diff --git a/docs/mathematics/differential-geometry/differential-manifolds.md b/docs/mathematics/differential-geometry/differential-manifolds.md index 3f71cea..5ec0ac1 100644 --- a/docs/mathematics/differential-geometry/differential-manifolds.md +++ b/docs/mathematics/differential-geometry/differential-manifolds.md @@ -1,6 +1,6 @@ # Differential manifolds -In the following sections of differential geometry we make use of the Einstein summation convention introduced in [vector analysis](/en/physics/mathematical-physics/vector-analysis/curvilinear-coordinates/) and $\mathbb{K} = \mathbb{R}$ or $\mathbb{K} = \mathbb{C}.$ +In the following sections of differential geometry we make use of the Einstein summation convention and $\mathbb{K} = \mathbb{R}$ or $\mathbb{K} = \mathbb{C}$. ## Definition @@ -38,4 +38,4 @@ The last axiom ensures that any chart is tacitly assumed to be already contained To clarify the definitions, a passive transformation corresponds only to a descriptive transformation. Whereas an active transformation corresponds to a transformation on the manifold $M$. -A passive transformation may also be given directly by $\phi_\beta \circ \phi_\alpha: x \mapsto y$ since $\psi = \mathrm{id}$ in this case. Note that the definitions could also have been given by the inverse as the transformations are all diffeomorphisms. \ No newline at end of file +A passive transformation may also be given directly by $\phi_\beta \circ \phi_\alpha: x \mapsto y$ since $\psi = \mathrm{id}$ in this case. Note that the definitions could also have been given by the inverse as the transformations are all diffeomorphisms. diff --git a/docs/mathematics/linear-algebra/dual-vector-spaces.md b/docs/mathematics/linear-algebra/dual-vector-spaces.md index 16d2a0f..c46fc45 100644 --- a/docs/mathematics/linear-algebra/dual-vector-spaces.md +++ b/docs/mathematics/linear-algebra/dual-vector-spaces.md @@ -1,6 +1,6 @@ # Dual vector spaces -We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n.$ In the following sections we make use of the Einstein summation convention introduced in [vector analysis](/en/physics/mathematical-physics/vector-analysis/curvilinear-coordinates/) and $\mathbb{K} = \mathbb{R} \lor\mathbb{K} = \mathbb{C}$. +We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n.$ In the following sections we make use of the Einstein summation convention introduced and $\mathbb{K} = \mathbb{R} \lor\mathbb{K} = \mathbb{C}$. > *Definition 1*: let $\mathbf{\hat f}: V \to \mathbb{K}$ be a **covector** or **linear functional** on $V$ if for all $\mathbf{v}_{1,2} \in V$ and $\lambda, \mu \in \mathbb{K}$ we have > @@ -54,4 +54,4 @@ From theorem 1 it follows that for each covector basis $\{\mathbf{\hat e}^i\}$ o ??? note "*Proof*:" - Will be added later. \ No newline at end of file + Will be added later. diff --git a/docs/mathematics/linear-algebra/eigenspaces.md b/docs/mathematics/linear-algebra/eigenspaces.md index b096b48..75a213c 100644 --- a/docs/mathematics/linear-algebra/eigenspaces.md +++ b/docs/mathematics/linear-algebra/eigenspaces.md @@ -42,7 +42,7 @@ Furthermore it follows from the definition that any linear combination of eigenv A \mathbf{x} - \lambda \mathbf{x} = (A - \lambda I) \mathbf{x} = \mathbf{0}, $$ - which implies that $(A - \lambda I)$ is singular and $\det(A - \lambda I) = 0$ by [definition](../determinants/#properties-of-determinants). + which implies that $(A - \lambda I)$ is singular and $\det(A - \lambda I) = 0$ by [definition](determinants.md#properties-of-determinants). The eigenvalues $\lambda$ may thus be determined from the **characteristic polynomial** of degree $n$ that is obtained from $\det (A - \lambda I) = 0$. In particular, the eigenvalues are the roots of this polynomial. @@ -110,7 +110,7 @@ The complex conjugate of an eigenvector of $A$ is also an eigenvector of $A$ wit ??? note "*Proof*:" - Let $A$ be a $n \times n$ matrix and let $\lambda_1, \dots, \lambda_n \in \mathbb{K}$ be the eigenvalues of $A$. It follows from the [fundamental theorem of algebra](../../number-theory/complex-numbers/#roots-of-polynomials) that + Let $A$ be a $n \times n$ matrix and let $\lambda_1, \dots, \lambda_n \in \mathbb{K}$ be the eigenvalues of $A$. It follows from the [fundamental theorem of algebra](../number-theory/complex-numbers.md#roots-of-polynomials) that $$ \det (A - \lambda I) = (\lambda_1 - \lambda)(\lambda_2 - \lambda) \cdots (\lambda_n - \lambda), @@ -182,7 +182,7 @@ for $k \in \mathbb{K}$. ### Hermitian case -The following section is for the special case that a matrix is [Hermitian](../matrices/matrix-arithmatic/#hermitian-matrix). +The following section is for the special case that a matrix is [Hermitian](matrices/matrix-arithmetic.md#hermitian-matrix). > *Theorem 7*: the eigenvalues of a Hermitian matrix are real. @@ -264,4 +264,4 @@ The factorization $A = U T U^H$ is often referred to as the *Schur decomposition ??? note "*Proof*:" - Will be added later. \ No newline at end of file + Will be added later. diff --git a/docs/mathematics/linear-algebra/orthogonality.md b/docs/mathematics/linear-algebra/orthogonality.md index 936a8c2..fd74c62 100644 --- a/docs/mathematics/linear-algebra/orthogonality.md +++ b/docs/mathematics/linear-algebra/orthogonality.md @@ -46,7 +46,7 @@ for all $\mathbf{v} \in S$, and hence $\mathbf{u}_1 + \mathbf{u}_2 \in S^\perp$. ### Fundamental subspaces -Let $V$ be an Euclidean inner product space $V = \mathbb{R}^n$ with its inner product defined by the [scalar product](../inner-product-spaces/#euclidean-inner-product-spaces). With this definition of the inner product on $V$ the following theorem may be posed. +Let $V$ be an Euclidean inner product space $V = \mathbb{R}^n$ with its inner product defined by the [scalar product](inner-product-spaces.md#euclidean-inner-product-spaces). With this definition of the inner product on $V$ the following theorem may be posed. > *Theorem 1*: let $A$ be an $m \times n$ matrix, then > @@ -118,7 +118,7 @@ Known as the fundamental theorem of linear algebra. Which can be used to prove t S^\perp = R(X^T)^\perp = N(X), $$ - from the [rank nullity theorem](../vector-spaces/#rank-and-nullity) it follows that + from the [rank nullity theorem](vector-spaces.md#rank-and-nullity) it follows that $$ \dim S^\perp = \dim N(X) = n - r. @@ -464,4 +464,4 @@ $$ \mathbf{\hat x} = (A^T A)^{-1} A^T \mathbf{b}, $$ -is the unique solution of the normal equations for $A$ nonsingular and consequently, the unique least squares solution of the system $A \mathbf{x} = \mathbf{b}$. \ No newline at end of file +is the unique solution of the normal equations for $A$ nonsingular and consequently, the unique least squares solution of the system $A \mathbf{x} = \mathbf{b}$. diff --git a/docs/mathematics/linear-algebra/tensors/tensor-formalism.md b/docs/mathematics/linear-algebra/tensors/tensor-formalism.md index 0b17f59..6a2723d 100644 --- a/docs/mathematics/linear-algebra/tensors/tensor-formalism.md +++ b/docs/mathematics/linear-algebra/tensors/tensor-formalism.md @@ -1,6 +1,6 @@ # Tensor formalism -We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n$ and a corresponding dual space $V^*$ with a basis $\{\mathbf{\hat e}^i\}.$ In the following sections we make use of the Einstein summation convention introduced in [vector analysis](/en/physics/mathematical-physics/vector-analysis/curvilinear-coordinates/) and $\mathbb{K} = \mathbb{R} \lor\mathbb{K} = \mathbb{C}.$ +We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n$ and a corresponding dual space $V^*$ with a basis $\{\mathbf{\hat e}^i\}.$ In the following sections we make use of the Einstein summation convention and $\mathbb{K} = \mathbb{R} \lor\mathbb{K} = \mathbb{C}$. ## Definition @@ -239,4 +239,4 @@ with $u^j = g^{ij} u_i$. > \mathbf{g}(\mathbf{e}^i) = g_{ij} \mathbf{\hat e}^j. > $$ -So far, a vector space $V$ and its associated dual space $V^*$ have been introduced as a priori independent entities. An inner product provides us with an explicit mechanism to construct a bijective linear mapping associated with each vector by virtue of the metric. \ No newline at end of file +So far, a vector space $V$ and its associated dual space $V^*$ have been introduced as a priori independent entities. An inner product provides us with an explicit mechanism to construct a bijective linear mapping associated with each vector by virtue of the metric. diff --git a/docs/physics/classical-mechanics/hamiltonian-mechanics/equations-of-hamilton.md b/docs/physics/classical-mechanics/hamiltonian-mechanics/equations-of-hamilton.md index 6e600a0..fc7d76e 100644 --- a/docs/physics/classical-mechanics/hamiltonian-mechanics/equations-of-hamilton.md +++ b/docs/physics/classical-mechanics/hamiltonian-mechanics/equations-of-hamilton.md @@ -78,7 +78,7 @@ To put it differently; a Hamiltonian of a conservative autonomous system is cons Will be added later. -It may be observed that the Hamiltonian $\mathcal{H}$ and [generalised energy](/en/physics/mechanics/lagrangian-mechanics/lagrange-generalizations/#the-generalized-energy) $h$ are identical. Note however that $\mathcal{H}$ must be expressed in $(\mathbf{q},\mathbf{p},t)$ which is not the case for $h$. +It may be observed that the Hamiltonian $\mathcal{H}$ and [generalised energy](../lagrangian-mechanics/lagrange-generalizations.md#the-generalized-energy) $h$ are identical. Note however that $\mathcal{H}$ must be expressed in $(\mathbf{q},\mathbf{p},t)$ which is not the case for $h$. > *Proposition 2*: a coordinate $q_j$ is cyclic if > diff --git a/docs/physics/classical-mechanics/hamiltonian-mechanics/hamiltonian-formalism.md b/docs/physics/classical-mechanics/hamiltonian-mechanics/hamiltonian-formalism.md index 0e34b65..6c0609d 100644 --- a/docs/physics/classical-mechanics/hamiltonian-mechanics/hamiltonian-formalism.md +++ b/docs/physics/classical-mechanics/hamiltonian-mechanics/hamiltonian-formalism.md @@ -1,8 +1,8 @@ # Hamiltonian formalism of mechanics -The Hamiltonian formalism of mechanics is based on the definitions posed by [Lagrangian mechanics](/en/physics/mechanics/lagrangian-mechanics/lagrangian-formalism) and the axioms, postulates and principles posed in the [Newtonian formalism](/en/physics/mechanics/newtonian-mechanics/newtonian-formalism/). +The Hamiltonian formalism of mechanics is based on the definitions posed by [Lagrangian mechanics](../lagrangian-mechanics/lagrangian-formalism.md) and the axioms, postulates and principles posed in the [Newtonian formalism](../newtonian-mechanics/newtonian-formalism.md). -Where the Lagrangian formalism used the [principle of virtual work](/en/physics/mechanics/lagrangian-mechanics/lagrange-equations/#principle-of-virtual-work) to derive the Lagrangian equations of motion, the Hamiltonian formalism will derive the Lagrangian equations with the stationary action principle. A derivative of Fermat's principle of least time. +Where the Lagrangian formalism used the [principle of virtual work](../lagrangian-mechanics/lagrange-equations.md#principle-of-virtual-work) to derive the Lagrangian equations of motion, the Hamiltonian formalism will derive the Lagrangian equations with the stationary action principle. A derivative of Fermat's principle of least time. In Hamilton's formulation the stationary action principle is referred to as Hamilton's principle. diff --git a/docs/physics/classical-mechanics/lagrangian-mechanics/lagrangian-formalism.md b/docs/physics/classical-mechanics/lagrangian-mechanics/lagrangian-formalism.md index 42ae32b..0102263 100644 --- a/docs/physics/classical-mechanics/lagrangian-mechanics/lagrangian-formalism.md +++ b/docs/physics/classical-mechanics/lagrangian-mechanics/lagrangian-formalism.md @@ -1,6 +1,6 @@ # Lagrangian formalism of mechanics -The Lagrangian formalism of mechanics is based on the axioms, postulates and principles posed in the [Newtonian formalism](/en/physics/mechanics/newtonian-mechanics/newtonian-formalism/). +The Lagrangian formalism of mechanics is based on the axioms, postulates and principles posed in the [Newtonian formalism](../newtonian-mechanics/newtonian-formalism.md). ## Configuration of a system @@ -72,4 +72,4 @@ for all $t \in \mathbb{R}$ (inexplicitly). ??? note "*Proof*:" - Will be added later. \ No newline at end of file + Will be added later. diff --git a/docs/physics/classical-mechanics/newtonian-mechanics/energy.md b/docs/physics/classical-mechanics/newtonian-mechanics/energy.md index afcca88..07a4dfa 100644 --- a/docs/physics/classical-mechanics/newtonian-mechanics/energy.md +++ b/docs/physics/classical-mechanics/newtonian-mechanics/energy.md @@ -2,7 +2,7 @@ ## Potential energy -> *Definition 1*: a force field $\mathbf{F}$ is conservative if it is [irrotational](../../mathematical-physics/vector-analysis/vector-operators/#potentials) +> *Definition 1*: a force field $\mathbf{F}$ is conservative if it is [irrotational](../../mathematical-physics/vector-analysis/vector-operators.md#potentials) > > $$ > \nabla \times \mathbf{F} = 0,