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From Mathsoc wiki

Ma224 Geometry

Lecturer: Prof. David Simms

Website: Link

These notes cover the 2007-2008 course; based on a glimpse of a blackboard the other day the course content may have now changed or at least been reordered. If you want more relevant notes, you can get some from Josh Tobin.

A brief explanation on the Einstein Summation Notation is given here.

1 Linear Operators
2 Linear Forms
- 2.1 Linear forms, dual space
- 2.2 Scalar Products
3 Adjoints
4 Tensors
- 4.1 Tensors
- 4.2 Skew-Symmetric Tensors, Determinants and Wedge Product
5 Pull-Back and Push-Forward
6 Orientation
- 6.1 Orientation
- 6.2 Hodge Star Operator
7 Continuity
8 Differentiability
- 8.1 Derivatives
- 8.2 Partial Derivatives
9 Coordinate Systems and Manifolds
10 Integration of Forms
- 10.1 Integration of Forms
- 10.2 Stokes' Theorem and the Poincare Lemma
11 Applications
12 References and Links

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Linear Operators

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K-vector Space, Linear Operators

$K$ -vector space: A finite dimensional vector space $M$ over a field $K$ is known as a $K$ -vector space.

Linear Operator: A map $T$ from $M$ to $M$ is a linear operator on $M$ if

$T(x +y) = T(x) + T(y)$
$T( \alpha x) = \alpha T(x)$

for all $x,y \in M$ and $\alpha \in K$

$K$ -algebra: The set of all linear operators on $M$ is called $\mbox{hom} \, M$ , and is a

i) $K$ -vector space

ii) ring

and thus is a $K$ -algebra - it has the following rules for addition, composition and scalar multiplication:

$\alpha (S + T) = \alpha S + \alpha T$
( $R + S)T = RT + ST$ , $R(S + T) = RS + RT$
$\alpha (ST) = (\alpha S)T = S (\alpha T)$

for $R,S,T \in \mbox{hom} M, \alpha \in K$ . An example of a $K$ -algebra would be that of $n \times n$ matrices.

Matrix of $T$ : Now let $M$ have finite dimension $n$ , and a basis $u_1, u_2, ... u_n$ (denoted $u_i$ for short), and let $T$ be a linear operator on $M$ . Then

$T u_j = \alpha_j^1 u_1 + \alpha_j^2 u_2 + ... + \alpha_j^n u_n = \alpha_j^i u_i$ where we will now use the Einstein summation convention - repeated indices are summed over.

We can thus form the matrix of $T$ with respect to the basis $u_i$ : $A = ( \alpha_j^i )$ , where $i$ signifies the row number and $j$ the column. The $j^{th}$ column of $A$ gives the coordinates of $Tu_j$ with respect to the basis $u_i$ . The mapping from $T$ to its matrix is a $K$ -algebra isomorphism, which depends on the choice of basis.

Transition Matrix: Suppose $w_i$ is a new basis, then $u_j = p_j^i w_i$ and $P = (p_j^i)$ is called the transition matrix from the old basis to the new basis. The $j^{th}$ column of $P$ consists of the new coordinates of the the old jth basis vector.

Conversely, we have $w_j = q_j^i u_i$ and a matrix $Q = (q_j^i) = P^{-1}$ . Now

$T w_j = T q_j^l u_l = q_j^l T u_l = q_j^l \alpha_l^k u_k = q_j^l \alpha_l^k p_k^i w_i = (p_k^i \alpha_l^k q_j^l)w_i$

Thus $T$ has the new matrix $PAP^{-1}$ with respect to the new basis $w_i$ .

Determinant: Note that

$\det PAP^{-1} = \det P \, \det A \, \det P^{-1} = \det A$

We thus define $\det T = \det A$ , well-defined and independent of the choice of basis.

Characteristic polynomial: The polynomial $p = \det (A - XI)$ is called the characteristic polynomial of $T$ , and is independent of choice of basis.

Theorem: (Hamilton-Cayley) The linear operator $T$ satisfies its own characteristic polynomial, i.e. $p(T) = 0$ .

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Eigenvectors and eigenspace

Eigenvector: A non-zero vector $x$ is called an eigenvector of $T$ if $Tx = \lambda x$ for $\lambda \in K$ an eigenvalue of $T$ corresponding to the eigenvector $x$ . Rearranging gives $(T - \lambda I)x = 0$ , hence $x$ is in the kernel of $(T - \lambda I)$ . This kernel is called the $\lambda$ -eigenspace if not equal to just the zero vector.

Generalised Eigenvector: We call $x$ a generalised eigenvector if there exists an integer $r>0$ with $(T - \lambda I)^r x = 0$ , and the kernel of $(T - \lambda I)^r$ is called a generalised eigenspace.

Diagonalisable: If $M$ has a basis $u_i$ of eigenvectors with eigenvalues $\lambda_i$ then

$T u_1 = \lambda_1 u_1 + 0 u_2 + \dots$

$T u_2 = 0 u_1 + \lambda_2 u_2 + \dots$

$\vdots$

$T u_n = 0 u_1 + 0 u_2 + \dots \lambda_n u_n$

and so $T$ has a diagonal matrix with the eigenvalues on the diagonal. With respect to the basis $u_i$ of eigenvalues $T$ is diagonalisable.

For a vector $x \in M$ we can write $x= \alpha^1 u_1 + \alpha^2 u_2 + \alpha^3 u_3 \dots$ $= x_1 + x_2 + \dots + x_k$ uniquely, where $x_i \in \lambda_i$ -eigenspace and $\lambda_1, \lambda_2 \dots \lambda_k$ is the set of distinct eigenvalues. We thus have the following in $M$ :

$(T - \lambda_1 I)(T - \lambda_2 I) \dots (T - \lambda_k I) x = (T - \lambda_1 I)(T - \lambda_2 I) \dots (T - \lambda_k I) [x_1 + x_2 + \dots + x_k] = 0$ , hence

$(T - \lambda_1 I)(T - \lambda_2 I) \dots (T - \lambda_k I)$ is a zero operator, denoted $f(T)$ where $f = (x - \lambda_1) \dots (x - \lambda_k)$

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Ideals and Polynomials

Ideal: Consider the $K$ -algebra $K[x]$ of polynomials. A subset $I$ of this algebra is called an ideal if:

i) $f,g \in I \Rightarrow f+g \in I$ (closed under addition)

ii) $f \in K[x], g \in I \Rightarrow fg \in I$ (closed under multiplication by polynomials in $K[x]$ )

We can choose the (non-zero) element of $I$ of minimal degree (denoting it by $q$ ) - then $q$ divides every element of $I$ and is the greatest common divisor (gcd) of $I$ .

Minimal Polynomial: Consider $T$ a linear operator on $M$ , then an ideal is:

$I = \lbrace f \in K[x] | f(T) = 0\rbrace$

with gcd $q$ . Then $q$ is the polynomial of minimal degree such that $q(T) = 0$ , and if $f(T) = 0$ then $q$ divides $f$ . $q$ is called the minimal polynomial of the operator $T$ . The eigenvalues of $T$ are the zeros of the minimal polynomial.

