## Calculus III

## Spring 2015

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.

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.

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.

Notes on center of mass in the plane.

Cylindrical and spherical coordinates.

Moment of inertia.

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

Conversions among rectangular/cylindrical/spherical coordinates.