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definition

Homogeneous quadratic equation $f: V \to \mathbb{Q}$ satisfying $f(a \cdot v) = a^2 f(v)$

Let $V$ be a vector space (or module) over the rational numbers $\mathbb{Q}$ . A homogeneous quadratic equation is a map $f : V \to \mathbb{Q}$ such that for any scalar $a \in \mathbb{Q}$ and any vector $v \in V$ , the function satisfies the homogeneity property $f(a \cdot v) = a^2 f(v)$ .

instance

Homogeneous quadratic equation as a function $ f: V \to \mathbb{Q} $

#instFun

This instance allows a homogeneous quadratic equation $ f \in \text{HomogeneousQuadratic } V $ to be treated as a function from a vector space $ V $ (over $ \mathbb{Q} $) to the rational numbers $ \mathbb{Q} $. It provides the mechanism to evaluate the equation at a vector $ v \in V $ to produce the value $ f(v) \in \mathbb{Q} $.

theorem

$f(a \cdot S) = a^2 f(S)$ for a homogeneous quadratic map $f$

#map_smul

Let $V$ be a vector space (or module) over the rational numbers $\mathbb{Q}$ . For any homogeneous quadratic map $f: V \to \mathbb{Q}$ , any scalar $a \in \mathbb{Q}$ , and any vector $S \in V$ , the identity $f(a \cdot S) = a^2 f(S)$ holds.

instance

Symmetric bilinear form as a function $V \to (V \to_\ell \mathbb{Q})$

#instFun

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$ . This instance allows a symmetric bilinear form $f \in \text{BiLinearSymm } V$ to be treated as a function from $V$ to the space of $\mathbb{Q}$ -linear maps from $V$ to $\mathbb{Q}$ (the dual space). Specifically, for any vector $v \in V$ , the evaluation $f(v)$ is a linear functional such that for any $w \in V$ , $f(v)(w)$ returns a value in $\mathbb{Q}$ .

definition

Symmetric bilinear map from first-factor linearity and $f(S, T) = f(T, S)$

#mk₂

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$ . Given a function $f : V \times V \to \mathbb{Q}$ , the definition `BiLinearSymm.mk₂` constructs a symmetric bilinear map on $V$ if $f$ satisfies linearity in its first argument and symmetry. Specifically, it requires: 1. Scalar multiplication in the first factor: $f(a \cdot S, T) = a \cdot f(S, T)$ for all $a \in \mathbb{Q}$ and $S, T \in V$ . 2. Addition in the first factor: $f(S_1 + S_2, T) = f(S_1, T) + f(S_2, T)$ for all $S_1, S_2, T \in V$ . 3. Symmetry (swap): $f(S, T) = f(T, S)$ for all $S, T \in V$ . The linearity in the second factor is automatically derived from these properties.

theorem

$f(a \cdot S, T) = a \cdot f(S, T)$ for symmetric bilinear forms

#map_smul₁

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$ . For any symmetric bilinear form $f$ on $V$ , any scalar $a \in \mathbb{Q}$ , and any vectors $S, T \in V$ , it holds that $f(a \cdot S, T) = a \cdot f(S, T)$ .

theorem

$f(S, T) = f(T, S)$ for symmetric bilinear forms

#swap

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$ . For any symmetric bilinear form $f$ on $V$ and for any vectors $S, T \in V$ , the identity $f(S, T) = f(T, S)$ holds.

theorem

$f(S, a \cdot T) = a \cdot f(S, T)$ for symmetric bilinear forms

#map_smul₂

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$ . For any symmetric bilinear form $f$ on $V$ , any scalar $a \in \mathbb{Q}$ , and any vectors $S, T \in V$ , the identity $f(S, a \cdot T) = a \cdot f(S, T)$ holds.

theorem

$f(S_1 + S_2, T) = f(S_1, T) + f(S_2, T)$ for symmetric bilinear forms

#map_add₁

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$ . For any symmetric bilinear form $f$ on $V$ and any vectors $S_1, S_2, T \in V$ , the identity $f(S_1 + S_2, T) = f(S_1, T) + f(S_2, T)$ holds.

