Mathematics

This lists the expected knowledge of mathematics and numerical methods. Course numbers between parenthesis list courses that cover this material to the required depth. These course numbers are for identification purposes only; there is no requirement to take these classes for the Ph.D. preliminary exam.

Mathematics questions will be phrased so that no knowledge specific to Mechanical Engineering is required for their solution.

Calculus (MAC 3311-3313 or EML 5060)

• Curve tracing
• Maxima and minima
• curvature, unit tangents, normals and binormals to curves in Cartesian, polar, and spherical coordinate systems
• Finding velocity and acceleration given a path in a Cartesian, polar, or spherical coordinate system
• Normals to surfaces given in explicit or implicit form
• Absolute and relative errors
• Taylor series expansions in one or more coordinates
• L'Hopital's rule
• Work, areas, volumes, centroids, moments of inertia in Cartesian/polar/spherical/general coordinate systems

Ordinary Differential Equations

• First order
  • Existence and uniqueness (MAP 3305 or EML 5060)
  • Phase plane (EML 5060)
  • Separation of variables (MAP 3305 or EML 5060)
  • Homogeneous equation (EML 5060)
  • Exact equations (MAP 3305 or EML 5060)
  • Bernoulli equation (EML 5060)
• First order systems (MAP 3305 or EML 5060)
• Reduction of order
  • X does not appear (EML 5060)
  • Y does not appear (EML 5060)
• Linear constant coefficients
  • Existence and uniqueness (MAP 3305 or EML 5060)
  • Characteristic polynomial (MAP 3305 or EML 5060)
  • Superposition (MAP 3305 or EML 5060)
  • Inhomogeneous equations by inspection (MAP 3305 or EML 5060)
  • Variation of parameters (MAP 3305 or EML 5060)
• Euler-Cauchy equation (EML 5060)

Partial Differential Equations (PDEs)

• Classification
  • Hyperbolic, parabolic, elliptic equations (EML 5060)
  • Initial vs. boundary value problems (EML 5060)
• Characteristics (EML 5060)
• Diagonal and canonical forms (EML 5060)
• Solution by separation of variables (MAP 3306 or EML 5060)
• Solution by Laplace transforms (EML 5060)
• Solution by D'Alembert's rule (EML 5060)

Linear Algebra

• Rectangular linear systems, subspaces, rank (EML 5060)
• Gaussian elimination (MAP 3305 or EML 5060)
• Reduction to canonical and row echelon forms (EML 5060)
• Matrix add, multiply, transpose, inverse (EML 5060)
• Basis, dimension, and change of basis (EML 5060)
• Determinants (MAP 3305 or EML 5060)
• Eigenvalue problems (MAP 3305 or EML 5060)
• Diagonalization (EML 5060)
• Symmetric/Hemitian matrices (EML 5060)
• Unitary matrices (EML 5060)
• Quadratic forms (EML 5060)
• Positive definite matrices (EML 5060)

Tensor Algebra(EGM5611)

• Vectors and tensors
• Indicial notation
• Kronecker delta and alternating tensor
• Generalized coordinate transformation
• Elementary tensor calculus