Reducible: A polynomial $f \in K[x]$ is called reducible if there exist $h,g \in K[x]$ such that $f = hg$ , with $h$ and $g$ having positive degree. Otherwise, it is called irreducible.

Unique Factorisation Theorem: For any $f \in K[x]$ with degree f > 0, then

$f = \alpha p_1 p_2 \dots p_k$

where $\alpha \in K$ and the $p_i$ are irreducible and monic (highest coefficient is 1). The Unique Factorisation Theorem for Polynomials in $K[x]$ states that this factorisation is unique (up to reordering the factors).

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Jordan Form

Primary Decomposition Theorem: If $T$ is a linear operator on $M$ which satisfies a polynomial equation with only linear factors

$(T - \lambda_1 1)^{r_1} \dots (T - \lambda_k 1)^{r_k} = 0$

then by the Primary Decomposition Theorem we can write $M$ as the direct sum of the generalised eigenspaces, that is:

$M = \mbox{ker} (T - \lambda_1 1)^{r_1} \oplus \dots \oplus \mbox{ker} (T - \lambda_k 1)^{r_k}$

Jordan string: A sequence of linearly independent vectors $u_1 \dots u_r$ is called a Jordan $\lambda$ -string if

$(T - \lambda 1) u_1 = u_2$

$(T - \lambda 1) u_2 = u_3$

$\vdots$

$(T - \lambda 1) u_{r-1} = u_r$

$(T - \lambda 1) u_r = 0$

Jordan Form: The above equations lead to a matrix of $T$ with $\lambda$ on the diagonal, ones just below the diagonal and elsewhere zero. This is called the $r \times r$ Jordan $\lambda$ -block. For each eigenvalue $\lambda$ the generalised $\lambda$ -eigenspace has a basis which is the union of a collection of Jordan $\lambda$ -strings. Now by the Primary Decomposition Theorem we can write $M$ as a direct sum of the generalised eigenspaces. Arranging the Jordan $\lambda$ -strings for each eigenvalue in descending order of length we can get a basis for $M$ with respect to which $T$ has a matrix $J$ with the Jordan blocks on the diagonal. $J$ is called the Jordan form of $T$ .

Diagonalisable: As a special case, an operator $T$ is diagonalisable if the space $M$ can be written as a direct sum of the eigenspaces (not generalised), and this is true if the minimal polynomial has no repeated factors.

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T-Invariance

T-Invariance: If $T$ is a linear operator on $M$ , and $N$ is a subspace of $M$ then $N$ is $T$ -invariant if $x \in N \Rightarrow Tx \in N$ . We denote by $T_N$ the map defined as $Tx : x \in N$ and called the restriction of $T$ to $N$ . If $T$ is diagonalisable, then so is $T_N$ .

Simultaneously Diagonalisable: We say that two operators $T$ and $S$ on $M$ are simultaneously diagonalisable if there is a basis of $M$ such that both $T$ and $S$ have diagonal matrices. It follows that $T$ and $S$ commute.

Conversely, if $S$ and $T$ are commuting linear operators on $M$ then each eigenspace of $S$ is $T$ -invariant, and vice versa, and if both $S$ and $T$ are diagonalisable then they are simultaneously diagonalisable.

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Linear Forms

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Linear forms, dual space

Linear Form: Consider $M$ an $n$ -dimensional $K$ -vector space. A scalar valued linear function $f$ on $M$ is called a linear form on $M$ .

Dual Space: We denote by $M^*$ the $K$ -vector space of all linear forms on $M$ and call it the dual space of $M$ . For $f \in M^*$ and $x \in M$ we write

$f(x) = \langle f,x \rangle$

for the value of $f$ on $x$ .

We have the following for $f, g \in M^*$ , $x,y \in M$ and $\alpha \in K$

$\langle f+g,x \rangle = \langle f,x \rangle + \langle g,x \rangle$

$\langle \alpha f,x \rangle = \alpha \langle f,x \rangle$

$\langle f, x + y \rangle = \langle f,x \rangle + \langle f,y \rangle$

$\langle f, \alpha x \rangle = \alpha \langle f,x \rangle$

The first two are due to the linearity of the first variable, and the latter two are due to the linearity of the second variable, giving a symmetry called duality. Hence the mapping $M^* \times M \rightarrow K$ is bilinear.

Coordinate functions: If $u_i$ a basis for $M$ then each $x \in M$ can be written as $x = \alpha^1 u_1 + \dots + \alpha^n u_n$ , so $\alpha^i$ is the $i^{th}$ coordinate with respect to the basis $u_i$ . We denote by $u^i$ the $i^{th}$ coordinate function with respect to the basis $u_i$ , given by $u^i \in M^*$ such that:

$\langle u^i, \alpha^1 u_1 + \dots + \alpha^n u_n \rangle = \alpha^i$ .

It follows that we can write any $x \in M$ as

$x = \langle u^i, x \rangle u_i$

summing over $i$ .

Basis for Dual Space: The $u^i$ form a basis for the dual space $M^*$ called the basis dual to $u_i$ .

Solution Space: Let $f^1, \dots ,f^n \in M^*$ . The set

$\lbrace x \in M: \langle f,x \rangle = 0, \dots \langle f_m, x \rangle = 0\rbrace$

is called the solution space of the system of homogeneous linear equations $f^1 = 0, \dots , f^m=0$ , and is a vector subspace of $M$ .

The space generated by $f^1, \dots ,f^n \in M^*$ is called the equation space, and its dimension is called the rank of the system of equations, and is equal to the number of linearly independent equations. If $M$ is an $n$ -dimensional vector space, then the dimension of the solution space is $n-r$ .

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Scalar Products

Scalar Product: A scalar product $( \, \cdot \, | \, \cdot \, )$ on $M$ is a map

$( \, \cdot \, | \, \cdot \, ): M\times M \rightarrow K$

which is linear in each variable. e.g. dot product.

Hermitian Scalar Product: If $M$ is a complex vector space, then we have instead a hermitian scalar product which is linear in the second variable, and has $\overline{(x\, |\, y)} = (y \,|\, x)$ (the overbar representing the complex conjugate). Hence also, $(x + y \,| \,z) = (x \,| \,z) + (y \,| \,z)$ and $( \alpha x\, | \,y) = \overline{\alpha} (x \,|\, y)$

Matrix of Scalar Product: If $u_i$ a basis for $M$ and $( \, \cdot \, | \, \cdot \, )$ a scalar product then $G = g_{ij}$ is the matrix of $( \, \cdot \, | \, \cdot \, )$ with respect to the basis $u_i$ , ie

$g_{ij} = ( u_i \, | \, u_j)$

Consider a new basis $w_i$ and transition matrix $P$ , with $P^{-1} = Q$ , then the matrix of the scalar product with respect to the new basis is given by

$G^' = Q^t G Q$ or $\overline{Q}^t G Q$ in the hermitian case.

Quadratic form: A quadratic form is the function $F: M \rightarrow K$ given by

$F(x) = ( x \,|\, x ) = g_{ij} u^i u^j$

Euclidean space: A real vector space $M$ is called a Euclidean space if a scalar product is given which is i) symmetric $( x \, |\, y) = (y \, |\, x)$ and ii) positive definite $(x \, | \, x) > 0$ for all $x \neq 0$ .