theorem

$f(S, T_1 + T_2) = f(S, T_1) + f(S, T_2)$ for symmetric bilinear forms

#map_add₂

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$ . For any symmetric bilinear form $f$ on $V$ and any vectors $S, T_1, T_2 \in V$ , the identity $f(S, T_1 + T_2) = f(S, T_1) + f(S, T_2)$ holds.

definition

The linear functional $f(\cdot, T)$ derived from a symmetric bilinear form $f$

#toLinear₁

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$ . For a symmetric bilinear form $f$ on $V$ and a fixed vector $T \in V$ , this definition represents the linear functional $V \to \mathbb{Q}$ mapping $S \mapsto f(S, T)$ .

theorem

$f(S, T) = f.\text{toLinear}_1(T)(S)$

#toLinear₁_apply

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$ . For any symmetric bilinear form $f$ on $V$ and any vectors $S, T \in V$ , the value of the form $f(S, T)$ is equal to the evaluation of the linear functional $f.\text{toLinear}_1(T)$ at the vector $S$ , where $f.\text{toLinear}_1(T)$ denotes the linear map $V \to \mathbb{Q}$ derived by fixing the second argument of $f$ as $T$ .

theorem

$f(\sum S_i, T) = \sum f(S_i, T)$ for Symmetric Bilinear Forms

#map_sum₁

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$ and let $f$ be a symmetric bilinear form on $V$ . For any natural number $n$ , any vector $T \in V$ , and any family of vectors $S_i \in V$ indexed by $i \in \{0, \dots, n-1\}$ , the following equality holds: $f\left(\sum_{i=0}^{n-1} S_i, T\right) = \sum_{i=0}^{n-1} f(S_i, T)$ where the left-hand side is the bilinear form applied to the sum of the vectors $S_i$ and the vector $T$ , and the right-hand side is the sum of the values of the form applied to each individual $S_i$ and $T$ .

theorem

$f(T, \sum S_i) = \sum f(T, S_i)$ for Symmetric Bilinear Forms

#map_sum₂

Let $V$ be a vector space (typically over $\mathbb{Q}$ in this context) and let $f$ be a symmetric bilinear form on $V$ . For any natural number $n$ , any vector $T \in V$ , and any family of vectors $S_i \in V$ indexed by $i \in \{0, \dots, n-1\}$ , the following equality holds: $f(T, \sum_{i=0}^{n-1} S_i) = \sum_{i=0}^{n-1} f(T, S_i)$ where the left-hand side is the bilinear form applied to $T$ and the sum of the vectors $S_i$ , and the right-hand side is the sum of the results of applying the form to $T$ and each individual $S_i$ .

definition

Homogeneous quadratic map $v \mapsto \tau(v, v)$ from symmetric bilinear form $\tau$

#toHomogeneousQuad

Let $V$ be a vector space over the field of rational numbers $\mathbb{Q}$ . Given a symmetric bilinear form $\tau$ on $V$ , this definition constructs a homogeneous quadratic map $f: V \to \mathbb{Q}$ defined by $f(v) = \tau(v, v)$ for every vector $v \in V$ .

theorem

$Q(S + T) = Q(S) + Q(T) + 2\tau(S, T)$ for the quadratic form $Q$ associated with symmetric bilinear form $\tau$

#toHomogeneousQuad_add

Let $V$ be a vector space over the rational numbers $\mathbb{Q}$ and let $\tau$ be a symmetric bilinear form on $V$ . Let $Q: V \to \mathbb{Q}$ be the associated homogeneous quadratic map defined by $Q(v) = \tau(v, v)$ . For any vectors $S, T \in V$ , the following identity holds: $Q(S + T) = Q(S) + Q(T) + 2\tau(S, T)$

definition

Homogeneous cubic map $V \to \mathbb{Q}$

#HomogeneousCubic

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$ . A homogeneous cubic equation (or map) is a function $f: V \to \mathbb{Q}$ such that for any scalar $a \in \mathbb{Q}$ and any element $v \in V$ , the condition $f(a \cdot v) = a^3 f(v)$ is satisfied.

instance

A homogeneous cubic map $f$ as a function $V \to \mathbb{Q}$

#instFun

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$ . A homogeneous cubic map $f$ (an element of the type `HomogeneousCubic V`) can be treated as a function $f : V \to \mathbb{Q}$ . This allows an instance of a cubic equation to be applied directly to an element $v \in V$ to obtain a value in $\mathbb{Q}$ .

