Choose from any of the topics below except 12.4. After that, think of a problem that is relevant to that topic. The problem should be new and not from any textbook example or other calculus text (not from examples in any textbook, website, or lecture, and not from any assignment in this or any other course). You should be able to explain your steps in full sentences. Explain your thought process for working out problems similar to your example.

10 Vectors 559 10.1 introduction to Cartesian Coordinates in Space 559 10.2 An Introduction to Vectors 574 10.3 The Dot Product . . . 588 10.4 The Cross Product . 601 10.5 Lines . 612 10.6 Planes 623 11 Vector Valued Functions 631 11.1 Vector-Valued Functions 631 11.2 Calculus and Vector-Valued Functions 637 11.3 The Calculus of Motion . . . . 651 11.4 Unit Tangent and Normal Vectors . . 664 11.5 The Arc Length Parameter and Curvature . 673 12 Functions of Several Variables 683 12.1 Introduction to Multivariable Functions 683 12.2 Limits and Continuity of Multivariable Functions 690 12.3 Partial Derivatives . . . . 700 12.4 Differentiability and the Total Differential 712 12.5 The Multivariable Chain Rule 721 12.6 Directional Derivatives 729 12.7 Tangent Lines, Normal Lines, and Tangent Planes 739 12.8 Extreme Values 749