Hilbert Space: A complex vector space $M$ is called a Hilbert space if a hermitian scalar product is given which is positive definite.

Norm: We write $||x|| = \sqrt{( x\, | \, x)}$ for the norm (or length) of $x$ .

We have the following properties of norms:

$|| \alpha x|| = \alpha||x||$
$\frac{x}{|| x||}$ has norm 1 (for $x \neq 0$ )
$|( x\,|\, y)| \leq ||x|| \, ||y||$ (Cauchy-Schwarz inequality)
$|| x + y ||\leq ||x|| + || y ||$ (triangle inequality)

Raising and Lowering the Index: If $M$ has a scalar product $( \, \cdot \, | \, \cdot \, )$ we have a map

$M \rightarrow M^*, x \mapsto ( \, x \, | \, \cdot \, ) = f$

the operation of taking the scalar product with $x$ , known as lowering the index. If $x$ has components $\alpha^i$ then $f$ has components $\alpha_i = \alpha^i g_{ij}$ . The $\alpha^i$ are called the contravariant components of $x$ and the $\alpha_i$ are called the covariant components of $x$ .

If the map $M \rightarrow M^*$ , $x \mapsto ( \, \cdot \, | \, \cdot \, )$ is bijective (i.e. has kernel zero, ie $( x \, | \, \cdot ) = 0 \Leftrightarrow x = 0$ ) then we say $( \, \cdot \, | \, \cdot \, )$ is non-degenerate. Thus $G$ has an inverse, $G^{-1} = (g^{ij})$ , and we have the reverse operation to lowering the index, called raising the index: $\alpha^j = \alpha_i g^{ij}$

Orthogonal Complement: If $M$ has a scalar product $( \, \cdot \, | \, \cdot \, )$ and $N$ a vector subspace, we have

i) $N^{\bot} = \lbrace x \in M : (x\, | \, y) = 0 \, \forall y \in N \rbrace$ , the orthogonal complement of $N$ in $M$

ii)the restriction of $( \, \cdot \, | \, \cdot \, )$ on N given by $(x \, | \, y )_N = ( x \, | \, y) \forall x,y \in N$

If $( \, \cdot \, | \, \cdot \, )$ a scalar product on a space $M$ and $N$ a vector subspace such that $( \, \cdot \, | \, \cdot \, )_N$ is non-degenerate, then $M = N \oplus N^{\bot}$ , the direct sum of $N$ and its orthogonal complement.

If $( \, \cdot \, | \, \cdot \, )$ a symmetric or hermitian scalar product on $M$ which is not identically zero, then there exists some $x \in M$ such that $( \, x\, | \, x \, )$ is non-zero.

If $( \, \cdot \, | \, \cdot \, )$ a symmetric or hermitian scalar product on $M$ then $M$ has a basis of mutually orthogonal vectors (i.e. they form a diagonal matrix).

Sylvester's Theorem: The numbers of plus and minus signs in the associated quadratic form of a scalar product when diagonalised are uniquely determined.

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Adjoints

Adjoint: Let $( \, \cdot | \, \cdot \, )$ be a non-degenerate scalar product on a finite dimensional vector space $M$ . Then for each linear operator $T$ on $M$ , the adjoint of $T$ is the operator $T^*$ on $M$ given by:

$( T x \, | \, y ) = ( x \, |\, T^* y )$ for all $x,y \in M$

If $T$ has a matrix $A$ with respect to an orthonormal basis then $T^*$ has matrix $A^t$ in the Euclidean case, and $\overline{A}^t$ in the Hilbert space case.

If $N$ is a subspace of $M$ , and $N$ is invariant under $T$ then the orthogonal complement of $N$ is invariant under $T^*$ .

Self-adjoint: $T = T^*$ , and then all the eigenvalues of $T$ are real numbers.

Normal Operator: A normal operator commutes with its adjoint, $TT^* = T^*T$

Spectral Theorem: If $T$ is either a self-adjoint operator on a finite dimensional Euclidean space or a normal operator on a Hilbert space, then $M$ has an orthonormal basis of eigenvectors of $T$ .

In Quantum Mechanics: the states of a physical system are represented by non-zero vectors $x$ in a Hilbert space $M$ . Each observable state (energy, momentum etc.) is represented by a self-adjoint operator on $M$ .

An experiment to determine the values of an observable $T$ when the system is in state $x$ , with $||x|| = 1$ , will give an average or expected value

$\langle T \rangle = (x\, |\, Tx)$

and the spread of values or variance is

$(\Delta T)^2 = \bigg( x \, | \, (T - \langle T \rangle I )^2 x \bigg)$

Heisenberg Uncertainty Relation: If $P,Q$ self-adjoint operators on a Hilbert space satisfying the commutation relation $PQ - QP = \alpha \mathbb{I}$ then

$(\Delta P)(\Delta Q) \geq \frac{1}{2} | \alpha |$

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Tensors

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Tensors

Tensor: A scalar valued function

$T: M_1 \times \dots \times M_k \rightarrow K$

where each $M_i$ is $M$ or $M^*$ is called a tensor over $M$ if it is multilinear (linear in each variable separately).

If (say) $T: M \times M^* \times M \rightarrow K$ is a tensor, and if $u_i$ is a basis for $M$ then the array of scalars

$\alpha_{i k}^j = T( u_i, u^j, u_k )$

is called the components of $T$ with respect to the basis $u_i$ . $T$ is uniquely determined by its components.

If $p_j^i$ the transition matrix to a new matrix $w_i$ , with inverse matrix $q_j^i$ then $T$ has new components

$T ( w_i, w^j, w_k ) = T (q_i^r u_r, p_s^j u^s, q_k^t u_t ) = q_i^r p_s^j q_k^t \alpha_{i k}^j$

hence each lower (covariant) index transforms by $Q$ , and each upper (contravariant) index transforms by $P$ .

Two tensors $S$ and $T$ have the same type if they are defined on the same set : $M_1 \times \dots \times M_k$ . The set of all such tensors is a $K$ -vector space of dimension $n^k$ .

Tensor Product: If $S: M_1 \times \dots \times M_k \rightarrow K$ , $T: M_{k+1} \times \dots \times M_l \rightarrow K$ tensors over $M$ , we define their tensor product $S \otimes T$ to be the tensor $S \otimes T: M_1 \times \dots M_l \rightarrow K$ given by

$S \otimes T (x_1 \dots x_k, x_{k+1} \dots x_l) = S(x_1 \dots x_k) T ( x_{k+1} \dots x_l)$

The components of $S \otimes T$ are the product of the components of $S$ and $T$ . The tensor product is bilinear and associative, but in general not commutative.

Contraction: Let $T: M_1 \times \dots M_r \times \dots \times M_s \times \dots \times M_k \rightarrow K$

be a tensor, with the $r^{th}$ index upper, $M_r = M^*$ , and the $s^{th}$ index lower, $M_s = M$ . Then we define a new tensor with $M_r$ and $M_s$ omitted by

$S( x_1 \dots x_{k-2} ) = T ( x_1 \dots u^i \dots u_i x_{k-2} )$

and we call $S$ the contraction of $T$ with respect to the $r^{th}$ and $s^{th}$ indices. Contraction is well-defined (independent of choice of basis).