theorem

$f(a \cdot S) = a^3 f(S)$ for a homogeneous cubic map $f$

#map_smul

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$ . For any homogeneous cubic map $f: V \to \mathbb{Q}$ , any scalar $a \in \mathbb{Q}$ , and any element $S \in V$ , the identity $f(a \cdot S) = a^3 f(S)$ holds.

instance

A symmetric trilinear form $f$ as a function $V \to (V \to_{\mathbb{Q}} V \to_{\mathbb{Q}} \mathbb{Q})$

#instFun

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$ . An element $f$ of the type `TriLinearSymm V`, representing symmetric trilinear forms on $V$ , can be treated as a function $f : V \to (V \to_{\mathbb{Q}} V \to_{\mathbb{Q}} \mathbb{Q})$ . This allows a symmetric trilinear form to be applied sequentially to elements of $V$ to obtain a linear map, and eventually a scalar in $\mathbb{Q}$ , reflecting the curried representation of a trilinear map.

definition

Symmetric trilinear map from first-factor linearity, $f(S, T, L) = f(T, S, L)$ , and $f(S, T, L) = f(S, L, T)$

#mk₃

Let $V$ be a vector space (or module) over the field of rational numbers $\mathbb{Q}$ . Given a function $f : V \times V \times V \to \mathbb{Q}$ , the constructor `TriLinearSymm.mk₃` defines a symmetric trilinear form on $V$ provided that $f$ satisfies linearity in its first argument and two specific swap symmetries. Specifically, it requires: 1. **Linearity in the first factor**: * Scalar multiplication: $f(a \cdot S, T, L) = a \cdot f(S, T, L)$ for all $a \in \mathbb{Q}$ and $S, T, L \in V$ . * Addition: $f(S_1 + S_2, T, L) = f(S_1, T, L) + f(S_2, T, L)$ for all $S_1, S_2, T, L \in V$ . 2. **Symmetry between the first and second factors**: $f(S, T, L) = f(T, S, L)$ for all $S, T, L \in V$ . 3. **Symmetry between the second and third factors**: $f(S, T, L) = f(S, L, T)$ for all $S, T, L \in V$ . Because the map is symmetric, linearity in the second and third arguments is automatically satisfied by the linearity in the first.

theorem

$f(S, T, L) = f(T, S, L)$ for symmetric trilinear forms

#swap₁

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$ . For any symmetric trilinear form $f$ on $V$ and any vectors $S, T, L \in V$ , the result of the form is invariant under the swapping of the first and second arguments, such that $f(S, T, L) = f(T, S, L)$ .

theorem

$f(S, T, L) = f(S, L, T)$ for symmetric trilinear forms

#swap₂

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$ . For any symmetric trilinear form $f$ on $V$ and any vectors $S, T, L \in V$ , the result of the form is invariant under the swapping of the second and third arguments, such that $f(S, T, L) = f(S, L, T)$ .

theorem

$f(S, T, L) = f(L, T, S)$ for symmetric trilinear forms

#swap₃

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$ . For any symmetric trilinear form $f$ on $V$ and any vectors $S, T, L \in V$ , the form is invariant under the swapping of the first and third arguments, such that $f(S, T, L) = f(L, T, S)$ .

theorem

$f(a \cdot S, T, L) = a f(S, T, L)$ for symmetric trilinear forms

#map_smul₁

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$ . For any symmetric trilinear form $f$ on $V$ , any scalar $a \in \mathbb{Q}$ , and any vectors $S, T, L \in V$ , the form satisfies the scalar homogeneity property in its first argument, such that $f(a \cdot S, T, L) = a f(S, T, L)$ .

