Mathematical Methods in Chemical Engineering - Vocabulary Flashcards

Comprehensive vocabulary flashcards covering fundamental mathematical concepts, methods, and terminology from 'Mathematical Methods in Chemical Engineering' by Arvind Varma and Massimo Morbidelli.

Minor

A determinant formed from an $n\text{th}$-order determinant by striking out an equal number $m$ of rows and $m$ columns, leaving a square array of order $n - m$.

Cofactor

The signed complementary minor of an element $a_{ij}$ in an $n\text{th}$-order determinant, defined as $A_{ij} = (-1)^{i+j} [\text{complement of } a_{ij}]$, obtained by deleting the $i\text{th}$ row and $j\text{th}$ column.

Alien Cofactor

The cofactor $A_{kj}$ of an element in row $k$ when used in an expansion with elements $a_{ij}$ of a different row $i$ ($i \neq k$), producing an expansion $\sum_{j=1}^n a_{ij} A_{kj} = 0$.

Rank of a Matrix

The order $r$ of the largest nonzero determinant that can be formed from an $(m \times n)$ matrix by deleting rows and columns, such that all determinants of order greater than $r$ are zero.

Characteristic Equation

The polynomial equation $|A - \lambda I| = 0$ obtained by expanding the characteristic determinant of a square matrix $A$, whose roots $\lambda_i$ are the eigenvalues of $A$.

Biorthogonality

The mutual orthogonality property between distinct eigenvectors $\{x_j\}$ and eigenrows $\{y_i^T\}$ of a square matrix $A$, stating that $y_i^T x_j = 0$ for $i \neq j$ and $y_i^T x_i \neq 0$.

Hamilton-Cayley Theorem

A theorem stating that every square matrix $A$ satisfies its own characteristic equation, such that $P_n(A) = 0$.

Sylvester's Formula

An expression that computes any matrix polynomial or analytic function $P(A)$ of an $n\text{th}$-order matrix $A$ with simple eigenvalues $\lambda_j$ in terms of a polynomial in $A$ of degree at most $(n - 1)$: $P(A) = \sum_{j=1}^n P(\lambda_j) \frac{\text{adj}(\lambda_j I - A)}{\prod_{k=1, k \neq j}^n (\lambda_j - \lambda_k)}$.

Positive Definite Quadratic Form

A quadratic form $Q = x^T A x$ with symmetric matrix $A$ that satisfies $Q > 0$ for all $x \neq 0$ and $Q = 0$ only when $x = 0$, which requires all eigenvalues of $A$ to be strictly positive.

Hessian Matrix

The symmetric matrix $F = [f_{ij}]$ of second-order partial derivatives of a multivariable function $f$, defined by $f_{ij} = \frac{\partial^2 f}{\partial x_i \partial x_j}$, whose definiteness determines the nature of an extremum.

Lipschitz Condition

A property satisfied by a function $f(x, y)$ in a domain $\mathcal{D}$ if there exists a positive constant $K$ such that $|f(x, y) - f(x, Y)| \le K |y - Y|$ for all points $(x, y)$ and $(x, Y)$ in $\mathcal{D}$.

Picard Iteration

A constructive method of successive approximations defined by $y_n(x) = y_0 + \int_{x_0}^x f(t, y_{n-1}(t))\,dt$ used to prove the existence and uniqueness of solutions to first-order differential equations.

Autonomous System

A system of ordinary differential equations $\frac{dy}{dt} = f(y)$ where the functions $f_i$ do not depend explicitly on the independent variable $t$.

Asymptotically Stable Steady State

A critical point $y_s$ of an autonomous system that is stable and has a neighborhood $\|y_0 - y_s\| < \delta_0$ such that every trajectory originating within it satisfies $\lim_{t \to \infty} y(t) = y_s$.

Fundamental Stability Theorem

A theorem stating that a steady state $y_s$ of an autonomous nonlinear system is asymptotically stable (or unstable) if the corresponding linearized system about $y_s$ is asymptotically stable (or unstable).

Saddle Point

An unstable critical point of a two-dimensional autonomous system characterized by real eigenvalues of the Jacobian matrix having opposite signs ($\lambda_1 < 0 < \lambda_2$).

Focus (or Spiral)

A steady state of a two-dimensional autonomous system where the eigenvalues of the Jacobian are complex conjugates with nonzero real parts ($\lambda = \alpha \pm i\beta$, $\alpha \neq 0$), around which trajectories spiral inward ($\alpha < 0$) or outward ($\alpha > 0$).

Center

A marginally stable steady state of a two-dimensional autonomous system characterized by purely imaginary eigenvalues ($\text{tr}\,A = 0$ and $|A| > 0$), surrounded by closed, concentric periodic trajectories.