If $T$ has components $\alpha^{i \,\,\, rs}_{jk}$ say then contracting the second and fourth indices gives a tensor with components $\beta^{i\,\,s}_k = \alpha^{i \,\,\, js}_{jk}$

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Skew-Symmetric Tensors, Determinants and Wedge Product

Permutation: A bijective map $\phi: \lbrace 1,2 \dots r \rbrace \rightarrow \lbrace 1,2 \dots r \rbrace$ is called a permutation of degree $r$ . The group $S_r$ of all permutations of degree $r$ is called the symmetric group of degree $r$ , order $r!$

If $\phi \in S_r$ , we put $\epsilon^{\phi} = \epsilon_{\phi} = \Bigg \lbrace \begin{matrix} 1 & \mbox{if} \, \phi \, \mbox{even} \\ -1 & \mbox{if} \, \phi \, \mbox{odd} \end{matrix}$

If $1 \leq i_1 \dots i_n \leq n$ , we have $\epsilon^{i_1 \dots i_n} = \epsilon_{i_1 \dots i_n} = \Bigg \lbrace \begin{matrix} 1 & \mbox{if} \,\, i_1 \dots i_n \, \mbox{even permutation of} \,\, 1 \dots n \\ -1 & \mbox{if}\, \, i_1 \dots i_n \, \mbox{odd permutation of} \,\, 1 \dots n \\ 0 & \mbox{otherwise} \end{matrix}$

We denote by $T^rM = M^* \otimes \dots \otimes M^*$ the space of all tensors $M \times \dots \times M \rightarrow K$ , where there are $r$ occurrences of $M^*$ and $M$ . If $u_i$ a basis for $M$ then $u^{i_1} \otimes \dots \otimes u^{i_r}$ is a basis for $T^rM$ .

For each $\phi \in S_r$ and each $T \in T^r M$ , we define $\phi \cdot T \in T^r M$ by

$(\phi \cdot T) (x_1 \dots x_r) = T ( x_{\phi(1)} \dots x_{\phi(r)} )$

Skew-Symmetric: A tensor $T \in T^r M$ is skew-symmetric if $\phi \cdot T = \epsilon^\phi T$ for all $\phi \in S_r$

Determinant: The unique skew-symmetric tensor $D: K^n \times \dots \times K^n \rightarrow K$ (n Ks) such that $D(e_1 \dots e_n) = 1$ is called the determinant on $K^n$

If $A = (\alpha_j^i) \in K^{n\times n}$ has columns $c_1 \dots c_n \in K^n$ , then $D(c_1 \dots c_n)$ is the called the determinant of the matrix A. We have that $D(A x_1 \dots A x_n) = \mbox{det} A \, D(x_1 \dots x_n)$ , if $A,B$ are $n \times n$ matrices then $\det(AB) = \det A \det B$ , $A$ is invertible iff $\det A$ is non-zero, and if $A$ invertible and $Ax = b$ , then

$x_i = \frac{D ( c_1 \dots b \dots c_n)}{\det A}$ (Cramer's rule, $b$ replacing the $i^{th}$ column.)

Skew-Symmetriser: The linear operator $A: T^r M \rightarrow T^r M$ defined by

$AT = \frac{1}{r!} \sum_{\phi \in S_r} \epsilon^\phi \phi \cdot T$

is called the skew-symmetriser. It satisfies:

$A[(AS)\otimes T] = A[ S \otimes T ] = A [S \otimes (AT) ]$

$A(S\otimes T) = (-1)^{st} A (T \otimes S)$

Wedge Product: If $S \in T^s M$ and $T \in T^t M$ we define their wedge product by:

$S \wedge T = \frac{1}{s! t!} \sum_{\phi \in S_{s+t}} \epsilon^{\phi} \phi \cdot (S \otimes T) = \frac{(s + t)!}{s! t!} A(S \otimes T)$

The wedge product is bilinear, associative and supercommutative: $S\wedge T = (-1)^{st} T \wedge S$ .

We denote by $M^{(r)}$ the vector space of all skew-symmetric tensors of the type $M \times \dots \times M \rightarrow K$ (r Ms), and by $M_{(r)}$ the vector space of all skew-symmetric tensors of the type $M^* \times \dots \times M^* \rightarrow K$ (r M*s). Then if $u_i$ is a basis for $M$ , we have that:

$M^{(r)} = \lbrace 0 \rbrace$ , $M_{(r)} = 0$ if $r > n$

$\lbrace u^{i_1} \wedge \dots \wedge u^{i_r} \rbrace_{i_1 < \dots < i_r}$ is a basis for $M^{(r)}$ and $\lbrace u_{i_1} \wedge \dots \wedge u_{i_r} \rbrace_{i_1 < \dots < i_r}$ is a basis for $M_{(r)}$

The dimension of $M^{(r)}$ is the same as that of $M_{(r)}$ , and equals $\frac{n!}{r!(n-r)!}$

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Pull-Back and Push-Forward

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Pull-Back and Push-Forward

Pull-back: Let $M,N$ be finite dimensional vector spaces, and $T$ a linear operator from $M$ to $N$ . We define the pull-back

$T^*: M^* \leftarrow N^*$

$\langle T^* f, x \rangle = \langle f, Tx \rangle$

We let

$T_*: M \rightarrow N$

be the pull-back of $T^*$ , then we have $T_* = T$

Push-Forward and Pull-Back: More generally, we define the push-forward

$T_*: M \! \otimes \dots \otimes \! M \rightarrow N \! \otimes \dots \otimes \! N$ (r of each vector space)

and the pull-back

$T^*: M^* \!\otimes \dots \otimes \! M^* \leftarrow N^* \! \otimes \dots \otimes \! N^*$ (r of each vector space)

$(T_*S) \left[ f^1 \dots f^r \right] = S \left[ T^* f_1, \dots , T^* f^r \right]$ (push-forward)

and

$(T^*S) \left[ x_1 \dots x_r \right] = S \left[ T x_1, \dots T x_r \right]$ (pull-back)

where $S$ is a tensor of rank $r$ .

Commutative Diagrams: If is a commutative diagram of linear maps then so

$\langle (UT)^*f, x \rangle = \langle f, UTx \rangle = \langle U^*f, Tx \rangle = \langle T^* U^*f, x \rangle$

meaning $(UT)^* = T^* U^*$ , so we again have a commutative diagram.

Hence the mapping

$T \mapsto T^*$

is a contravariant functor from the category $K$ -vect fd of finite dimensional vector spaces to the category $K$ -vect fd.

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Categories & Functors

Category: A category is defined as:

a collection of objects
for each ordered pair of objects $(M,N)$ a set Hom $(M,N)$ is given, called the morphism from $M$ to $N$ .
for each morphism $f$ from $L$ to $M$ , $g$ from $M$ to $N$ a composition $(gf)$ from $L$ to $N$ is given and

is called a commutative digram.

composition of morphisms is associative
for each object $M$ , there is a morphism from $M$ to $M$ called the identity on $M$ .