theorem

$f(S, a \cdot T, L) = a f(S, T, L)$ for symmetric trilinear forms

#map_smul₂

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$ . For any symmetric trilinear form $f$ on $V$ , any scalar $a \in \mathbb{Q}$ , and any vectors $S, T, L \in V$ , the form satisfies the scalar homogeneity property in its second argument: $f(S, a \cdot T, L) = a f(S, T, L).$

theorem

$f(S, T, a \cdot L) = a f(S, T, L)$ for symmetric trilinear forms

#map_smul₃

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$ . For any symmetric trilinear form $f$ on $V$ , any scalar $a \in \mathbb{Q}$ , and any vectors $S, T, L \in V$ , the form satisfies the scalar homogeneity property in its third argument, such that: $f(S, T, a \cdot L) = a f(S, T, L).$

theorem

Additivity of Symmetric Trilinear Forms in the First Argument

#map_add₁

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$ . For any symmetric trilinear form $f$ on $V$ and any vectors $S_1, S_2, T, L \in V$ , the form is additive in its first argument, such that $f(S_1 + S_2, T, L) = f(S_1, T, L) + f(S_2, T, L).$

theorem

Additivity of Symmetric Trilinear Forms in the Second Argument

#map_add₂

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$ . For any symmetric trilinear form $f$ on $V$ and any vectors $S, T_1, T_2, L \in V$ , the form is additive in its second argument, such that $f(S, T_1 + T_2, L) = f(S, T_1, L) + f(S, T_2, L).$

theorem

Additivity of Symmetric Trilinear Forms in the Third Argument

#map_add₃

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$ . For any symmetric trilinear form $f$ on $V$ and any vectors $S, T, L_1, L_2 \in V$ , the form is additive in its third argument, such that $f(S, T, L_1 + L_2) = f(S, T, L_1) + f(S, T, L_2).$

definition

Linear map $S \mapsto f(S, T, L)$ from a symmetric trilinear form $f$

#toLinear₁

Let $V$ be a vector space over the field of rational numbers $\mathbb{Q}$ . Given a symmetric trilinear form $f$ on $V$ and two fixed vectors $T, L \in V$ , the definition $f.\text{toLinear}_1(T, L)$ is the linear map $V \to \mathbb{Q}$ that sends a vector $S \in V$ to the scalar value $f(S, T, L)$ . This is obtained by fixing the second and third arguments of the trilinear form.

theorem

$f(S, T, L) = (f.\text{toLinear}_1(T, L))(S)$

#toLinear₁_apply

Let $V$ be a vector space over the field of rational numbers $\mathbb{Q}$ . For any symmetric trilinear form $f$ on $V$ and any vectors $S, T, L \in V$ , the value $f(S, T, L)$ is equal to the value of the linear map $f.\text{toLinear}_1(T, L)$ applied to $S$ . Here, $f.\text{toLinear}_1(T, L)$ represents the linear map $V \to \mathbb{Q}$ obtained by fixing $T$ and $L$ as the second and third arguments of the trilinear form.

theorem

$f(\sum S_i, T, L) = \sum f(S_i, T, L)$

#map_sum₁

Let $V$ be a vector space (or module) over the field of rational numbers $\mathbb{Q}$ . For any symmetric trilinear form $f$ on $V$ , any fixed vectors $T, L \in V$ , and any finite collection of vectors $S_i \in V$ indexed by $i \in \{0, \dots, n-1\}$ , the form satisfies linearity in its first argument: $f\left(\sum_{i} S_i, T, L\right) = \sum_{i} f(S_i, T, L)$ Here, the application $f(S, T, L)$ represents the evaluation of the trilinear form on the three vectors.

theorem

Linearity of symmetric trilinear forms in the second argument: $f(T, \sum S_i, L) = \sum f(T, S_i, L)$

#map_sum₂

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$ . For any symmetric trilinear form $f$ on $V$ , any fixed vectors $T, L \in V$ , and any finite collection of vectors $S_i \in V$ indexed by $i \in \{0, \dots, n-1\}$ , the form satisfies linearity in its second argument: $f\left(T, \sum_{i} S_i, L\right) = \sum_{i} f(T, S_i, L)$ Here, the application $f(T, S, L)$ represents the evaluation of the trilinear form on the three vectors.

theorem

Linearity of symmetric trilinear forms in the third argument: $f(T, L, \sum S_i) = \sum f(T, L, S_i)$

