Quickfire review questions for the CP3 and CP4 examinations

The following is a series of quick questions which should cover some of the possible topics on the CP3 and CP4 final.

Calculus

  • What is $\sinh{x}$ and $\cosh{x}$?
  • Write $f(x)^{g(x)}$ as an exponential
  • What is Leibniz's theorem for the nth derivative of a product $v(x)u(x)$?
  • Given a curve $y(x)$, what is arclength between $a\leq x\leq b$?
  • State first two terms of series expansions of $e^x$, $\sin x$, $\cos x$, $(1+x)^a$ where a is real.
  • State two first terms of series for $1/\cos x$ by bringing the series in denominator to the numerator
  • Expand $f(x + \delta x,y +\delta y)$ for small $\delta$ to two terms
  • State the general solution of $u_{tt} = c^2 u_{xx}$ in terms of general functions, $f$, and $g$.
  • State the conditions for a max, min, and saddle point of $f(x,y)$
  • Given a relation $f(P, V, T) = 0$, prove that

\begin{gather} \left(\pd{P}{V}\right)_T\left(\pd{V}{T}\right)_P\left(\pd{T}{P}\right)_V = -1, \\ \left(\pd{P}{V}\right)_T = 1/\left(\pd{V}{P}\right)_T \end{gather}

  • Given $df = P(x,y,z)dx + Q(x,y,z)dy + R(x,y,z) dz$, how do we check if it is exact differential?

Vectors and Matrices

  • State definition of linear independence of vectors
  • State definition of vector space
  • What is a basis of a vector space? What is the dimension of a vector space?
  • What is the span of vectors?
  • Given that $\mb{v} = c_1 \mb{u}_1 + \ldots c_n \mb{u}_n$, where $\mb{u}_i$ is orthogonal, state an expression for $c_i$
  • Let $B = \{\mb{b}_1, \ldots, \mb{b}_n\}$ and $C = \{\mb{c}_1, \ldots, \mb{c}_n\}$ be two bases. There is a unique matrix $P$ such that turns vectors in $B$ to vectors in $C$. How do you find $P$?

\[ [\mb{x}]_C = P [\mb{x}]_B. \]

  • State scalar triple product and vector triple product
  • Give two forms for vector equations of lines
  • Give two forms for equation of plane
  • Give expression (or idea of derivation) for shortest distance between two lines
  • Give expression (or idea of derivation) for shortest distance between point and line
  • Give expression (or idea of derivation) for shortest distance between line and a plane
  • Give expression (or idea of derivation) for shortest distance between point and plane
  • Define Hermitian, anti-Hermitian, singular, unitary, orthogonal, and normal matrices
  • What is the inverse of $\begin{pmatrix} a & b \\ c & d\end{pmatrix}$?
  • Given a general $n\times n$ matrix, write down an expression for the inverse using the cofactor matrix
  • Write down the general formula for the determinant of a matrix
  • What is the kernal of a matrix?
  • What is the rank of a matrix?
  • Explain how to use Gram-Schmidt procedure to find ortho-normal basis.
  • Given $A\mb{x} = \mb{b}$, state Cramer's method for finding $\mb{x}$
  • State the rotation matrix around the $y$-axi about an angle $\theta$
  • State the definition of eigenvalue, eigenvector
  • How do you diagonalize a matrix, $A$? What is special about a normal matrix? If a matrix can be diagonalized, what is $A^n$?

ODEs and Complex Numbers

  • State de Moivre's theorem
  • Solve $z^{n/m} = a$ where $a, z \in \mathbb{C}$ and $n/m$ is reduced to lowest terms.
  • Solve $y' + p(x)y = r(x)$
  • Given $m x'' + m\gamma x' + m \omega_0^2 x = 0$, state conditions for overdamping, critical damping, and underdamping. Sketch in all cases
  • What is the Q-Factor?

Multiple integrals and vector calculus

  • Given a region, $S$, in the $xy$ plane with density $\rho$, write down formulae for centre of mass and moments, $I_{xx}$, $I_{yy}$, $I_{xy}$
  • Given a transformation $u = u(x,y)$ and $v = v(x,y)$, write down the Jacobian $\de{x} \de{y} = J \de{u} \de{v}$. How do you calculate $J$ if you can't invert the relations?
  • State the Jacobian in polar coordinates, cylindrical coordinates, spherical coordinates
  • State the directional derivative of $f(x,y)$
  • Given a surface $z = f(x,y)$, how do you find the normal to the surface?
  • State three ways to calculate $\int_S \mb{F} \cdot \de\mb{S}$
  • Given a general curve in space, how do you calculate $\int_C \mb{F} \cdot \de\mb{r}$?
  • State the Divergence Theorem and Stokes' Theorem

Normal modes and waves

  • State the kinetic energy and potential energy of a string with displacement $y(x)$ in $x \in [0, L]$
  • What is d'Alembert's solution of the wave equation?
  • How do you solve the wave equation $u_{tt} = c^2 u_{xx}$ on a finite interval, $x \in [0, L]$ given boundary and initial conditions?
  • What is phase velocity? What is group velocity? How do you derive the expression for each?
  • (Need questions about transmission reflection of waves and power)