Appendix C Notation
The following table defines the notation used in this book. Page numbers or references refer to the first appearance of each symbol.
| Symbol | Description | Location |
|---|---|---|
| \(x'(t) = f(t, x), x(0) = x_0\) | first-order initial value problem | Subsection 1.1.1 |
| \(\dfrac{dP}{dt} = k \left( 1 - \dfrac{P}{N} \right) P\) | logistic population model | Subsection 1.1.2 |
| \(mx'' + bx' + kx = 0\) | simple damped harmonic oscillator | Subsection 1.1.3 |
| \(\dfrac{dy}{dx} =M(x) N(y)\) | separable differential equation | Subsection 1.2.1 |
| \(\dfrac{dx}{dt} + p(t) x = q(t)\) | first-order linear differential equation | Subsection 1.5.2 |
| \(\dfrac{dx}{dt} = f_\lambda(x)\) | one-parameter family | Subsection 1.7.2 |
| \({\mathbf x}(t)\) | vector-valued function | Subsection 2.2.2 |
| \(\dfrac{d {\mathbf x}}{dt} = {\mathbf f}(t, {\mathbf x}), {\mathbf x}(t_0) = {\mathbf x}_0\) | vector form of a system | Subsection 2.3.2 |
| \(\dfrac{dx}{dt} = f(x),\dfrac{dy}{dt} = g(x, y)\) | partially coupled system | Subsection 2.4.1 |
| \(\dfrac{d \mathbf x}{dt} = A {\mathbf x}\) | matrix notation for a system | Section 3.1 |
| \(A^{-1}\) | inverse of a matrix \(A\) | Subsection 3.1.1 |
| \(\det(A)\) | determinant of \(A\) | Subsection 3.1.1 |
| \(\mathbf x^T\) | matrix transpose | Subsection 3.1.2 |
| \(\det(A - \lambda I) = \lambda^2 - (a + d) \lambda + (ad - bc)\) | characteristic polynomial | Subsection 3.1.3 |
| \(\trace(A)\) | trace of \(A\) | Exercises 3.1.6 |
| \(I\) | identity matrix | Exercises 3.1.6 |
| \(e^{i \beta t} = \cos \beta t + i \sin \beta t\) | Euler’s formula | Subsection 3.4.1 |
| \({ \mathbf x}_{\text{Re}}\) | real part of a complex number or vector | Subsection 3.4.1 |
| \({ \mathbf x}_{\text{Im}}\) | imaginary part of a complex number or vector | Subsection 3.4.1 |
| \(\overline{\lambda}\) | complex conjugate | Subsection 3.4.3 |
| \(e^A\) | matrix exponential | Subsection 3.9.1 |
| \(e^A\) | matrix exponential | Subsection 3.9.1 |
| \(a(t) x'' + b(t) x' + c(t) x = g(t)\) | second-order linear differential equation | Section 4.1 |
| \(p(\lambda) = \det(A - \lambda I) = \lambda^2 + \frac{b}{a} \lambda + \frac{c}{a}\) | characteristic polynomial | Subsection 4.1.2 |
| \(W[f, g](t)\) | Wronskian | Exercises 4.2.7 |
| \(\omega_0\) | natural frequency | Subsection 4.4.1 |
| \(\omega\) | driving frequency | Subsection 4.4.1 |
| \(\overline{\omega}\) | mean frequency | Subsection 4.4.2 |
| \(\delta\) | half difference | Subsection 4.4.2 |
| \(H(\lambda)\) | transfer function | Subsection 4.4.3 |
| \(G(\omega)\) | gain | Subsection 4.4.3 |
| \(J\) | Jacobian matrix | Subsection 5.1.1 |
| \(H\) | Hamiltonian function | Subsection 5.2.2 |
| \({\mathcal L}(f)(s)\) | Laplace transform | Subsection 6.1.1 |
| \(u_a(t) = u(t - a)\) | Heaviside function | Example 6.1.5 |
| \({\mathcal L}^{-1}(F(s))(t)\) | inverse Laplace transform | Subsection 6.1.3 |
| \(\Gamma(x)\) | gamma function | Exercises 6.1.7 |
| \(\delta\) | Dirac delta function | Subsection 6.3.1 |
| \(f*g\) | convolution product | Subsection 6.4.1 |