#map_sum₃

Let $V$ be a module over the rational numbers $\mathbb{Q}$ . For any symmetric trilinear form $f$ on $V$ , and for any vectors $T, L \in V$ and a finite collection of vectors $S_i \in V$ indexed by $i \in \{0, \dots, n-1\}$ , the form satisfies linearity in its third argument: $f(T, L, \sum_{i} S_i) = \sum_{i} f(T, L, S_i)$ Here, the curried application $f(T, L, S)$ represents the evaluation of the trilinear form on the three vectors.

theorem

$f(\sum S_i, \sum T_k, \sum L_l) = \sum_i \sum_k \sum_l f(S_i, T_k, L_l)$ for symmetric trilinear forms

#map_sum₁₂₃

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$ . For any symmetric trilinear form $f$ on $V$ and any finite collections of vectors $\{S_i\}_{i < n_1}$ , $\{T_k\}_{k < n_2}$ , and $\{L_l\}_{l < n_3}$ in $V$ , the form satisfies multilinearity across all three arguments simultaneously: $f\left(\sum_{i} S_i, \sum_{k} T_k, \sum_{l} L_l\right) = \sum_{i} \sum_{k} \sum_{l} f(S_i, T_k, L_l)$ where the application $f(S, T, L)$ represents the evaluation of the trilinear form on the three vectors.

definition

Homogeneous cubic map $f(S) = \tau(S, S, S)$ from a symmetric trilinear form $\tau$

#toCubic

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$ . Given a symmetric trilinear form $\tau$ on $V$ , this definition constructs a homogeneous cubic map $f: V \to \mathbb{Q}$ defined by evaluation on the diagonal: $f(S) = \tau(S, S, S)$ for any $S \in V$ . This map $f$ satisfies the homogeneity condition $f(a \cdot S) = a^3 f(S)$ for all scalars $a \in \mathbb{Q}$ and vectors $S \in V$ .

theorem

$f(S + T) = f(S) + f(T) + 3\tau(S, S, T) + 3\tau(T, T, S)$ for homogeneous cubic map $f$

#toCubic_add

Let $V$ be a module over the field of rational numbers $\mathbb{Q}$ . Let $\tau$ be a symmetric trilinear form on $V$ , and let $f: V \to \mathbb{Q}$ be the associated homogeneous cubic map defined by $f(X) = \tau(X, X, X)$ . For any elements $S, T \in V$ , the following identity holds: $f(S + T) = f(S) + f(T) + 3\tau(S, S, T) + 3\tau(T, T, S)$

Physlib.Mathematics.LinearMaps

Homogeneous quadratic equation $f: V \to \mathbb{Q}$ satisfying $f(a \cdot v) = a^2 f(v)$

Homogeneous quadratic equation as a function \( f: V \to \mathbb{Q} \)

$f(a \cdot S) = a^2 f(S)$ for a homogeneous quadratic map $f$

Symmetric bilinear form as a function $V \to (V \to_\ell \mathbb{Q})$

Symmetric bilinear map from first-factor linearity and $f(S, T) = f(T, S)$

$f(a \cdot S, T) = a \cdot f(S, T)$ for symmetric bilinear forms

$f(S, T) = f(T, S)$ for symmetric bilinear forms

$f(S, a \cdot T) = a \cdot f(S, T)$ for symmetric bilinear forms

$f(S_1 + S_2, T) = f(S_1, T) + f(S_2, T)$ for symmetric bilinear forms

$f(S, T_1 + T_2) = f(S, T_1) + f(S, T_2)$ for symmetric bilinear forms

The linear functional $f(\cdot, T)$ derived from a symmetric bilinear form $f$

$f(S, T) = f.\text{toLinear}_1(T)(S)$

$f(\sum S_i, T) = \sum f(S_i, T)$ for Symmetric Bilinear Forms

$f(T, \sum S_i) = \sum f(T, S_i)$ for Symmetric Bilinear Forms

Homogeneous quadratic map $v \mapsto \tau(v, v)$ from symmetric bilinear form $\tau$

$Q(S + T) = Q(S) + Q(T) + 2\tau(S, T)$ for the quadratic form $Q$ associated with symmetric bilinear form $\tau$