An example is the category $K$ -vector field of $K$ -vector spaces, for which the morphisms are the linear operators acting on and between the vector spaces.

Isomorphic: Two objects $M,N$ in a category are isomorphic if for a morphism $f$ between $M$ and $N$ , there exists an inverse morphism between them (i.e. if the morphism is bijective).

Functor: A covariant/contravariant functor $F$ from a category $C_1$ to a category $C_2$ is a map

$M \mapsto F(M)$

from the objects of the first category to the objects of the second, and a map $T$ to $\begin{cases} T^* \\ T_* \end{cases}$ from the morphisms of $C_1$ to the morphisms of $C_2$ such that

$T\!: \! M \rightarrow N$

maps to

$T_* \!\!: \!F(M) \rightarrow F(N)$ (covariant case)

$T^* \!\!: \!F(M) \leftarrow F(N)$ (contravariant case)

such that

the identity maps to the identity.
If

is a commutative diagram then

(covariant case) is a commutative diagram, i.e. $(UT)_* = U_* T_*$ (note order preserved)

and

(contravariant case) is a commutative diagram i.e. $(UT)^* = T^* U^*$ (note order reversed)

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Properties of the Pull-Back and Push-Forward

Is a Functor: If $T: M \rightarrow N$ a linear map on $M,N$ finite dimensional vector spaces, then for each integer $r \geq 1$ , the push-forward

$T_* \!: M \! \otimes \dots \otimes \! M \rightarrow N \! \otimes \dots \otimes \! N$ (r of each space)

is a covariant functor from K-vect fd to K-vect fd, and the pull-back

$T_* \!: M^* \! \otimes \dots \otimes \! M^* \leftarrow N^* \! \otimes \dots \otimes \! N^*$ (r of each space)

is a contravariant functor.

Preservation: The pull-back and push-forward preserve tensor products, skew-symmetry and wedge products.

Given a vector space $M$ of finite dimension $n$ , we have that $M_{(r)}$ is the space of skew-symmetric tensors of the type

$M^* \! \times \dots \times \! M^* \rightarrow K$

and we take the direct sum

$M_{(*)} = M_{(0)} \oplus M_{(1)} \oplus \dots \oplus M_{(r)} \oplus \dots \oplus M_{(n)}$

a vector space of dimension $2^n$ . The wedge product makes $M_{(*)}$ into an associative K-algebra, and each linear operator from $M$ to $N$ gives a homomorphism of $K$ -algebras.

Also, on $M_{(n)}$ the action of the push-forward $T_*$ is multiplication by $\det T$ , as is the action of the pull-back $T^*$ on $M^{(n)}$ .

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Orientation

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Orientation

Orientation: Let $M$ be a finite dimensional real vector space, and $P = ( p_j^i)$ be the transition matrix from basis $u_1 \dots u_n$ to basis $w_1 \dots w_n$ , so $u_j = p_j^i w_i$ , then

$u_1 \wedge \dots \wedge u_n = \det P \, w_1 \wedge \dots \wedge w_n$ .

We say that $u_1 \dots u_n$ has the same orientation as $w_1 \dots w_n$ if $\det P>0$ ; otherwise they have opposite orientation. "Same orientation as" is an equivalence relation on the set of all bases of $M$ and there are exactly two equivalence classes.

Oriented: We call $M$ oriented if one of the equivalence classes is chosen as positively oriented, and the other as negatively oriented. e.g. the usual orientation of $\mathbb{R}^n$ designates $e_1 \dots e_n$ as positive.

Standard Basis: A basis is a standard basis if it is positively oriented, and the matrix of the scalar product has plus or minus one on the diagonal, and zero elsewhere.

Volume Form: The $n$ -form $u^1 \wedge \dots \wedge u^n$ is independent of the choice of standard basis $u_1 \dots u_n$ , and is called the volume form (vol) of $M$ . Thus,

$\mbox{vol}(a_1 \dots a_n) =$ volume of the parellopiped spanned by the vectors $a_1 \dots a_n$ and if the scalar product has components $g_{ij} = ( w_i \, | \, w_j )$ with respect to a positively oriented basis $w_1 \dots w_n$ , then

$\mbox{vol} = \sqrt{| \det g_{ij} |} w^1 \wedge \dots \wedge w^n$

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Hodge Star Operator

Hodge Star Operator: Let $M$ be a real $n$ -dimensional oriented vector space with a non-degenerate symmetric scalar product. Then for each $0 \leq r \leq n$ we define the Hodge Star operator

$*: M^{(r)} \rightarrow M^{(n-r)} \,\,\, , \,\,\, \omega \mapsto * \omega$

$* \omega (v_1 \dots v_{n-r}) = \frac{1}{r!} \omega (u_{i_1} \dots u_{i_r} ) \mbox{vol} (u^{i_1} \dots u^{i_r} u^{i_{r+1}}\dots u^{i_n}) ( u_{i_{r+1}} \, | \, v_1 ) \dots (u_{i_n} \, | \, v_{n-r} )$

If $u_1 \dots u_n$ a standard basis for $M$ , then

$* u^1 \wedge \dots \wedge u^r = s_{r+1} \dots s_n u^{r+1} \wedge \dots \wedge u^n$

where $s_i = ( u_i | u_i ) = \pm 1$

The Hodge star may also be defined by contraction as

$(\star \omega)_{i_1 \dots i_{n-r}} = \frac{1}{r!} \varepsilon_{j_1 \dots j_r i_1 \dots i_{n-r}} \omega^{j_1 \dots j_r}$

where $\varepsilon_{i_1 \dots i_n}$ is the completely antisymmetric epsilon symbol ( $\epsilon_{12\dots n} = 1$ and other components by skew-symmetry). Note that here you would use a metric tensor to raise and lower the indices - this is equivalent to the inner products used in the above definition to introduce signs corresponding to the metric signature.

What this means: The dual tensor on the left has $n-r$ indices, which fix $n-r$ indices in the Levi-Civita epsilon tensor. The remaining $r$ indices are summed over, and so all permutations of these occur on the right hand side - that is, $r!$ permutations. However all of these permutations can be transformed back to an even permutation of the $r$ indices, giving a sum of $r!$ copies of an element of the original tensor $\omega$ , this element determined by the permutation of indices. (This works if the space is oriented, I think this is necessary as you need two sign changes coming from odd permutations to get all positive terms in the sum... i.e. not only does the epsilon tensor change sign but you have some basis element implied by the indices of the tensor, which reorients... I think this makes sense. Actually it probably doesn't.)

So the action of the Hodge star can be thought of as a cyclic permutation of the indices of the tensor.

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Continuity

Norm: A norm (length function) on $M$ is a function mapping from $M$ to the real numbers such that:

$|| x + y || \leq || x || + ||y||$ (triangle inequality)
$||\alpha x|| = |\alpha| ||x||$
$||x|| \geq 0$
$||x|| = 0 \Leftrightarrow x =0$

A real or complex vector space with a chosen norm is called a normed vector space.

