A Tutorial for Calculus II.

TDP

 

 

• 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 Areas.
• 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.
• About this document ...