Homogeneous cubic map $V \to \mathbb{Q}$

A homogeneous cubic map $f$ as a function $V \to \mathbb{Q}$

$f(a \cdot S) = a^3 f(S)$ for a homogeneous cubic map $f$

A symmetric trilinear form $f$ as a function $V \to (V \to_{\mathbb{Q}} V \to_{\mathbb{Q}} \mathbb{Q})$

Symmetric trilinear map from first-factor linearity, $f(S, T, L) = f(T, S, L)$ , and $f(S, T, L) = f(S, L, T)$

$f(S, T, L) = f(T, S, L)$ for symmetric trilinear forms

$f(S, T, L) = f(S, L, T)$ for symmetric trilinear forms

$f(S, T, L) = f(L, T, S)$ for symmetric trilinear forms

$f(a \cdot S, T, L) = a f(S, T, L)$ for symmetric trilinear forms

$f(S, a \cdot T, L) = a f(S, T, L)$ for symmetric trilinear forms

$f(S, T, a \cdot L) = a f(S, T, L)$ for symmetric trilinear forms

Additivity of Symmetric Trilinear Forms in the First Argument

Additivity of Symmetric Trilinear Forms in the Second Argument

Additivity of Symmetric Trilinear Forms in the Third Argument

Linear map $S \mapsto f(S, T, L)$ from a symmetric trilinear form $f$

$f(S, T, L) = (f.\text{toLinear}_1(T, L))(S)$

$f(\sum S_i, T, L) = \sum f(S_i, T, L)$

Linearity of symmetric trilinear forms in the second argument: $f(T, \sum S_i, L) = \sum f(T, S_i, L)$

Linearity of symmetric trilinear forms in the third argument: $f(T, L, \sum S_i) = \sum f(T, L, S_i)$

$f(\sum S_i, \sum T_k, \sum L_l) = \sum_i \sum_k \sum_l f(S_i, T_k, L_l)$ for symmetric trilinear forms

Homogeneous cubic map $f(S) = \tau(S, S, S)$ from a symmetric trilinear form $\tau$

$f(S + T) = f(S) + f(T) + 3\tau(S, S, T) + 3\tau(T, T, S)$ for homogeneous cubic map $f$

Physlib.Mathematics.LinearMaps

Homogeneous quadratic equation f:V→Qf: V \to \mathbb{Q}f:V→Q satisfying f(a⋅v)=a2f(v)f(a \cdot v) = a^2 f(v)f(a⋅v)=a2f(v)

Homogeneous quadratic equation as a function \( f: V \to \mathbb{Q} \)

f(a⋅S)=a2f(S)f(a \cdot S) = a^2 f(S)f(a⋅S)=a2f(S) for a homogeneous quadratic map fff

Symmetric bilinear form as a function V→(V→ℓQ)V \to (V \to_\ell \mathbb{Q})V→(V→ℓ​Q)

Symmetric bilinear map from first-factor linearity and f(S,T)=f(T,S)f(S, T) = f(T, S)f(S,T)=f(T,S)

f(a⋅S,T)=a⋅f(S,T)f(a \cdot S, T) = a \cdot f(S, T)f(a⋅S,T)=a⋅f(S,T) for symmetric bilinear forms

f(S,T)=f(T,S)f(S, T) = f(T, S)f(S,T)=f(T,S) for symmetric bilinear forms

f(S,a⋅T)=a⋅f(S,T)f(S, a \cdot T) = a \cdot f(S, T)f(S,a⋅T)=a⋅f(S,T) for symmetric bilinear forms

f(S1+S2,T)=f(S1,T)+f(S2,T)f(S_1 + S_2, T) = f(S_1, T) + f(S_2, T)f(S1​+S2​,T)=f(S1​,T)+f(S2​,T) for symmetric bilinear forms

f(S,T1+T2)=f(S,T1)+f(S,T2)f(S, T_1 + T_2) = f(S, T_1) + f(S, T_2)f(S,T1​+T2​)=f(S,T1​)+f(S,T2​) for symmetric bilinear forms

