely to ensure the directional derivative remains consistent across bases.^[22] Similarly, in classical mechanics with generalized coordinates, the momentum conjugate to position, often denoted p_i, carries covariant indices, pairing naturally with the contravariant displacement dq^i to form work as a scalar.^[23] The force, interpreted as the rate of change of this momentum, can thus be viewed in its covariant form, emphasizing its role as a covector in dual pairings.^[24] A key application of these pairs is in constructing invariants through tensor contraction, where the summation over paired indices yields quantities unchanged under coordinate transformations. For a contravariant vector v^i and a covariant vector \omega_i, the contraction v^i \omega_i produces a scalar invariant, as the opposing transformation rules cancel out, preserving the inner product.^[23] This operation leverages the duality between vector spaces, ensuring that physical laws expressed in such forms remain coordinate-independent.^[22] For higher-rank tensors of type (p, q), where p contravariant and q covariant indices are present, contractions can pair specific indices to reduce the rank and form invariants. By contracting one contravariant index with one covariant index, the resulting tensor maintains the overall transformation properties while yielding a lower-rank object that is scalar under the paired transformation; repeated contractions can fully reduce to scalars for complete invariance.^[24] This process is essential in formulating invariant expressions in multidimensional systems, such as stress-energy tensors where multiple pairs contribute to conserved quantities.^[25] In special relativity, a prominent example involves the 4-velocity u^\mu, a contravariant 4-vector tangent to the worldline, paired with the covariant 4-gradient \partial_\mu. Their contraction u^\mu \partial_\mu yields the proper time derivative along the trajectory, a Lorentz scalar invariant under boosts and rotations, crucial for describing relativistic kinematics.^[26] This pairing underscores how covariant-contravariant structures preserve fundamental physical scalars in spacetime transformations.^[25]

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