Calculus volume 3

OpenStax

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0.1 About openstax

0.2 About openstax's resources

0.3 Customization

0.6 About Calculus Volume 3

0.7 Coverage and scope

0.8 Pedagogical foundation

0.9 Assessments that reinforce key concepts

0.10 Early or late transcendentals

0.11 Comprehensive art program

0.12 Assessments that reinforce key concepts

0.13 Ancillaries

0.15 About our team

0.16 Senior contributing authors

0.17 Contributing authors

1 Table of integrals

2 Table of derivatives

3 Review of pre-calculus

Parametric equations and polar coordinates

1 Parametric equations

2 Calculus of parametric curves

3 Polar coordinates

3.1 Defining polar coordinates

3.2 Polar curves

3.3 Symmetry in polar coordinates

3.4 Key concepts

4 Area and arc length in polar coordinates

5 Conic sections

Vectors in space

1 Vectors in the plane

1.1 Vector representation

1.2 Combining vectors

1.3 Vector components

1.4 Proof of commutative property

1.5 Proof of distributive property

1.6 Unit vectors

1.7 Applications of vectors

1.8 Key concepts

2 Vectors in three dimensions

2.1 Three-dimensional coordinate systems

2.2 Writing equations in ℝ 3

2.3 Working with vectors in ℝ 3

2.4 Key concepts

2.5 Key equations

3 The dot product

3.1 The dot product and its properties

3.3 Using the dot product to find the angle between two vectors

3.6 Projections

3.8 Key concepts

3.9 Key equations

4 The cross product

4.1 The cross product and its properties

4.4 Determinants and the cross product

4.5 Using the cross product

4.7 The triple scalar product

4.10 Applications of the cross product

4.11 Key concepts

4.12 Key equations

5 Equations of lines and planes in space

5.1 Equations for a line in space

5.2 Distance between a point and a line

5.3 Relationships between lines

5.4 Equations for a plane

5.5 Parallel and intersecting planes

5.6 Key concepts

5.7 Key equations

6 Quadric surfaces

6.1 Identifying cylinders

6.2 Quadric surfaces

6.3 Key concepts

7 Cylindrical and spherical coordinates

Vector-valued functions

1 Vector-valued functions and space curves

1.1 Definition of a vector-valued function

1.2 Graphing vector-valued functions

1.3 Limits and continuity of a vector-valued function

1.4 Key concepts

1.5 Key equations

2 Calculus of vector-valued functions

2.1 Derivatives of vector-valued functions

2.3 Tangent vectors and unit tangent vectors

2.4 Integrals of vector-valued functions

2.5 Key concepts

2.6 Key equations

3 Arc length and curvature

3.1 Arc length for vector functions

3.2 Arc-length parameterization

3.5 The normal and binormal vectors

3.6 Key concepts

3.7 Key equations

4 Motion in space

Differentiation of functions of several variables

1 Functions of several variables

2 Limits and continuity

3 Partial derivatives

4 Tangent planes and linear approximations

5 The chain rule

5.1 Chain rules for one or two independent variables

5.3 The generalized chain rule

5.4 Implicit differentiation

5.5 Key concepts

5.6 Key equations

6 Directional derivatives and the gradient

7 Maxima/minima problems

8 Lagrange multipliers

Multiple integration

1 Double integrals over rectangular regions

2 Double integrals over general regions

3 Double integrals in polar coordinates

4 Triple integrals

4.1 Integrable functions of three variables

4.2 Triple integrals over a general bounded region

4.3 Changing the order of integration

4.4 Average value of a function of three variables

4.5 Key concepts

4.6 Key equations

5 Triple integrals in cylindrical and spherical coordinates

5.1 Review of cylindrical coordinates

5.2 Integration in cylindrical coordinates

5.3 Review of spherical coordinates

5.4 Integration in spherical coordinates

5.5 Key concepts

5.6 Key equations

6 Calculating centers of mass and moments of inertia

7 Change of variables in multiple integrals

7.1 Planar transformations

7.3 Change of variables for double integrals

7.4 Change of variables for triple integrals

7.5 Key concepts

7.6 Chapter review exercises

Vector calculus

1 Vector fields

2 Line integrals

3 Conservative vector fields

4 Green’s theorem

4.1 Extending the fundamental theorem of calculus

4.2 Circulation form of green’s theorem

4.4 Flux form of green’s theorem

4.6 Green’s theorem on general regions

4.7 Key concepts

4.8 Key equations

5 Divergence and curl

6 Surface integrals

6.1 Parametric surfaces

6.2 Surface area of a parametric surface

6.3 Surface integral of a scalar-valued function

6.4 Orientation of a surface

6.5 Surface integral of a vector field

6.6 Key concepts

6.7 Key equations

7 Stokes’ theorem

8 The divergence theorem

8.1 Overview of theorems

8.2 Stating the divergence theorem

8.4 Using the divergence theorem

8.5 Application to electrostatic fields

8.7 Key concepts

8.8 Key equations

8.9 Chapter review exercises

Second-order differential equations

1 Second-order linear equations

1.1 Homogeneous linear equations

1.2 Second-order equations with constant coefficients

1.3 Distinct real roots

1.4 Single repeated real root

1.5 Complex conjugate roots

1.6 Summary of results

1.7 Initial-value problems and boundary-value problems

1.8 Key concepts

1.9 Key equations

2 Nonhomogeneous linear equations

4 Series solutions of differential equations

Source: OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2

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