The linear functional f(⋅,T)f(\cdot, T)f(⋅,T) derived from a symmetric bilinear form fff

f(S,T)=f.toLinear1(T)(S)f(S, T) = f.\text{toLinear}_1(T)(S)f(S,T)=f.toLinear1​(T)(S)

f(∑Si,T)=∑f(Si,T)f(\sum S_i, T) = \sum f(S_i, T)f(∑Si​,T)=∑f(Si​,T) for Symmetric Bilinear Forms

f(T,∑Si)=∑f(T,Si)f(T, \sum S_i) = \sum f(T, S_i)f(T,∑Si​)=∑f(T,Si​) for Symmetric Bilinear Forms

Homogeneous quadratic map v↦τ(v,v)v \mapsto \tau(v, v)v↦τ(v,v) from symmetric bilinear form τ\tauτ

Q(S+T)=Q(S)+Q(T)+2τ(S,T)Q(S + T) = Q(S) + Q(T) + 2\tau(S, T)Q(S+T)=Q(S)+Q(T)+2τ(S,T) for the quadratic form QQQ associated with symmetric bilinear form τ\tauτ

Homogeneous cubic map V→QV \to \mathbb{Q}V→Q

A homogeneous cubic map fff as a function V→QV \to \mathbb{Q}V→Q

f(a⋅S)=a3f(S)f(a \cdot S) = a^3 f(S)f(a⋅S)=a3f(S) for a homogeneous cubic map fff

A symmetric trilinear form fff as a function V→(V→QV→QQ)V \to (V \to_{\mathbb{Q}} V \to_{\mathbb{Q}} \mathbb{Q})V→(V→Q​V→Q​Q)

Symmetric trilinear map from first-factor linearity, f(S,T,L)=f(T,S,L)f(S, T, L) = f(T, S, L)f(S,T,L)=f(T,S,L), and f(S,T,L)=f(S,L,T)f(S, T, L) = f(S, L, T)f(S,T,L)=f(S,L,T)

f(S,T,L)=f(T,S,L)f(S, T, L) = f(T, S, L)f(S,T,L)=f(T,S,L) for symmetric trilinear forms

f(S,T,L)=f(S,L,T)f(S, T, L) = f(S, L, T)f(S,T,L)=f(S,L,T) for symmetric trilinear forms

f(S,T,L)=f(L,T,S)f(S, T, L) = f(L, T, S)f(S,T,L)=f(L,T,S) for symmetric trilinear forms

f(a⋅S,T,L)=af(S,T,L)f(a \cdot S, T, L) = a f(S, T, L)f(a⋅S,T,L)=af(S,T,L) for symmetric trilinear forms

f(S,a⋅T,L)=af(S,T,L)f(S, a \cdot T, L) = a f(S, T, L)f(S,a⋅T,L)=af(S,T,L) for symmetric trilinear forms

f(S,T,a⋅L)=af(S,T,L)f(S, T, a \cdot L) = a f(S, T, L)f(S,T,a⋅L)=af(S,T,L) for symmetric trilinear forms

Additivity of Symmetric Trilinear Forms in the First Argument

Additivity of Symmetric Trilinear Forms in the Second Argument

Additivity of Symmetric Trilinear Forms in the Third Argument

Linear map S↦f(S,T,L)S \mapsto f(S, T, L)S↦f(S,T,L) from a symmetric trilinear form fff

f(S,T,L)=(f.toLinear1(T,L))(S)f(S, T, L) = (f.\text{toLinear}_1(T, L))(S)f(S,T,L)=(f.toLinear1​(T,L))(S)

f(∑Si,T,L)=∑f(Si,T,L)f(\sum S_i, T, L) = \sum f(S_i, T, L)f(∑Si​,T,L)=∑f(Si​,T,L)

Linearity of symmetric trilinear forms in the second argument: f(T,∑Si,L)=∑f(T,Si,L)f(T, \sum S_i, L) = \sum f(T, S_i, L)f(T,∑Si​,L)=∑f(T,Si​,L)

Linearity of symmetric trilinear forms in the third argument: f(T,L,∑Si)=∑f(T,L,Si)f(T, L, \sum S_i) = \sum f(T, L, S_i)f(T,L,∑Si​)=∑f(T,L,Si​)

f(∑Si,∑Tk,∑Ll)=∑i∑k∑lf(Si,Tk,Ll)f(\sum S_i, \sum T_k, \sum L_l) = \sum_i \sum_k \sum_l f(S_i, T_k, L_l)f(∑Si​,∑Tk​,∑Ll​)=∑i​∑k​∑l​f(Si​,Tk​,Ll​) for symmetric trilinear forms

