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Multilinear form

In mathematics, particularly in the field of multilinear algebra, a multilinear form (or k-linear form) on a vector space V over a field F (such as the real or complex numbers) is a map T: V^k \to F that is linear in each of its k arguments when the others are held fixed.^[1]^[2] This generalizes the concept of a linear functional, which corresponds to the case k=1, where T belongs to the dual space V^*.^[3] The space of all k-linear forms on V, denoted L^k(V; F) or (V^*)^{\otimes k}, forms a vector space of dimension n^k, where n = \dim V, with a natural basis constructed from tensor products of dual basis elements.^[1]^[4] Multilinear forms play a foundational role in several areas of mathematics, serving as building blocks for tensor algebra and multilinear maps between vector spaces.^[2] They can be classified into subtypes based on symmetry properties under permutation of arguments: symmetric multilinear forms, which are invariant under any permutation of the arguments, and alternating (or skew-symmetric) forms, which change sign under odd permutations and vanish if any two arguments are identical.^[5]^[1] The latter are central to the exterior algebra, where the wedge product \wedge equips the space \Lambda^k V^* with dimension \binom{n}{k}.^[3] A prominent example is the determinant function on an n-dimensional space, which is the unique (up to scalar multiple) alternating n-linear form normalized such that \det(e_1, \dots, e_n) = 1 for a basis \{e_i\}.^[4]^[5] Beyond pure linear algebra, multilinear forms extend to applications in differential geometry and physics, where differential k-forms on manifolds are smooth sections of the exterior bundle, representing skew-symmetric multilinear functionals on tangent spaces at each point.^[3] These enable the formulation of integration theories, such as Stokes' theorem, which relates the integral of a form over a domain to that of its exterior derivative over the boundary.^[3] In tensor analysis, multilinear forms underpin the description of higher-rank tensors, facilitating computations in general relativity and continuum mechanics.^[2]

Definition and Properties

Formal Definition

A multilinear form of order k, where k is a positive integer, on vector spaces V_1, \dots, V_k over a field F is a map \phi: V_1 \times \cdots \times V_k \to F that is linear in each argument V_i while holding the others fixed.^[6] This linearity condition requires that, for each i = 1, \dots, k, scalars \lambda, \mu \in F, vectors v_1 \in V_1, \dots, v_k \in V_k, and u, w \in V_i,

\phi(v_1, \dots, v_{i-1}, \lambda u + \mu w, v_{i+1}, \dots, v_k) = \lambda \phi(v_1, \dots, v_{i-1}, u, v_{i+1}, \dots, v_k) + \mu \phi(v_1, \dots, v_{i-1}, w, v_{i+1}, \dots, v_k).

^[7] The domain is the Cartesian product of the vector spaces, which may be the same space V repeated k times, yielding \phi: V^k \to F.^[8] This construction generalizes the notion of a linear functional, which corresponds to the case k=1 where \phi: V \to F is simply linear.^[6]

Linearity and Continuity

A multilinear form \phi: V_1 \times \cdots \times V_k \to \mathbb{F}, where each V_i is a vector space over the field \mathbb{F} and \phi is linear in each argument separately, exhibits additivity and homogeneity independently in each slot. Specifically, for fixed vectors v_2, \dots, v_k \in V_2, \dots, V_k, the map v_1 \mapsto \phi(v_1, v_2, \dots, v_k) is additive, meaning \phi(v_1 + w_1, v_2, \dots, v_k) = \phi(v_1, v_2, \dots, v_k) + \phi(w_1, v_2, \dots, v_k) for all v_1, w_1 \in V_1, and homogeneous, so \phi(\lambda v_1, v_2, \dots, v_k) = \lambda \phi(v_1, v_2, \dots, v_k) for all \lambda \in \mathbb{F}. This property extends analogously to each subsequent argument when the others are held fixed.^[9] A key consequence of multilinearity is that \phi is uniquely determined by its values on basis tuples. If \{e^{(i)}_j\}_{j=1}^{n_i} is a basis for each V_i with \dim V_i = n_i < \infty, then for arbitrary v_i = \sum_{j_i=1}^{n_i} a^{(i)}_{j_i} e^{(i)}_{j_i} in V_i, multilinearity yields