Ball: Let $X \subset M$ , $a \in X$ , $r > 0$ , then we define the ball of radius $r$ in $X$ centred at $a$ by:

$B_X (a, r) = \lbrace x \in X | \, ||x - a || < r \rbrace$

Open: A set $V \subset X$ is called open in $X$ if for each $a \in V$ there is a positive $r$ such that $B_X (a,r) \subset V$ i.e. each point of $V$ is an interior point. We have that $B_X (a,r)$ is open in $X$ .

Continuous: A map $f: X \rightarrow Y$ , where $X \subset M, Y \subset N$ both normed vector spaces, is called continuous at $a \in X$ if for each $\epsilon > 0$ there is a $\delta > 0$ such that:

$f \left[ B_X (a,\delta) \right] \subset B_Y ( f(a), \epsilon)$ i.e. all the points in the ball in $X$ map into the ball in $Y$ .

If $f: M \supset X \rightarrow Y \subset N$ then $f$ is continuous at $a$ if and only if for each $V$ open in $Y$ such that $f(a) \in V$ there exists a $W$ open in $X$ such that $fW \subset V$ .

Usual Topology: The collections of open sets of $X$ a subset of a finite dimensional real or complex vector space is called the usual topology on $X$ and is independent of choice of norm.

Topology on X: Let $X$ be a set. A topology on $X$ is a collection of subsets of $X$ called the open sets of the topology such that

the empty set and $X$ are both open
If $\lbrace V_i \rbrace _{i \in I}$ a collection of sets $V_i$ open in $X$ then the union of the sets $U V_i$ is open in $X$ .
If $V_1 \dots V_k$ a finite collection of sets $V_i$ open in $X$ then $V_1 \cap V_2 \cap \dots \cap V_k$ is open in $X$ .

Topological Space: A set $X$ together with a topology on $X$ is called a topological space.

Continuous at $a$ : A map $f$ from $X$ to $Y$ , both topological spaces with $a \in X$ is continuous at $a$ if for each $V$ open in $Y$ such that $f(a) \in V$ there exists a $W$ open in $X$ such that $a \in W$ and $fW \subset V$ .

$f$ is continuous if it is continuous at $a$ for all $a \in X$ .

If $X$ a topological space and if $V \subset X$ , and if for each $a \in V$ there exists an open $W$ such that $a \in W_a \subset V$ then $V$ is open.

$f$ is continuous if and only if $V$ open in $Y$ implies $f^{-1} V$ open in $X$

$f,g$ continuous implies $gf$ continuous.

We thus have a category top whose objects are topological spaces and whose morphisms are the continuous maps $f$ . In the calculus, we consider the category whose objects are open subsets of finite dimensional real or complex vector spaces, and whose morphisms are continuous maps. The isomorphisms of these categories are known as homeomorphisms.

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Differentiability

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Derivatives

Differentiability: Let $f$ map from $V \subset M$ to $W \subset N$ where $V,W$ are open subsets of finite dimensional real or complex spaces $M,N$ . Let $a \in V$ then $f$ is differentiable at $a$ if there exists a linear operator

$f'(a): M \rightarrow N$

called the derivative of $f$ at $a$ , such that

$f(a + h) = f(a) + f'(a) h + \phi (h)$

where $\frac{|| \phi(h) ||}{||h||} \rightarrow 0$ as $||h|| \rightarrow 0$ . Here $f'(a) h$ is a linear approximation to $f(a+h) - f(a)$ and $\phi(h)$ is a remainder term.

The derivative $f'(a)$ is uniquely determined by the formula

$f'(a) h = \lim_{t \rightarrow 0} \frac{f(a+th) - f(a)}{t} = \frac{d}{dt} f(a+th) \Bigg|_{t=0}$

which is the directional derivatve of $f$ at $a$ along $h$

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Partial Derivatives

Function of $n$ independent variables: Let $f: R^n \supset V \rightarrow R$ with $V$ open in $R^n$ , then we call $f$ a real-valued function of $n$ independent variables.

Partial Derivative: If $a = (a_1, \dots , a_n) \in V$ and $x = (x_1, \dots ,x_n)$ , $f(x) = f(x_1, \dots , x_n)$ , then we define

$\frac{\partial f}{\partial x^j} (a) = \frac{\partial f}{\partial x^j} (a_1, \dots, a_n) = \lim_{t \rightarrow 0} \frac{f (a_1, \dots a_j + t , \dots a_n) - f( a_1 \dots a_j \dots a_n)}{t} = \lim_{t \rightarrow 0} \frac{f(a + t e_j) - f(a)}{t} = \frac{d}{dt} f(a + t e_j) \Bigg|_{t=0}$

= the directional derivative of $f$ at $a$ along $e_j$ . This is called the partial derivative of $f$ at $a$ with respect to the $j^{th}$ usual coordinate function $x^j$ .

If $f$ is an operator from an open subset of $R^n$ to $R^m$ , then $f'(a): R^n \rightarrow R^m$ is the $m \times n$ matrix

$f'(a) = \left(\frac{\partial f^i}{\partial x^j} (a) \right)$

where $i$ runs from $1$ to $m$ and $j$ from $1$ to $n$ .

Operator Norms: Let $T: M \rightarrow N$ be a linear operator on $M,N$ finite dimensional normed vector spaces. We write

$||T|| = \sup_{||u|| = 1} ||Tu||$

called the operator norm of $T$ . This satisfies:

$||S + T|| \leq ||S|| + ||T||$
$||\alpha T|| = |\alpha| ||T||$
$||Tx|| \leq ||T|| ||x||$
$||ST|| \leq ||S|| ||T||$
$||T|| \geq 0$ with equality if and only if $T = 0$ .

Chain Rule for Functions on Finite Dimensional Real or Complex Vector Space: Let $g: U \rightarrow V$ , $f: V \rightarrow W$ and $f \cdot g: U \rightarrow W$ where $U,V,W$ are open subsets of finite dimensional real or complex vector spaces. If $g$ is differentiable at $a$ and $f$ is differentiable at $g(a)$ then $f \cdot g$ is differentiable at $a$ and

$(f \cdot g)' (a) = f'\Big(g(a)\Big) g'(a)$

which is the chain rule for functions on finite dimensional real or complex vector spaces. Note that the order cannot be interchanged as this is an operator product.

C^r: If $f$ is $r$ times differentiable and its $r^{th}$ derivative is continuous then we say $f$ is a $C^r$ function. If the $r^{th}$ derivative exists for all $r$ , then we say that $f$ is $C^{\infty}$ or smooth.

For each $r$ we have a category whose objects are open subsets of finite dimensional real or complex vector spaces, and whose morphisms are $C^r$ functions. The isomorphisms in this category are called $C^r$ diffeomorphisms.

Given $f: R^n \supset V \rightarrow W \subset R^m$ where $f(x_1 \dots x_n) = \Big( f^1 (x_1 \dots x_n) \dots f^m (x_1 \dots x_n) \Big)$ then $f$ is $C^1$ if and only if $f' = \left( \frac{\partial f^i}{\partial x^j}\right)$ (this an $m \times n$ matrix) exists and is continuous, ie $\frac{\partial f^i}{\partial x^j}$ all exist and are continuous.