Homogeneous cubic map f(S)=τ(S,S,S)f(S) = \tau(S, S, S)f(S)=τ(S,S,S) from a symmetric trilinear form τ\tauτ

f(S+T)=f(S)+f(T)+3τ(S,S,T)+3τ(T,T,S)f(S + T) = f(S) + f(T) + 3\tau(S, S, T) + 3\tau(T, T, S)f(S+T)=f(S)+f(T)+3τ(S,S,T)+3τ(T,T,S) for homogeneous cubic map fff

Homogeneous quadratic equation $f: V \to \mathbb{Q}$ satisfying $f(a \cdot v) = a^2 f(v)$

$f(a \cdot S) = a^2 f(S)$ for a homogeneous quadratic map $f$

Symmetric bilinear form as a function $V \to (V \to_\ell \mathbb{Q})$

Symmetric bilinear map from first-factor linearity and $f(S, T) = f(T, S)$

$f(a \cdot S, T) = a \cdot f(S, T)$ for symmetric bilinear forms

$f(S, T) = f(T, S)$ for symmetric bilinear forms

$f(S, a \cdot T) = a \cdot f(S, T)$ for symmetric bilinear forms

$f(S_1 + S_2, T) = f(S_1, T) + f(S_2, T)$ for symmetric bilinear forms

$f(S, T_1 + T_2) = f(S, T_1) + f(S, T_2)$ for symmetric bilinear forms

The linear functional $f(\cdot, T)$ derived from a symmetric bilinear form $f$

$f(S, T) = f.\text{toLinear}_1(T)(S)$

$f(\sum S_i, T) = \sum f(S_i, T)$ for Symmetric Bilinear Forms

$f(T, \sum S_i) = \sum f(T, S_i)$ for Symmetric Bilinear Forms

Homogeneous quadratic map $v \mapsto \tau(v, v)$ from symmetric bilinear form $\tau$

$Q(S + T) = Q(S) + Q(T) + 2\tau(S, T)$ for the quadratic form $Q$ associated with symmetric bilinear form $\tau$

Homogeneous cubic map $V \to \mathbb{Q}$

A homogeneous cubic map $f$ as a function $V \to \mathbb{Q}$

$f(a \cdot S) = a^3 f(S)$ for a homogeneous cubic map $f$

A symmetric trilinear form $f$ as a function $V \to (V \to_{\mathbb{Q}} V \to_{\mathbb{Q}} \mathbb{Q})$

Symmetric trilinear map from first-factor linearity, $f(S, T, L) = f(T, S, L)$ , and $f(S, T, L) = f(S, L, T)$

$f(S, T, L) = f(T, S, L)$ for symmetric trilinear forms

$f(S, T, L) = f(S, L, T)$ for symmetric trilinear forms

$f(S, T, L) = f(L, T, S)$ for symmetric trilinear forms

$f(a \cdot S, T, L) = a f(S, T, L)$ for symmetric trilinear forms

$f(S, a \cdot T, L) = a f(S, T, L)$ for symmetric trilinear forms

$f(S, T, a \cdot L) = a f(S, T, L)$ for symmetric trilinear forms

Linear map $S \mapsto f(S, T, L)$ from a symmetric trilinear form $f$

$f(S, T, L) = (f.\text{toLinear}_1(T, L))(S)$

$f(\sum S_i, T, L) = \sum f(S_i, T, L)$

Linearity of symmetric trilinear forms in the second argument: $f(T, \sum S_i, L) = \sum f(T, S_i, L)$

Linearity of symmetric trilinear forms in the third argument: $f(T, L, \sum S_i) = \sum f(T, L, S_i)$

$f(\sum S_i, \sum T_k, \sum L_l) = \sum_i \sum_k \sum_l f(S_i, T_k, L_l)$ for symmetric trilinear forms

Homogeneous cubic map $f(S) = \tau(S, S, S)$ from a symmetric trilinear form $\tau$

$f(S + T) = f(S) + f(T) + 3\tau(S, S, T) + 3\tau(T, T, S)$ for homogeneous cubic map $f$