\phi(v_1, \dots, v_k) = \sum_{j_1=1}^{n_1} \cdots \sum_{j_k=1}^{n_k} \left( \prod_{i=1}^k a^{(i)}_{j_i} \right) \phi(e^{(1)}_{j_1}, \dots, e^{(k)}_{j_k}),

so \phi is fully specified by the finite set of values \{\phi(e^{(1)}_{j_1}, \dots, e^{(k)}_{j_k})\}, which can be chosen arbitrarily to define any multilinear form on these spaces.^[9] In the context of normed spaces over \mathbb{R} or \mathbb{C}, continuity of multilinear forms depends on dimensionality. When all V_i are finite-dimensional, every multilinear form is continuous with respect to the product topology on V_1 \times \cdots \times V_k, as it follows from the continuity of linear maps on finite-dimensional spaces and the finite composition of such maps.^[10] In infinite-dimensional normed spaces, however, multilinearity alone does not imply continuity; separate verification is required, though continuous multilinear forms are uniformly continuous on compact subsets of the domain by the Heine-Cantor theorem applied to the product space.^[11] Continuity is equivalent to boundedness for multilinear forms on normed spaces. A multilinear form \phi is bounded if there exists K \geq 0 such that |\phi(v_1, \dots, v_k)| \leq K \|v_1\| \cdots \|v_k\| for all v_i \in V_i, and this holds if and only if \phi is continuous at the origin (and thus everywhere). The induced norm is given by

\|\phi\| = \sup \{ |\phi(v_1, \dots, v_k)| : \|v_i\| \leq 1 \ \forall i = 1, \dots, k \},

which is finite precisely when \phi is continuous.^[10]^[11]

Algebraic Construction

Tensor Product Spaces

The tensor product of vector spaces provides the algebraic foundation for multilinear maps by serving as the universal object that linearizes multilinear constructions. For two vector spaces V_1 and V_2 over a field K, their tensor product V_1 \otimes V_2 is a vector space equipped with a bilinear map \beta: V_1 \times V_2 \to V_1 \otimes V_2, (v_1, v_2) \mapsto v_1 \otimes v_2, satisfying the universal property: for any vector space W and any bilinear map \phi: V_1 \times V_2 \to W, there exists a unique linear map \overline{\phi}: V_1 \otimes V_2 \to W such that \phi = \overline{\phi} \circ \beta. This property ensures that V_1 \otimes V_2 is unique up to isomorphism. The construction extends naturally to the multilinear case: for vector spaces V_1, \dots, V_k over K, the tensor product V_1 \otimes \cdots \otimes V_k is a vector space with a k-linear map \beta: V_1 \times \cdots \times V_k \to V_1 \otimes \cdots \otimes V_k, (v_1, \dots, v_k) \mapsto v_1 \otimes \cdots \otimes v_k, such that for any vector space W and k-linear map \phi: V_1 \times \cdots \times V_k \to W, there exists a unique linear map \overline{\phi}: V_1 \otimes \cdots \otimes V_k \to W with \phi = \overline{\phi} \circ \beta. The tensor product can be constructed explicitly as the quotient of the free vector space on the set of symbols \{v_1 \otimes \cdots \otimes v_k \mid v_i \in V_i\} by the subspace generated by the multilinearity relations. For the bilinear case (k=2), these relations include elements of the form (v_1 + v_1', v_2) - (v_1 \otimes v_2 + v_1' \otimes v_2), (v_1, v_2 + v_2') - (v_1 \otimes v_2 + v_1 \otimes v_2'), and (\lambda v_1, v_2) - \lambda (v_1 \otimes v_2) = (v_1, \lambda v_2) - \lambda (v_1 \otimes v_2) for \lambda \in K. This quotient enforces the required multilinearity, yielding a vector space where the images of the symbols satisfy the bilinear (or multilinear) properties. Such a construction guarantees the existence of the tensor product satisfying the universal property. When the vector spaces V_i are finite-dimensional with \dim(V_i) = n_i, the tensor product V_1 \otimes \cdots \otimes V_k is also finite-dimensional, with dimension \prod_{i=1}^k n_i. If \{e_{i,j} \mid 1 \leq j \leq n_i\} is a basis for each V_i, then the set \{e_{1,j_1} \otimes \cdots \otimes e_{k,j_k} \mid 1 \leq j_i \leq n_i \ \forall i\} forms a basis for V_1 \otimes \cdots \otimes V_k. The tensor product operation exhibits several key properties that facilitate its use in algebraic constructions. It is associative, with natural isomorphisms V \otimes (W \otimes U) \cong (V \otimes W) \otimes U for vector spaces V, W, U, induced by the universal property via the bilinear maps (v, w \otimes u) \mapsto (v \otimes w) \otimes u and (v \otimes w, u) \mapsto v \otimes (w \otimes u). For identical spaces, the tensor product admits a natural symmetry via the isomorphism V \otimes V \to V \otimes V given by v_1 \otimes v_2 \mapsto v_2 \otimes v_1, which extends to a flip map interchanging factors in more general tensor products of distinct spaces.

