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 real-world 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 n-space
• 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 non-gradient 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 re-iteration in Cartesian coordinates
• Parametric domains and surfaces; polar coords
• Area-scaling for scalars and vectors
• Surface integrals for scalar fields
• Flux integrals

Triple Integrals

• Interpretation/Evaluation by Inspection/Average Values
• Inspection, iteration, re-iteration
• 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