Order of differentiation: If $f: R^n \supset V \rightarrow W \subset R$ is $C^2$ then

$\frac{\partial^2 f}{\partial x^i \partial x^j} = \frac{\partial^2 f}{\partial x^j \partial x^i}$

Mean Value Theorem for Functions on Finite Dimensional Normed Spaces: Let $f: M \supset V \rightarrow N$ be $C^1$ , and

$x,y \in V$ such that

$\left[ x,y \right] = \lbrace tx + (1 -t)y | 0 \leq t \leq 1 \rbrace \subset V$

Let $||f'\left[tx + (1 -t)y\right] || \leq k$ for all t between 0 and 1, then $||f(x) - f(y)|| \leq k ||x - y||$

Inverse Function Theorem: Let $f:M \supset V \rightarrow N$ be a $C^r$ function on an open subset $V$ , where $M,N$ are finite dimensional real or complex vector spaces. Let $a \in V$ at which $f'(a): M \rightarrow N$ is invertible, then there exists an open neighbourhood $W$ of $a$ such that $f: W \rightarrow f(W)$ is a $C^r$ diffeomorphism onto open $f(W)$ in $N$ .

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Coordinate Systems and Manifolds

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Coordinate Systems and Manifolds

Coordinate System: Let $V$ be an open set of a topological space $X$ . An $n$ -dimensional coordinate system on $X$ with domain $V$ is a homeomorphism $y$ of $V$ onto on open set $y(V)$ in $R^n$ ,

$y: X \supset V \rightarrow y(V) \subset R^n$

$y(x) = \Big(y^1(x), \dots, y^n(x)\Big)$

and $y^i: V \rightarrow R$ is called the $i^{th}$ coordinate function of the coordinate system $y$ .

Manifold: A topological space $X$ is called an $n$ -dimensional manifold if for each $a \in X$ there exists an $n$ -dimensional coordinate system with domain a neighbourhood of $a$ .

Let $y = \Big(y^1(x), \dots, y^n(x)\Big)$ be coordinates with domain $V$ and let $f: V \rightarrow R$ . Then there exists a unique $F$ with domain $y(V)$ such that

$f(x) = F \Big(y^1(x), \dots, y^n(x) \Big) \forall x \in V$

thus each function $f$ on $V$ can be written as a unique function of the coordinates. $f$ is called a continuous/ $C^r$ / smooth function if $F$ is continuous/ $C^r$ / smooth.

$C^r$ compatible: Let $y=(y^i)$ with domain $V$ and $z=(z^i)$ with domain $W$ be $n$ -dimensional coordinate systems on $X$ . We say that $y$ and $z$ are $C^r$ -compatible if $y^i$ is a $C^r$ function of $z^1 \dots z^n$ and $z^i$ is a $C^r$ function of $y^1 \dots y^n$ on $V \cap W$

$C^r$ manifold: A topological space $X$ is called a $C^r$ -manifold if a collection of mutually $C^r$ -compatible coordinate systems are given whose domains cover $X$ . Such a collection is called an atlas on $X$ . Similarly for $C^{\infty}$ or smooth manifold.

From now on, manifold means smooth manifold and coordinate system means a system $C^{\infty}$ -compatible with the system's given atlas.

Implicit Function Theorem: Let $f= \Big(f^1 \dots f^l\Big)$ be $C^r$ real-valued functions on an open set $V$ in $R^n$ , so

$f: R^n \supset V \rightarrow R^l$

and

$f^{\prime}: R^n \rightarrow R^l$ an $l \times n$ matrix, and let

$X =\lbrace x \in V | f(x) = 0 \rbrace$

be the space of solution of the $l$ equations, $f^1 =0, \dots , f^l =0$ . Let $a \in X$ be a point at which rank $f^{\prime}(a) = l$ with (say) the first $l$ columns of $f^{\prime}(a)$ being linearly independent, then there exists an open neighbourhood $U$ of $a$ in $X$ such that

$x^{l+1}, \dots , x^n$

are coordinates on $X$ with domain $U$ and $x^1 \dots x^l$ are $C^r$ functions of $x^{l+1} \dots x^n$ on $U$

Thus, if $X = \lbrace x | f(x) = 0, \mbox{rank} \, f^{\prime} (x) = l \rbrace$ then $X$ is an $n-l$ dimensional $C^r$ manifold.

Coordinates at $a$ : Let $X$ be a smooth manifolds and $a \in X$ . Coordinates $y=(y^i)$ are called coordinates at $a$ if their domain is a neighbourhood of $a$ .

Smooth at $a$ : A real-valued function $f$ is called smooth at $a$ if its domain is a neighbourhood of $a$ and there exists coordinates $y^i$ at $a$ such $f= F(y^1,\dots, y^n)$ on a neighbourhood of $a$ with $F$ $C^{\infty}$ . $f$ is called smooth if it is smooth at $a$ for all $a \in V$ , where $V$ is an open subset of the manifold $X$ .

Partial Derivative: If $f$ is smooth at $a$ , $y^i$ coordinates at $a$ with $f=F(y^1, \dots , y^n)$ on a neighbourhood of $a$ we write

$\frac{\partial f}{\partial y^j}(a) = \frac{\partial F}{\partial x^j} \left( y^1(a), \dots , y^n(a) \right)$

called the partial derivative of $f$ at $a$ with respect to the $j^{th}$ coordinate with respect to coordinates $y^1, \dots , y^n$ .

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Tangent Vectors

Parametrised Path: A parametrised path $\alpha$ in $X$ , domain $U$ open in $\mathbb{R}$ ,

$\alpha: \mathbb{R} \supset U \rightarrow X, t \mapsto \alpha(t)$

is called smooth if $y^i\Big(\alpha(t)\Big)$ is a $C^{\infty}$ function of $t$ (when defined) for any coordinates $(y^i)$ on $X$ .

Tangent Vector: For each parameter $t \in U$ define an operator $\dot{\alpha}(t)$ acting on functions $f$ smooth at $\alpha(t)$ by

$\dot{\alpha}(t) f = \frac{d}{dt} f \Big(\alpha(t)\Big)$

= rate of change of $f$ along $\alpha$ at parameter $t$ . $\dot{\alpha}(t)$ is called a tangent vector to $X$ at the point $\alpha(t)$ .

Tangent Space: The set of all tangent vectors to $X$ at $a$ is called the tangent space to $X$ at $a$ , denoted $T_aX$ . Thus $\dot{\alpha}(t) \in T_{\alpha(t)}X$ , and is called the velocity vector of $\alpha$ at parameter $t$ .

If $y=(y^i)$ coordinates at $a$ , then

$\frac{\partial}{\partial y^1_{\,\,a}} \dots \frac{\partial}{\partial y^n_{\,\,\, a}}$

is a basis for $T_aX$ .

If $z^i$ another coordinate system at $a$ then the transition matrix on changing from $y^i$ to $z^i$ is given by $\frac{\partial z^i}{\partial y^j}(a)$ .

