## Calculus III

## Fall 2014

Syllabus.

Solutions distributed in class.

Day-by-day summary of the course so far.

The Greek Alphabet.

A review of some trigonometry.

Topics to review from Calc I and Calc II.

Practice sheet: converting to polar coordinates.

Example 7 on p 797.

Dot products and angles.

Lines in R^2 and the dot product.

Cauchy-Schwarz and Triangle Inequalities.

One determinant computed six ways.

Determinants and the Laplace expansion.

The cross product.

The distance between two lines in three-dimensional space.

A brief summary of conic sections.

A brief summary of quadric surfaces.

A diagram of the conic sections.

Partial proof of Rule 5 (page 850).

Speed and arclength.

Reparametrization by arclength: an example.

Curvature and acceleration (tweaked).

Curvature and acceleration (version distributed in class).

Comparing Calc I and Calc III limit laws.

Examples involving path limits.

Galileo, Kepler, and Newton.

Some basic facts about inequalities.

Linearization and differentials: the one-dimensional case (text sec. 3.10).

The derivative of a function of two variables.

A summary comparison of differentiablity for y=f(x) and z=f(x,y).

The Chain Rule in higher dimensions.

Gradients and level sets.

Comparing Calc I and Calc III max/min rules.

The Extreme Value Theorem in the Plane.

Further properties of the gradient.

Proof of Clairaut's Theorem.

The "chart method" for certain integrals.

The center of mass of a lamina.

Cylindrical and spherical coordinates.

Moment of inertia.

The area of surface z=f(x,y).

Conversions among rectangular/cylindrical/spherical coordinates.