Liapunov Function

A scalar, positive definite function $V(x)$ with continuous first partial derivatives whose total time derivative $\dot{V}(x) = (\nabla V)^T f(x)$ is negative definite or semidefinite, establishing stability without solving the ODE system.

Hopf Bifurcation

A local bifurcation in which a steady state transitions between stability and instability as a parameter crosses a critical value, marked by a pair of complex conjugate eigenvalues crossing the imaginary axis and the emergence or disappearance of a limit cycle.

Limit Cycle

An isolated closed trajectory in the phase plane of an autonomous system representing a self-sustained periodic solution.

Fundamental Set of Solutions

A collection of $n$ linearly independent solutions $f_1(x), f_2(x), \dots, f_n(x)$ of an $n\text{th}$-order homogeneous linear ordinary differential equation $L[y] = 0$ on an interval $[a, b]$.

Wronskian Determinant

The determinant $W(x)$ constructed from $n$ solutions of an $n\text{th}$-order linear ODE and their derivatives up to order $(n - 1)$, which is nonzero on $[a, b]$ if and only if the solutions are linearly independent.

One-Sided Green's Function

The kernel $\frac{W(x, t)}{a_0(t)W(t)}$ depending solely on the linear differential operator $L[\,]$, used via the method of variation of parameters to construct particular solutions of nonhomogeneous initial value problems.

Self-Adjoint Differential Operator

A linear differential operator $L[\,]$ that equals its adjoint operator $L^*[\,]$, requiring $a_1(x) = a_0'(x)$ for a second-order operator $a_0(x)y'' + a_1(x)y' + a_2(x)y$.

Fredholm Alternative

A theorem stating that a nonhomogeneous linear boundary value problem $L[y] = -f$ with boundary conditions $U_i[y] = \gamma_i$ possesses a unique solution if and only if the corresponding homogeneous problem admits only the trivial solution.

Robin Boundary Condition

A third-kind boundary condition that prescribes a linear combination of the function value and its normal derivative at a boundary, such as $-k \frac{dy}{dx} + h y = 0$.

Rayleigh Quotient

An expression giving the eigenvalues $\lambda$ of a self-adjoint Sturm-Liouville problem in terms of integrals of the eigenfunction and its derivative: $\lambda = \frac{-p y y'|_a^b + \int_a^b [p (y')^2 + q y^2]\,dx}{\int_a^b \rho y^2\,dx}$.

Sturm's Oscillation Theorem

A theorem establishing that for a regular Sturm-Liouville eigenvalue problem ordered by increasing eigenvalues $\lambda_1 < \lambda_2 < \dots$, the $n\text{th}$ eigenfunction $y_n(x)$ has exactly $(n - 1)$ zeroes in the open interval $(a, b)$.

Finite Fourier Transform

A linear integral transform defined by $\mathcal{S}[f(x)] = \int_a^b \rho(x) \phi_n(x) f(x)\,dx$, projecting a function $f(x)$ onto the normalized eigenfunctions $\phi_n(x)$ of a self-adjoint Sturm-Liouville boundary value problem.

Ordinary Point

A point $x_0$ of a second-order linear ODE $y'' + p(x)y' + q(x)y = 0$ at which both coefficient functions $p(x)$ and $q(x)$ are analytic.

Regular Singular Point

A singular point $x_0$ of a linear ODE $y'' + p(x)y' + q(x)y = 0$ where $p(x)$ and $q(x)$ fail to be analytic, but $(x - x_0)p(x)$ and $(x - x_0)^2 q(x)$ are analytic at $x_0$.

Indicial Equation

The algebraic quadratic equation $r(r - 1) + p_0 r + q_0 = 0$ obtained by balancing the lowest-power term when applying the Frobenius method about a regular singular point at $x = 0$.

Legendre Polynomials

A sequence of orthogonal polynomials $P_n(x)$ of degree $n$ that satisfy Legendre's differential equation $(1 - x^2)y'' - 2x y' + n(n + 1)y = 0$ on the interval $[-1, 1]$.

Bessel Function of the First Kind

A nonsingular solution $J_\alpha(x)$ of Bessel's equation $x^2 y'' + x y' + (x^2 - \alpha^2)y = 0$, defined by the convergent power series $J_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m (x/2)^{2m+\alpha}}{m! \Gamma(m + \alpha + 1)}$.

Bessel Function of the Second Kind

A second linearly independent solution $Y_\alpha(x)$ of Bessel's equation, which is unbounded as $x \to 0$ and involves a logarithmic singularity when $\alpha$ is an integer.

Modified Bessel Functions

Solutions $I_\alpha(x)$ (first kind) and $K_\alpha(x)$ (second kind) of the modified Bessel equation $x^2 y'' + x y' - (x^2 + \alpha^2)y = 0$, which exhibit non-oscillatory, exponential-like growth and decay.