Identification with Tensors

In multilinear algebra, the dual space V_i^* of a vector space V_i over a field F consists of all linear functionals \alpha_i: V_i \to F.^[12] These dual spaces play a central role in identifying multilinear forms with tensors, as the space of k-linear forms on V_1 \times \cdots \times V_k (with values in F) corresponds bijectively to the linear functionals on the tensor product space V_1 \otimes \cdots \otimes V_k, which form the dual space (V_1 \otimes \cdots \otimes V_k)^*.^[6] This identification leverages the universal property of the tensor product, which linearizes multilinear maps.^[12] Furthermore, there exists a canonical isomorphism (V_1 \otimes \cdots \otimes V_k)^* \cong V_1^* \otimes \cdots \otimes V_k^*, establishing that multilinear forms are precisely the elements of the tensor product of the dual spaces.^[6] This isomorphism preserves the algebraic structure, allowing multilinear forms to be expressed as sums of pure tensors from the duals.^[13] The explicit construction of this map sends a pure tensor \alpha_1 \otimes \cdots \otimes \alpha_k, with each \alpha_i \in V_i^*, to the multilinear form \phi: V_1 \times \cdots \times V_k \to F defined by

\phi(v_1, \dots, v_k) = \prod_{i=1}^k \alpha_i(v_i)

for v_i \in V_i, and extends linearly to arbitrary elements of V_1^* \otimes \cdots \otimes V_k^*.^[13] This mapping is bijective and respects the multilinearity in each argument.^[12] Under change of basis, multilinear forms transform according to the dual representation, meaning their components in the new basis are obtained by contracting with the basis change matrix for each covariant index, which defines the covariant transformation law.^[13] This contrasts with contravariant tensors, which arise from tensor products of the original spaces V_i and transform with the inverse of the basis change matrix.^[13]

Special Cases

Bilinear Forms

A bilinear form on a vector space V over a field F is a map \phi: V \times V \to F that is linear in each argument separately.^[14] Such forms arise as the special case of multilinear maps with two inputs and play a central role in linear algebra and geometry.^[15] Bilinear forms are classified into several types based on additional symmetries. A bilinear form \phi is symmetric if \phi(v, w) = \phi(w, v) for all v, w \in V.^[16] It is skew-symmetric (or alternating) if \phi(v, w) = -\phi(w, v) for all v, w \in V, which implies \phi(v, v) = 0.^[17] Over fields of characteristic not equal to 2, skew-symmetric and alternating forms coincide.^[15] A bilinear form is non-degenerate if the only vector v \in V satisfying \phi(v, w) = 0 for all w \in V is v = 0, or equivalently, if the associated linear map V \to V^* is an isomorphism. With respect to a basis \{e_1, \dots, e_n\} of V, any bilinear form \phi has a matrix representation: if v = \sum v_i e_i and w = \sum w_j e_j, then \phi(v, w) = \mathbf{v}^T A \mathbf{w}, where A = (a_{ij}) with a_{ij} = \phi(e_i, e_j) is the Gram matrix of \phi.^[18] For symmetric bilinear forms, A is symmetric (A = A^T); for skew-symmetric forms, A is skew-symmetric (A = -A^T).^[15] Changes of basis transform A via congruence: if P is the change-of-basis matrix, the new matrix is P^T A P.^[15] The classification of bilinear forms depends on the field F. Over the real numbers \mathbb{R}, every symmetric bilinear form is diagonalizable by an orthogonal change of basis, meaning there exists an orthonormal basis in which the Gram matrix is diagonal with real entries.^[19] Over algebraically closed fields such as the complex numbers \mathbb{C} (of characteristic not 2), symmetric bilinear forms are similarly diagonalizable by congruence to a diagonal matrix.^[20] For general (non-symmetric) bilinear forms over algebraically closed fields, the classification involves a Jordan canonical form under strict equivalence, decomposing into blocks corresponding to the Jordan structure of the associated linear operators.^[21] Symmetric bilinear forms are closely linked to quadratic forms. Given a symmetric bilinear form \phi, the associated quadratic form is q(v) = \phi(v, v), which is homogeneous of degree 2: q(\lambda v) = \lambda^2 q(v).^[22] Conversely, over fields of characteristic not 2, any quadratic form q determines a unique symmetric bilinear form via the polarization identity:

\phi(v, w) = \frac{1}{4} \left( q(v + w) - q(v - w) \right).

This identity recovers \phi fully from q, establishing a one-to-one correspondence between symmetric bilinear forms and quadratic forms.^[23] Prominent examples include the standard dot product on \mathbb{R}^n, defined by \phi(v, w) = v_1 w_1 + \cdots + v_n w_n, which is symmetric, positive definite, and non-degenerate, with Gram matrix the n \times n identity.^[24] Another key example is the Minkowski metric on \mathbb{R}^{1,3}, given by \phi(v, w) = v_0 w_0 - v_1 w_1 - v_2 w_2 - v_3 w_3, an indefinite symmetric non-degenerate form essential in special relativity for measuring spacetime intervals.^[25]

Multilinear Forms on Finite-Dimensional Spaces

In finite-dimensional vector spaces, multilinear forms exhibit particularly tractable algebraic properties due to the existence of bases. Let V be a vector space over a field F with \dim V = n < \infty. A k-linear form on V is a multilinear map \phi: V^k \to F. The space of all such k-linear forms, often denoted \mathrm{Hom}(V^{\otimes k}, F) or T^k(V^*), forms a vector space itself. Given a basis \{e_1, \dots, e_n\} for V, any k-linear form \phi is uniquely determined by its values on the basis tuples (e_{i_1}, \dots, e_{i_k}), of which there are n^k such tuples, and these values can be assigned arbitrarily. Thus, \dim T^k(V^*) = n^k.^[5] A prominent example of a multilinear form arises in the determinant function, which operates on n-tuples of vectors in V. Specifically, \det: (V^n) \to F is defined such that for column vectors represented by an n \times n matrix A with respect to a basis, \det(A) is the standard determinant. This map is n-linear in the columns (or rows) of A, and when restricted to alternating forms—those satisfying \det(v_1, \dots, v_i, \dots, v_j, \dots, v_n) = -\det(v_1, \dots, v_j, \dots, v_i, \dots, v_n) for i \neq j—it is non-degenerate, meaning \det(v_1, \dots, v_n) = 0 if and only if \{v_1, \dots, v_n\} is linearly dependent. The determinant is unique up to scalar multiple among such alternating n-linear forms that evaluate to 1 on the standard basis.^[7]^[26] In the context of oriented finite-dimensional spaces, volume forms provide a canonical class of top-degree alternating multilinear forms. For an oriented n-dimensional space V equipped with an inner product, a volume form \omega \in \mathrm{Alt}^n(V^*) (the space of alternating n-linear forms) is characterized by \omega(e_1, \dots, e_n) = 1, where \{e_1, \dots, e_n\} is a positively oriented orthonormal basis; since \dim \mathrm{Alt}^n(V^*) = 1, such \omega is unique up to positive scalar multiple. This form assigns to any parallelepiped spanned by vectors v_1, \dots, v_n the signed oriented volume \det(v_1, \dots, v_n); the unsigned volume is |\omega(v_1, \dots, v_n)|. For the bilinear case (k=2) in n=2 dimensions, such volume forms are non-degenerate alternating forms, distinct from the non-degenerate symmetric forms associated with inner products.^[9] Under a change of basis represented by an invertible linear map g: V \to V, multilinear forms transform via the pullback: for a k-linear form \phi, the transformed form is \phi' = g^* \phi, defined by \phi'(v_1, \dots, v_k) = \phi(g v_1, \dots, g v_k). For alternating forms like the determinant or volume forms, this simplifies to \phi' = \det(g) \cdot \phi, reflecting the orientation-preserving or reversing nature of g \in \mathrm{GL}(V). This transformation law ensures that volumes scale by |\det(g)| under basis changes.^[9]

