Contents

• Contents
• Analytic geometry in the 3-dimensional space.
• Planes and lines.
• Dot product.
• Cross product.
• The box product.
• Curves in the plane and in the space.
• Vector-valued functions.
• Limits and Continuity.
• Derivatives.
• Surfaces.
• Quadric surfaces.
• Parametrized surfaces.
• Multivariable functions.
• Domains.
• Level curves.
• Limits.
• Continuous functions.
• Partial derivatives; the differential of a function.
• First partial derivatives.
• Differentiability, linearization.
• Partial derivatives of higher order.
• The chain rule.
• One independent variable.
• Two independent variables.
• Implicit differentiation.
• Applications of the derivative.
• Taylor's formula.
• Extrema of a function and saddle points.
• Directional derivative and Gradient vector.
• Functions of 2 variables.
• Functions of 3 variables.
• Lagrange's multipliers.
• Curl of a plane vector field.
• The divergence of a vector field in the plane.
• The curl of a vector field in the space.
• The divergence of a vector field in the space.
• Multiple integrals.
• Double integrals.
• Double integrals over a rectangle.
• Double integrals over a bounded region: the general case.
• Applications: areas, moments, center of mass in the plane.
• Double integrals in polar coordinates.
• Triple integrals.
• Triple integrals: definition and first properties.
• Evaluation of a triple integral.
• Average value of a function over a domain in space.
• Applications: masses, moments, center of mass in the three dimensional space.
• Triple integrals in cylindrical coordinates.
• Triple integrals in spherical coordinates.
• Substitution in multiple integrals.
• Substitution in double integrals.
• Substitution in triple integrals.
• Integration and Vector Fields.
• Line integrals.
• Definition and properties.
• Mass of a wire, moments and center of mass.
• Vector Fields, Work, Circulation and Flux.
• Gradient field.
• Work of a force.
• Flow, circulation and flux.
• Green's theorem in the plane.
• Surface integrals and Surface area.
• Surface area.
• Surface integral.
• Special formulas for Surface Area.
• Mass of a thin shell, moments and center of mass.
• Orientation of a surface.
• Flux across an oriented surface.
• The Divergence theorem.
• Stoke's theorem.
• Conservative Fields, Potential Function.

This short tutorial is a companion material to the course ``Calculus II'' given at JCT (first year - spring term). It is intended to help the student, but will replace neither personal lecture notes nor a good textbook. In particular, proofs are generally not given. Sometimes, we give an outline of the proof.

References to books are given on the course home page.

From time to time, detailed handouts (with complete proofs and solutions to exercises) are given to the students in classroom.

Constructive remarks are welcome. Please e-mail to dana@mail.jct.ac.il.