Gamma Function

The special function defined by the improper integral $\Gamma(a) = \int_0^\infty t^{a-1} e^{-t}\,dt$ for $a > 0$, which extends the factorial function via the relation $\Gamma(a + 1) = a \Gamma(a)$.

Quasilinear PDE

A partial differential equation that is linear in all of its highest-order derivatives, with coefficients that may depend on the independent variables, the unknown function, and its lower-order derivatives.

Characteristic Curves

Curves in the space of independent variables along which a partial differential equation reduces to a system of ordinary differential equations.

Hyperbolic PDE

A second-order two-dimensional PDE $A z_{xx} + B z_{xy} + C z_{yy} + \dots = 0$ satisfying $B^2 - 4AC > 0$, which admits two distinct real families of characteristic curves.

Parabolic PDE

A second-order two-dimensional PDE $A z_{xx} + B z_{xy} + C z_{yy} + \dots = 0$ satisfying $B^2 - 4AC = 0$, possessing a single real family of characteristic curves.

Elliptic PDE

A second-order two-dimensional PDE $A z_{xx} + B z_{xy} + C z_{yy} + \dots = 0$ satisfying $B^2 - 4AC < 0$, having no real characteristic curves.

Kinematic Wave Equation

A first-order PDE of the form $\frac{\partial y}{\partial t} + \frac{\partial \phi}{\partial x} = 0$ where $\phi = \phi(y)$, in which constant values of $y$ propagate at the wave velocity $c(y) = \frac{d\phi}{dy}$.

Expansion Wave

A simple wave solution of a first-order reducible PDE where the characteristics diverge in the direction of propagation, causing the profile to become broader and less steep over time.

Compression Wave

A simple wave solution of a first-order reducible PDE where the characteristics converge, causing the profile to progressively steepen as it propagates.

Shock

A propagating jump discontinuity in the weak solution of a first-order PDE that forms when characteristics intersect, resolving an otherwise multivalued and nonphysical solution.

Jump Condition

The conservation relation $V_s (y^+ - y^-) = \phi(y^+) - \phi(y^-)$ equating the shock propagation velocity $V_s$ to the ratio of the net flux jump to the state jump across a discontinuity.

Shock Layer (Constant Pattern)

A continuous, traveling wave solution of a higher-order dissipative PDE that propagates at constant velocity without changing shape, which approaches a shock discontinuity as the dissipation coefficient tends to zero.

Duhamel's Formula

An integral formula that expresses the solution of a linear PDE with time-dependent boundary conditions as a convolution of the time-dependent boundary input with the system's impulse response.

Finite Hankel Transform

An integral transform defined by $\mathcal{H}_0[f(r)] = \int_0^a r J_0(\lambda_n r) f(r)\,dr$, utilized to eliminate the radial space variable in cylindrical boundary value problems.

Transfer Function

An $(n \times m)$ matrix $G(s) = -(A - sI)^{-1}B$ in Laplace transform space that relates the transformed output vector $Y(s)$ to the transformed input vector $F(s)$ via $Y(s) = G(s)F(s)$.

Heaviside Expansion Theorem

An algebraic inversion formula for rational Laplace transforms $F(s) = \frac{p(s)}{q(s)}$ based on partial fraction expansions evaluated at the distinct and repeated roots of the denominator polynomial $q(s)$.

Asymptotic Sequence

A sequence of functions $\{u_n(\epsilon)\}$ satisfying $u_n(\epsilon) = o[u_{n-1}(\epsilon)]$ as $\epsilon \to 0$, such that each successive term decays faster than the preceding term.

Secular Term

An unbounded term (such as $t \sin t$ or $t^n$) appearing in the coefficients of a regular perturbation series on an infinite domain, which destroys the uniformity of the asymptotic expansion as $t \to \infty$.

Boundary Layer

A narrow subregion in a singular perturbation problem where the dependent variable undergoes rapid variation to satisfy a boundary condition that is lost when the perturbation parameter $\epsilon$ is set to zero.

Method of Strained Coordinates

A singular perturbation technique that removes secular terms by expanding both the dependent variable and the independent variable as power series in $\epsilon$, thereby adjusting the coordinate scale.

Matching Principle

The asymptotic rule stating that the $m$-term outer expansion of the $k$-term inner solution must equal the $k$-term inner expansion of the $m$-term outer solution in the intermediate overlapping domain.

Composite Expansion

A single uniformly valid approximation constructed on the entire domain by summing the inner and outer expansions and subtracting their common part in the overlapping region: $y_c(x) = y^o(x) + y^i(x) - (\text{common part})$.

WKB Method

A singular perturbation technique for linear differential equations where the highest derivative is multiplied by a small parameter, representing the solution as an exponential of an asymptotic series: $y(x; \epsilon) \sim \exp\left[\frac{1}{\delta}\sum_{n=0}^\infty \delta^n S_n(x)\right]$.