Alternating Forms

Definition and Antisymmetry

An alternating multilinear form on a vector space V over a field K is a k-linear map \phi: V^k \to K that satisfies \phi(v_{\sigma(1)}, \dots, v_{\sigma(k)}) = \operatorname{sgn}(\sigma) \phi(v_1, \dots, v_k) for all permutations \sigma \in S_k and all v_1, \dots, v_k \in V, where \operatorname{sgn}(\sigma) is the sign of the permutation (+1 for even, -1 for odd).^[27] This condition extends the multilinearity property by imposing a symmetry constraint under reordering of arguments. Equivalently, \phi is alternating if \phi(v_1, \dots, v_i, \dots, v_j, \dots, v_k) = -\phi(v_1, \dots, v_j, \dots, v_i, \dots, v_k) for any adjacent transposition (i, j) with i = j-1, or more generally, if \phi vanishes whenever any two arguments are equal: \phi(v_1, \dots, v_i, \dots, v_i, \dots, v_k) = 0 for i \neq j.^[28]^[8] The antisymmetry of alternating forms is closely related to skew-symmetry, where a multilinear form \psi satisfies \psi(v_1, \dots, v_i, \dots, v_j, \dots, v_k) = -\psi(v_1, \dots, v_j, \dots, v_i, \dots, v_k) for all pairs i < j. Every alternating form is skew-symmetric, as the permutation condition implies sign changes under transpositions.^[27] However, the converse holds only over fields of characteristic not equal to 2: if \operatorname{char}(K) \neq 2, then every skew-symmetric form is alternating. In characteristic 2, skew-symmetric forms may not vanish on repeated arguments, distinguishing the concepts.^[28]^[7] The space of alternating k-forms arises as the image of the alternation operator \operatorname{Alt}: (V^*)^{\otimes k} \to \operatorname{Alt}^k(V^*), defined by

\operatorname{Alt}(\phi) = \frac{1}{k!} \sum_{\sigma \in S_k} \operatorname{sgn}(\sigma) \, \phi \circ \sigma,

which projects any multilinear form \phi onto the subspace of alternating forms by averaging over signed permutations.^[27] This operator is idempotent, meaning \operatorname{Alt}(\operatorname{Alt}(\phi)) = \operatorname{Alt}(\phi), and its kernel consists of forms that average to zero under signed permutations. For a finite-dimensional vector space V of dimension n, the space of alternating k-forms on V has dimension \binom{n}{k}, corresponding to the basis elements formed by wedging distinct dual basis vectors.^[7]^[29]

Exterior Product

The exterior algebra of a vector space V over a field K of characteristic not equal to 2 is constructed as the quotient of the tensor algebra T(V) by the two-sided ideal I generated by all elements of the form v \otimes v for v \in V. Denoted \wedge(V) = T(V)/I, this algebra realizes the universal property for alternating multilinear maps from V, enforcing antisymmetry by setting squares to zero in each homogeneous component. The grading on T(V) descends to \wedge(V), yielding a direct sum \wedge(V) = \bigoplus_{k \geq 0} \wedge^k(V), where \wedge^0(V) = K and \wedge^k(V) is spanned by equivalence classes of decomposable tensors modulo the relations from I. The multiplication in \wedge(V) is induced by the tensor product, resulting in the wedge product \wedge : \wedge(V) \times \wedge(V) \to \wedge(V). For homogeneous elements \alpha \in \wedge^p(V) and \beta \in \wedge^q(V), the wedge product is defined as \alpha \wedge \beta = \mathrm{Alt}(\alpha \otimes \beta), where \mathrm{Alt} denotes the alternator map projecting onto the alternating subspace via \mathrm{Alt}(T) = \frac{1}{k!} \sum_{\sigma \in S_k} \mathrm{sgn}(\sigma) \sigma(T) for a k-tensor T. This operation is bilinear, associative, and graded-commutative, satisfying \alpha \wedge \beta = (-1)^{pq} \beta \wedge \alpha. If \{e_1, \dots, e_n\} is a basis for an n-dimensional vector space V, then a basis for \wedge^k(V) consists of the elements e_{i_1} \wedge \cdots \wedge e_{i_k} where $1 \leq i_1 < i_2 < \cdots < i_k \leq n, with dimension \binom{n}{k}. These basis elements are the images of the corresponding ordered tensor products under the quotient map, and any element of \wedge^k(V) expands uniquely in this basis due to the antisymmetry relations. The space of alternating k-linear forms on V, denoted \mathrm{Alt}^k(V; K), is naturally isomorphic to the dual space (\wedge^k V)^*. Under this duality, for \alpha \in \mathrm{Alt}^k(V; K) and v_1 \wedge \cdots \wedge v_k \in \wedge^k V, the pairing is given by \langle \alpha, v_1 \wedge \cdots \wedge v_k \rangle = \frac{1}{k!} \alpha(v_1, \dots, v_k). This identification preserves the algebraic structure, allowing multilinear forms to act on exterior products compatibly with the wedge operation.