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Differentials, Push-forward and Pull-Back, Tensor Fields

Differential: If $f$ smooth at $a \in X$ a smooth manifold then the differential of $f$ at $a$ , denoted $df_a$ is the linear form on $T_aX$ given by

$\langle df_a,v \rangle = vf = \dot{\alpha}(t) f = \frac{d}{dt} f\Big(\alpha(t)\Big)$

for $v \in T_aX, v = \alpha(t)$ say. Thus $df_a \in T_a^*X$ the dual of $T_aX$ , and measures the rate of change of $f$ at $a$ . If $y^i$ coordinates at $a$ then $dy^1_{\,\,\, a}, \dots dy^n_{\,\,\, a}$ are the basis of the dual space, dual to the basis $\frac{\partial}{\partial y^1_{\,\,\ a}} \dots \frac{\partial}{\partial y^n_{\,\,\, a}}$ of $T_aX$ .

Scalar Field: Let $X$ be a smooth manifold, $V$ open in $X$ . A smooth real-valued function $f: x \mapsto f(x)$ with domain $V$ is called a scalar field.

We denote by $C^{\infty}(V)$ the set of all scalar fields with domain $V$ , and this is an $\mathbb{R}$ -algebra.

Smooth Map and Pull-Back: A map $\phi: X \rightarrow Y$ of smooth manifolds is called smooth if i) $V$ open in $Y \Rightarrow \phi^{-1} V$ open in $X$ , and ii) $f \in C^{\infty}(V) \Rightarrow \phi^*f = f \circ \phi \in C^{\infty}(\phi^{-1}V)$ .

$\phi^* f = f \circ \phi$ is called the pull-back of $f$ under $\phi$

Differentiable 1-form: Let $X$ be a smooth manifold, $V$ open in $X$ . A differentiable 1-form $\omega$ is a function on $V$ such that

$x \mapsto \omega_x \in T_x^* X$

If $\omega$ a 1-form with domain $V$ , $y^i$ coordinates with domain $W$ then for each $x \in V \cap W$ , we have $dy^1_x , \dots , d y^n_x$ a basis for $T_x^* X$ , ie

$\omega = \omega_1 dy^1 + \dots + \omega_n dy^n$ on $V\cap W$

We denote by $\Omega^1(V)$ the set of 1-forms with domain $V$ , and for each $f \in C^{\infty}(V)$ have $df \in \Omega^1(V)$ so we have a linear map

$d: C^{\infty}(V) \rightarrow \Omega^1(V)$ called the differential.

Tangent Bundle: If $X$ a smooth manifold we write

$TX = \bigcup_{x \in X} T_xX$

the set of all tangent vectors to $X$ , called the tangent bundle of $X$ .

Push-forward: If $\phi: X \rightarrow Y$ smooth we define the pushforward

$\phi_*: TX \rightarrow TY$

$[ \phi_* v ] f= \frac{d}{dt} f \Big(\phi(\alpha(t))\Big) = \frac{d}{dt} (\phi^* f) (\alpha(t)) = \dot{\alpha}(t) [ \phi^* f ] = v [\phi^* f]$

for all $f$ smooth at $\phi(\alpha)$ , $v = \dot{\alpha}(t) \in T_x X$ , $x\in X$ . So we have

$\phi_*$ [ velocity vector of $\alpha(t)$ ] = velocity vector of $\phi\Big(\alpha(t)\Big)$

Pull-back of 1-forms: If $\phi: X \rightarrow Y$ a smooth map, $\omega$ a differential 1-form on $Y$ with domain $V$ open, we define $\phi^* \omega$ to be the differential 1-form on $X$ with domain $\phi^{-1}V$ , given by

$\langle (\phi^* \omega)_x, v \rangle = \langle \omega_{\phi(x)}, \phi_* v \rangle$

for each $v \in T_xX$ , $x \in X$ . This is called the pull-back of $\omega$ under $\phi$ .

The pull-back commutes with differentials, $\phi^* df = d \phi^* f$ .

Chain Rule for Maps of Manifolds: We have

$(\psi \cdot \phi)_* = \psi_* \cdot \phi_*$

$(\psi \cdot \phi)^{\prime} (x) = \psi^{\prime} (\phi(x)) \phi^{\prime}(x)$

Tensor Fields: A tensor field $S$ on a smooth manifold $X$ with domain $V$ open in $X$ is a function which to each $x \in V$ assigns a tensor $S_{x}$ over $T_{x}X$ , of fixed type.

Differential (Exterior Derivative): Let $y^i$ be coordinates with domain $V$ on a smooth $n$ -dimensional manifold $X$ , then for each integer $r\geq 0$ define the linear operator

$d: \Omega^r (V) \rightarrow \Omega^{r+1}(V)$

as follows: i) if $f$ a scalar field, $f \in \Omega^0(V)$ then $df \in \Omega^1 (V)$

ii) if $\omega \in \Omega^r(V)$ ,

$\omega = \sum_{i_i < \dots < i_r} \omega_{i_1 \dots i_r} dy^{i_1} \wedge \dots \wedge dy^{i_r}$

say, then,

$d\omega = \sum_{i_i < \dots < i_r}d \omega_{i_1 \dots i_r}\wedge dy^{i_1} \wedge \dots \wedge dy^{i_r}$

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Submanifolds

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Integration of Forms

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Integration of Forms

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Stokes' Theorem and the Poincare Lemma

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Applications

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References and Links

The main references would be Simms' lecture notes from class and the notes for his previous courses:

Some people used Linear Algebra and its Applications by Gilbert Strang for revision of first year material. The books Modern Geometry - Methods and Applications (Part I) by Dubrovin, Fomenko and Novikov, Tensor Analysis on Manifolds by Bishop and Goldberg and Calculus on Manifolds by Spivak might possibly be slightly useful, or at least interesting. There is a very good explanation of tensors, forms and the wedge product in these notes by Prof. J. Binney of Oxford. Examination-wise, it's all about the past papers, and this might be useful:

A Sample Exam Generator

and these could be helpful too:

Jordan Form Solutions

Don't forget that any number of questions may be attempted...

Retrieved from "http://www.maths.tcd.ie/~mathsoc/wiki/2321_2322"

Categories: Group II courses | Mathematics courses

2321 2322

From Mathsoc wiki

Contents

Linear Operators

K-vector Space, Linear Operators

Eigenvectors and eigenspace

Ideals and Polynomials

Jordan Form

T-Invariance

Linear Forms

Linear forms, dual space

Scalar Products

Adjoints

Tensors

Tensors

Skew-Symmetric Tensors, Determinants and Wedge Product

Pull-Back and Push-Forward

Pull-Back and Push-Forward

Categories & Functors

Properties of the Pull-Back and Push-Forward

Orientation

Orientation

Hodge Star Operator

Continuity

Differentiability

Derivatives

Partial Derivatives

Coordinate Systems and Manifolds

Coordinate Systems and Manifolds

Tangent Vectors

Differentials, Push-forward and Pull-Back, Tensor Fields

Submanifolds

Integration of Forms

Integration of Forms

Stokes' Theorem and the Poincare Lemma

Applications

Vector Analysis in R^3

Geometry of Classical Mechanics

Geometry of Complex Analysis

Geometry of Critical Points of Functions Subject to Constraints

References and Links

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