Calculus III

Fall 2014

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.