Applications

Differential Forms

A differential k-form on a smooth manifold M is defined as a smooth section of the kth exterior power of the cotangent bundle, \wedge^k T^*M. At each point p \in M, this assigns an element of \wedge^k (T_p M)^*, i.e., a linear map \omega_p : \wedge^k T_p M \to \mathbb{R}, or equivalently an alternating multilinear map \omega_p : (T_p M)^k \to \mathbb{R} that is linear in each argument and antisymmetric under exchange of any two vectors.^[30]^[31] This construction extends the algebraic notion of alternating multilinear forms to the geometric setting of manifolds by associating them pointwise to the tangent spaces. The foundation for higher k-forms begins with 1-forms, which are sections of the cotangent bundle T^*M. The cotangent space T^*_p M at p \in M is the dual vector space to the tangent space T_p M, consisting of all linear functionals on T_p M. In local coordinates (x^1, \dots, x^n) around p, where \{\partial/\partial x^i\} forms a basis for T_p M, the dual basis for T^*_p M is \{dx^i\}, satisfying dx^i(\partial/\partial x^j) = \delta^i_j. Thus, any 1-form \omega near p can be expressed as \omega = \sum_i f_i \, dx^i for smooth functions f_i.^[32]^[33] For a smooth map f: N \to M between manifolds, the pullback f^*\omega defines an induced k-form on N. Specifically, for q \in N and vectors v_1, \dots, v_k \in T_q N, (f^*\omega)_q(v_1, \dots, v_k) = \omega_{f(q)}(df_q(v_1), \dots, df_q(v_k)), where df_q: T_q N \to T_{f(q)} M is the differential of f. This operation ensures that differential forms transform covariantly under maps, preserving their multilinear structure.^[34] An n-form on an n-dimensional manifold M is a volume form if it is nowhere zero, providing a consistent notion of oriented volume. The existence of such a form requires M to be orientable, meaning there is a consistent choice of orientation across M, typically via an atlas where transition maps have positive Jacobian determinants. Volume forms thus encode both the topological orientation and a local scaling for integration purposes.^[35]^[36]

Integration and Stokes' Theorem

The integration of a differential k-form over an oriented k-dimensional manifold is defined locally using coordinate charts and extended globally via an atlas compatible with the orientation. For a compact oriented k-manifold K without boundary and a smooth k-form \omega with compact support, the integral \int_K \omega is computed in a coordinate chart (U, \phi) where \phi: U \to \mathbb{R}^k is an orientation-preserving parametrization, pulling back \omega to \phi^* \omega = f \, dx^1 \wedge \cdots \wedge dx^k on the parameter domain V = \phi(K \cap U), yielding \int_V f \, du^1 \cdots du^k. Consistency across charts follows from the change-of-variables formula, where the Jacobian determinant is positive due to orientation preservation, ensuring the integral is well-defined and independent of the atlas.^[37] This construction generalizes to integration over singular chains, which provide a topological framework for manifolds and their substructures. A k-chain C in a manifold M is a formal integer linear combination of singular k-simplices (continuous maps from the standard k-simplex to M), and the integral \int_C \omega for a k-form \omega is defined by linearity: for a simplex \sigma: \Delta^k \to M, \int_\sigma \omega = \int_{\Delta^k} \sigma^* \omega, computed via the standard affine parametrization of \Delta^k and the resulting multiple integral over [0,1]^k. The boundary operator \partial on chains satisfies \partial^2 = 0 and is defined face-wise for simplices, inducing the oriented boundary \partial M for a manifold with boundary via the fundamental class.^[38] Stokes' theorem unifies these notions by relating the exterior derivative to boundaries: for a compact oriented k-manifold M with boundary (possibly empty) and a smooth (k-1)-form \omega on M,

