MATH 253 for Mech 2
Course Level Learning Goals
When the student approaches the final examination,
s/he should be able to

Translate physical problems into mathematical notation,
using several independent (input) variables and
several dependent (output) variables if necessary;

Construct linear approximations for multivariable relationships
(usually based on derivatives),
and use these to estimate function values and predict trends,

Formulate macroscropic relationships using definite integrals
(single, double, or triple), by reasoning with
simple linear relationships at the microscopic scale,

Calculate, without electronic assistance, the partial
derivatives, antiderivatives, and algebraic solutions
necessary to solve problems in mathematics, physics, and engineering;

Calculate, with electronic assistance,
more elaborate partial derivatives, integrals, and solutions
to differential equations in order to solve complex problems
(technical proficiency with Matlab is important here),
and

Translate mathematical results into physical
predictions for realworld systems.
Topic Level Learning Goals
Geometry and Visualization
 Lines and planes in 2 or 3 space dimensions
 Dot products, orthogonality, projections
 Applications: Reflection and refraction
 Hypersurfaces; lines and hyperplanes in nspace
 Spheres, ellipsoids, hyperboloids, paraboloids;
level surfaces.
 Functions and relations; graph mode versus contour mode
Rates of Change

Directional Derivatives

The Gradient
 Gradients and Scalar Potentials
 Higher Derivatives and Clairaut's Theorem
Linearization
 Differentiability
 Tangent hyperplanes; normal lines
 Implicit Differentiation
Single Integrals
 Interpretations/Evaluation by Inspection/Average Values
 Inverse substitution
 Parametrizing curves
 Interpretations: velocity, speed, acceleration
 Polar Coordinates
 Line integrals (scalar fields, vector fields)
Work and Conservative Vector Fields
 Quick discard test for nongradient fields
 Antiderivative of a gradient field
 Fundamental Theorem for line integrals
 Path Independence; Fine Points
Double Integrals
 Interpretations/Evaluation by Inspection/Average Values
 Iteration and reiteration in Cartesian coordinates
 Parametric domains and surfaces; polar coords
 Areascaling for scalars and vectors
 Surface integrals for scalar fields
 Flux integrals
Triple Integrals
 Interpretation/Evaluation by Inspection/Average Values
 Inspection, iteration, reiteration
 Inverse substitution, determinants [if time permits]
 Cylindrical and spherical coordinates
 Divergence Theorem, Applications
Optimization and Approximation
 Directional Derivatives and the Gradient
 Quadratic approximations
 Local maxima and minima; classifying critical points
 Optimization on restricted domains