Here are the fully comprehensive and concise notes on all the essential material from the sources, structured for maximum memorisation and exam preparation.
Part 1: Linear Algebra
1. Vector Spaces & Bases
- Vector Space Properties: Closed under addition and scalar multiplication 1-3.
- Linear Independence: Vectors \(u_1, u_2, ..., u_m\) are linearly independent if \(a_1u_1 + a_2u_2 + ... + a_mu_m = 0\) is only satisfied when all scalars \(a_i = 0\) 4, 5.
- Basis & Dimension: A basis is a linearly independent set of vectors that spans the space; the number of vectors in the basis is the dimension of the space 5. Crucial Fact: Any \(n\) vectors of dimension \(d\) where \(d < n\) must be linearly dependent 6.
2. Matrices & Determinants
- Matrix Multiplication: \((AB)_{ij} = \sum_k A_{ik} B_{kj}\) 7, 8. Matrices generally do not commute: \(AB \neq BA\) 9.
- Transpose Properties: \((A^T)_{ij} = A_{ji}\) 10. Important rule to memorise: \((AB)^T = B^T A^T\) 11.
- Trace: \(\text{Tr}(A)\) is the sum of diagonal elements. Exhibits cyclic permutation: \(\text{Tr}(ABC) = \text{Tr}(BCA) = \text{Tr}(CAB)\) 12, 13.
- Hermitian Matrices: A matrix with complex entries is Hermitian if \(A = A^\dagger\), where \(A^\dagger\) is the complex conjugate of the transpose 13.
Determinant Properties
- 3x3 determinant equals the scalar triple product of its columns 14.
- Can be calculated using the Levi-Civita symbol: \(\det A = \sum_{ijk} \epsilon_{ijk} A_{1i} A_{2j} A_{3k}\) 15, 16.
- Row Operations: Swapping two rows flips the sign; multiplying a row by \(\lambda\) multiplies the determinant by \(\lambda\); adding a multiple of one row to another leaves the determinant unchanged 17.
- \(\det(A) = \det(A^T)\) and \(\det(AB) = \det(A)\det(B)\) 18, 19.
Inverse
- Defined by \(A^{-1}A = I\) 20.
- Formula using cofactors \((C_{i,j})\): \((A^{-1})_{ij} = C_{j,i} / \det(A)\) 21.
- Property to memorise: \((A^T)^{-1} = (A^{-1})^T\) 22.
3. Linear Simultaneous Equations (\(Ax = y\)) & The Kernel
- Kernel: The set of vectors \(x\) where \(Ax = 0\) 23. The kernel is non-trivial, meaning it contains non-zero solutions, if and only if the columns of \(A\) are linearly dependent. For a square matrix this means \(\det A = 0\) 24, 25.
- Existence of Solutions: If the rows of \(A\) are linearly independent, no equations are inconsistent and at least one solution always exists 26, 27.
- Uniqueness of Solutions: If the columns of \(A\) are linearly dependent, the kernel is non-trivial, so you will have either multiple solutions or no solutions. You only get unique solutions if columns are linearly independent 28, 29.
4. Linear Maps & Orthogonal Matrices
- Linear Maps: A map \(f\) where \(f(v_1 + v_2) = f(v_1) + f(v_2)\) and \(f(\lambda v) = \lambda f(v)\). Any \(m \times n\) matrix represents a linear map from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) 30, 31.
- Orthogonal Matrices: Defined by \(O^T O = I\), hence \(O^{-1} = O^T\). Their determinants are strictly \(\pm 1\), and their columns form an orthonormal set of vectors 32, 33. They geometrically represent rotations and reflections 34, 35.
5. Eigenvalues & Eigenvectors
- Defined by \(Av = \lambda v\) 36.
- Characteristic Polynomial: Found by solving \(P_A(\lambda) = \det(A - \lambda I) = 0\) 37.
- Trace and Determinant Rules: For an \(n \times n\) matrix, \(\det(A) = \lambda_1 \lambda_2 ... \lambda_n\) and \(\text{Tr}(A) = \lambda_1 + \lambda_2 + ... + \lambda_n\) 38.
Real Symmetric Matrices: Essential Memorisation
- All eigenvalues are purely real 39, 40.
- Eigenvectors corresponding to different eigenvalues are completely orthogonal 40, 41.
- You can always form an orthonormal basis from the eigenvectors, allowing the matrix to be diagonalised: \(A = S D S^T\), where \(S\) is orthogonal and \(D\) is diagonal 42-45.
6. The Hessian Matrix (\(H\))
Used to classify stationary points in multivariable functions 46, 47. Look at the eigenvalues of \(H\):
- Local minimum: All eigenvalues of \(H\) are positive 48, 49.
- Local maximum: All eigenvalues of \(H\) are negative 48, 49.
- Saddle point: \(H\) has a mix of positive and negative eigenvalues 48, 49.
Part 2: Partial Differential Equations (PDEs)
1. The Big Three PDEs To Recognise
- Diffusion / Heat Equation: \(\frac{\partial \phi}{\partial t} = D\nabla^2\phi\) 50, 51.
- Laplace's Equation: \(\nabla^2\phi = 0\), describing steady-state diffusion/heat 52.
- Wave Equation: \(\frac{\partial^2 \phi}{\partial t^2} - c_0^2\nabla^2\phi = 0\), where \(c_0\) is the wave speed 52.
2. Classification of 2nd Order Linear PDEs
For a general PDE of form: \(a \frac{\partial^2\psi}{\partial x^2} + 2b \frac{\partial^2\psi}{\partial x\partial y} + c \frac{\partial^2\psi}{\partial y^2} = 0\):
- Elliptic: \(b^2 - ac < 0\), e.g. Laplace's equation 53, 54.
- Hyperbolic: \(b^2 - ac > 0\), e.g. Wave equation 54.
- Parabolic: \(b^2 - ac = 0\), e.g. Diffusion equation 55.
3. Boundary Conditions & Superposition
- Dirichlet: Prescribes the value of the function on the boundary 56.
- Neumann: Prescribes the normal derivative \(\vec{n} \cdot \nabla f\) on the boundary 56.
- Homogeneous: The prescribed value is zero. Principle of Superposition: If \(\phi_1\) and \(\phi_2\) both solve a linear homogeneous equation and satisfy homogeneous boundary conditions, their sum \(\phi_1 + \phi_2\) also does 56, 57.
4. Solving PDEs: Key Techniques
- Characteristic Variables: For homogeneous PDEs without \(t\), solve the quadratic \(ap^2 + 2bp + c = 0\) for roots \(p_+\) and \(p_-\). The general solution is \(f(y + p_+x) + g(y + p_-x)\) 58, 59.
- Separation of Variables: Assume a separable solution, e.g. \(\phi(x,y) = X(x)Y(y)\) 60.
- Substitute into the PDE and separate variables so each side depends only on one variable, setting them equal to a constant \(k\) 60, 61.
- Determine the sign of \(k\) based on the homogeneous boundary conditions, e.g. you need sines/cosines to hit zero at boundaries 62-64.
- Use the Principle of Superposition to sum over all valid solutions, usually an infinite series 65, 66.
- Use the inhomogeneous boundary condition and Fourier series methods to find the specific constants of the general solution 66-68.
Wave Solutions
- Travelling Waves (Infinite string): Expressible as \(y(x,t) = f(x - ct) + g(x + ct)\), representing waves moving right and left respectively without changing shape 69, 70.
- Standing Waves (Finite string): Formed by superimposing left and right travelling waves, evaluated via Fourier series using initial displacement \(Y(x)\) and initial velocity \(V(x)\) 71-73.