\int_M d\omega = \int_{\partial M} \omega,

where the induced orientation on \partial M ensures the right-hand side respects the outward normal convention. In the chain formulation, this extends to \int_C d\omega = \int_{\partial C} \omega for any k-chain C, proven by verifying it on standard simplices using the fundamental theorem of calculus and Fubini's theorem in local coordinates.^[37]^[38] A key application arises in de Rham cohomology, which measures the topological invariants of M through the sequence of differential forms and the exterior derivative d. The k-th de Rham cohomology group is H^k_{dR}(M) = \ker(d: \Omega^k(M) \to \Omega^{k+1}(M)) / \operatorname{im}(d: \Omega^{k-1}(M) \to \Omega^k(M)), where closed forms (d\omega = 0) represent cohomology classes. Stokes' theorem implies that for a closed k-form \omega, the integral \int_K \omega over a compact oriented k-cycle K (with \partial K = 0) depends only on the de Rham class of \omega, as differing representatives \omega and \omega + d\eta yield the same integral by \int_K d\eta = \int_{\partial K} \eta = 0. This duality links de Rham cohomology to singular homology, establishing de Rham's theorem that H^k_{dR}(M) \cong H_k(M; \mathbb{R}).^[39]

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The existence of the map ¯b satisfying the above conditions is called the universal property of the tensor product. This definition is quite abstract. It is not ...Missing: hoffman kunze
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[PDF] Chapter V. Tensors and Multilinear Forms
May 25, 2019 · We generalize the idea of bilinear forms to the idea of multilinear forms. We define the tensor product of vector spaces and tensors, which are ...<|control11|><|separator|>
[14]
bilinear form in nLab
Bilinear forms ; It is called symmetric if · y · ⟨ y , x ⟩ ; It is called antisymmetric or skew-symmetric if · y · − ⟨ y , x ⟩ ; It is called alternating if · x · 0 ; Let ...Definitions · Examples · Generalizations
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[PDF] BILINEAR FORMS The geometry of Rn is controlled algebraically by ...
In fact, we now show that a skew-symmetric bilinear form is just another name for a symmetric or an alternating bilinear form, depending on whether or not the ...
[16]
[PDF] Linear Algebra: Non-degenerate Bilinear Forms - DPMMS
Let ψ : V × V → F be a bilinear form. We assume that either ψ is symmetric, i.e. ψ(u, v) = ψ(v, u) for all u, v ∈ V , or ψ is skew-symmetric, i.e. ψ(u, v) = −ψ ...
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[PDF] Bilinear Forms
Feb 28, 2005 · Definition A bilinear form B on a vector space V is called alternate (or skew-symmetric if char F 6= 2) if B(v, v) = 0 for all w, v ∈ V. If ...
[18]
[PDF] Chapter 3. Bilinear forms - Lecture notes for MA1212
A bilinear form on a real vector space V is a function f : V × V → R which assigns a number to each pair of elements of V in such a way that f is linear in ...
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[PDF] Contents 5 Bilinear and Quadratic Forms - Evan Dummit
Then a bilinear form on V is diagonalizable if and only if it is symmetric. ◦ Proof: The forward direction was established above. For the reverse, we show the ...
[20]
[PDF] Symmetric bilinear forms
... quadratic ⇔ F⊥ is quadratic ⇔. F ⊕ F⊥ = V . As a simple application, one finds that any non-degenerate symmetric bilinear form B on V can be 'diagonalized'.
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[PDF] arXiv:1311.0565v4 [math.RA] 19 Nov 2013
Nov 19, 2013 · Abstract. We present an alternative account of the problem of classifying and finding normal forms for arbitrary bilinear forms.<|control11|><|separator|>
[22]
[PDF] Lecture 4.7. Bilinear and quadratic forms - Purdue Math
Apr 9, 2020 · A bilinear form is a function linear in each vector. A quadratic form is a function of one vector derived from a symmetric bilinear form.
[23]
polarization identity in nLab
May 24, 2024 · The polarization identity reconstructs the bilinear form from the quadratic form. More generally, starting from any bilinear form, the polarization identity ...Idea · Statements · Examples · Higher order
[24]
[PDF] Bilinear Forms over a field F Let V be a vector space.
Every real symmetric matrix, and every complex hermitian matrix, is diagonalizable as a linear transfor- mation. That is, V has a basis of eigenvectors for A, ...
[25]
[PDF] Bilinear forms - Purdue Math
Mar 18, 2023 · A symmetric bilinear form can be recovered from its quadratic form ... quadratic form (which is called the Minkowski metric) is. Q(u) = u2. 0 ...
[26]
[PDF] Whence the Determinant?
δA is called the determinant of A. A bit more work reveals that the determinant is the unique (up to a scalar multiple) alternating multilinear form. Many ...
[27]
[PDF] Alternating multilinear forms - Jordan Bell
Aug 21, 2018 · Definition 1. A map f ∈ Lp(E; R) is called alternating if (x1,...,xp) ∈ Ep with xi = xi+1 for some 1 ≤ i<p implies f(x1,...,xp) = 0.Missing: operator | Show results with:operator
[28]
[PDF] Lecture 3.3: Alternating multilinear forms
Definition. A k-linear form is alternating if f (x1,..., xk ) = 0 whenever xi = xj for some i 6= j.
[29]
[PDF] Math1410: Crash Course on Forms and Cohomology - Brown Math
If turns out that every alternating k-tensor is a linear combination of the examples given. Therefore, the vector space of alternating k-tensors has di- mension ...
[30]
Differential k-Form -- from Wolfram MathWorld
A differential k-form is a tensor of tensor rank k that is antisymmetric under exchange of any pair of indices.
[31]
differential form in nLab
Mar 12, 2025 · A differential form is a geometrical object on a manifold that can be integrated, a section of the exterior algebra of the cotangent bundle.Idea · Definitions · Standard definition · Operations on differential forms<|control11|><|separator|>
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[PDF] Differential Forms and Integration - UCLA Mathematics
In the language of forms, this is asserting that any one-dimensional form f(x)dx ... can be viewed as sections of the cotangent bundle T*Rn, and similarly 2-forms ...
[33]
cotangent bundle in nLab
Nov 23, 2017 · Idea. Given a differentiable manifold X X , the cotangent bundle T * ( X ) T^*(X) of X X is the dual vector bundle over X X dual to the ...
[34]
pullback of a differential form in nLab
Jun 21, 2024 · In terms of push-forward of vector fields. If differential forms are defined as linear duals to vectors then pullback is the dual operation to ...Definition · In terms of coordinate expression · Compatibility with the de...
[35]
Volume form - Encyclopedia of Mathematics
Jun 29, 2014 · The volume form is a special differential form defined on oriented Riemannian manifolds and which introduces a natural concept of measure on the manifold.On vector spaces · On Riemannian manifolds · Volume measure
[36]
[PDF] LECTURE 23: THE STOKES FORMULA 1. Volume Forms
Definition 1.2. A nowhere vanishing smooth n-form µ on an n-dimensional smooth manifold M is called a volume form. Remark. If M is orientable, and µ is a ...
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7.3: C- Differential Forms and Stokes' Theorem
Sep 5, 2021 · Integrating it over a domain (an n -dimensional submanifold) gives the standard volume integral. More generally one defines integration of k - ...
[38]
Differential Forms and Chains
An integral of a 1-form (respectively 2-form) over a 1-chain (respectively 2 ... 7.4 Stokes' Theorem. The main theorem of the course is: Theorem 1: (Stokes' ...
[39]
[PDF] proof of de rham's theorem - Harvard University
The natural isomorphism will be given by a version of Stokes' theorem, which describes a duality between de Rham cohomology and singular homology.