# Calculus III

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About *Calculus III:*

Here is a listing of all the material that is currently available online.

Three Dimensional Space

This is the only chapter that exists in two places in my notes. When I originally wrote these notes all of these topics were covered in Calculus II however, we have since moved several of them into Calculus III. So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes. Many of the sections not covered in Calculus III will be used on occasion there anyway and so they serve as a quick reference for when we need them.

The 3-D Coordinate System We will introduce the concepts and notation for the three dimensional coordinate system in this section.

Equations of Lines In this section we will develop the various forms for the equation of lines in three dimensional space.

Equations of Planes Here we will develop the equation of a plane.

Quadric Surfaces In this section we will be looking at some examples of quadric surfaces.

Functions of Several Variables A quick review of some important topics about functions of several variables.

Vector Functions We introduce the concept of vector functions in this section. We concentrate primarily on curves in three dimensional space. We will however, touch briefly on surfaces as well.

Calculus with Vector Functions Here we will take a quick look at limits, derivatives, and integrals with vector functions.

Tangent, Normal and Binormal Vectors We will define the tangent, normal and binormal vectors in this section.

Arc Length with Vector Functions In this section we will find the arc length of a vector function.

Velocity and Acceleration In this section we will revisit a standard application of derivatives. We will look at the velocity and acceleration of an object whose position function is given by a vector function.

Curvature We will determine the curvature of a function in this section.

Cylindrical Coordinates We will define the cylindrical coordinate system in this section. The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.

Spherical Coordinates In this section we will define the spherical coordinate system. The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.

Partial Derivatives

Limits Taking limits of functions of several variables.

Partial Derivatives In this section we will introduce the idea of partial derivatives as well as the standard notations and how to compute them.

Interpretations of Partial Derivatives Here we will take a look at a couple of important interpretations of partial derivatives.

Higher Order Partial Derivatives We will take a look at higher order partial derivatives in this section.

Differentials In this section we extend the idea of differentials to functions of several variables.

Chain Rule Here we will look at the chain rule for functions of several variables.

Directional Derivatives We will introduce the concept of directional derivatives in this section. We will also see how to compute them and see a couple of nice facts pertaining to directional derivatives.

Applications of Partial Derivatives

Tangent Planes and Linear Approximations We’ll take a look at tangent planes to surfaces in this section as well as an application of tangent planes.

Gradient Vector, Tangent Planes and Normal Lines In this section we’ll see how the gradient vector can be used to find tangent planes and normal lines to a surface.

Relative Minimums and Maximums Here we will see how to identify relative minimums and maximums.

Absolute Minimums and Maximums We will find absolute minimums and maximums of a function over a given region.

Lagrange Multipliers In this section we’ll see how to use Lagrange Multipliers to find the absolute extrema for a function subject to a given constraint.

Multiple Integrals

Double Integrals We will define the double integral in this section.

Iterated Integrals In this section we will start looking at how we actually compute double integrals.

Double Integrals over General Regions Here we will look at the most general double integral.

Double Integrals in Polar Coordinates In this section we will take a look at evaluating double integrals using polar coordinates.

Triple Integrals Here we will define the triple integral as well as how we evaluate them.

Triple Integrals in Cylindrical Coordinates We will evaluate triple integrals using cylindrical coordinates in this section.

Triple Integrals in Spherical Coordinates In this section we will evaluate triple integrals using spherical coordinates.

Change of Variables In this section we will look at change of variables for double and triple integrals.

Surface Area Here we look at the one real application of double integrals that we’re going to look at in this material.

Area and Volume Revisited We summarize the area and volume formulas from this chapter.

Line Integrals

Vector Fields In this section we introduce the concept of a vector field.

Line Integrals Part I Here we will start looking at line integrals. In particular we will look at line integrals with respect to arc length.

Line Integrals Part II We will continue looking at line integrals in this section. Here we will be looking at line integrals with respect to x, y, and/or z.

Line Integrals of Vector Fields Here we will look at a third type of line integrals, line integrals of vector fields.

Fundamental Theorem for Line Integrals In this section we will look at a version of the fundamental theorem of calculus for line integrals of vector fields.

Conservative Vector Fields Here we will take a somewhat detailed look at conservative vector fields and how to find potential functions.

Green’s Theorem We will give Green’s Theorem in this section as well as an interesting application of Green’s Theorem.

Curl and Divergence In this section we will introduce the concepts of the curl and the divergence of a vector field. We will also give two vector forms of Green’s Theorem.

Surface Integrals

Parametric Surfaces In this section we will take a look at the basics of representing a surface with parametric equations. We will also take a look at a couple of applications.

Surface Integrals Here we will introduce the topic of surface integrals. We will be working with surface integrals of functions in this section.

Surface Integrals of Vector Fields We will look at surface integrals of vector fields in this section.

Stokes’ Theorem We will look at Stokes’ Theorem in this section.

Divergence Theorem Here we will take a look at the Divergence Theorem.

Three Dimensional Space

This is the only chapter that exists in two places in my notes. When I originally wrote these notes all of these topics were covered in Calculus II however, we have since moved several of them into Calculus III. So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes. Many of the sections not covered in Calculus III will be used on occasion there anyway and so they serve as a quick reference for when we need them.

The 3-D Coordinate System We will introduce the concepts and notation for the three dimensional coordinate system in this section.

Equations of Lines In this section we will develop the various forms for the equation of lines in three dimensional space.

Equations of Planes Here we will develop the equation of a plane.

Quadric Surfaces In this section we will be looking at some examples of quadric surfaces.

Functions of Several Variables A quick review of some important topics about functions of several variables.

Vector Functions We introduce the concept of vector functions in this section. We concentrate primarily on curves in three dimensional space. We will however, touch briefly on surfaces as well.

Calculus with Vector Functions Here we will take a quick look at limits, derivatives, and integrals with vector functions.

Tangent, Normal and Binormal Vectors We will define the tangent, normal and binormal vectors in this section.

Arc Length with Vector Functions In this section we will find the arc length of a vector function.

Velocity and Acceleration In this section we will revisit a standard application of derivatives. We will look at the velocity and acceleration of an object whose position function is given by a vector function.

Curvature We will determine the curvature of a function in this section.

Cylindrical Coordinates We will define the cylindrical coordinate system in this section. The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.

Spherical Coordinates In this section we will define the spherical coordinate system. The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.

Partial Derivatives

Limits Taking limits of functions of several variables.

Partial Derivatives In this section we will introduce the idea of partial derivatives as well as the standard notations and how to compute them.

Interpretations of Partial Derivatives Here we will take a look at a couple of important interpretations of partial derivatives.

Higher Order Partial Derivatives We will take a look at higher order partial derivatives in this section.

Differentials In this section we extend the idea of differentials to functions of several variables.

Chain Rule Here we will look at the chain rule for functions of several variables.

Directional Derivatives We will introduce the concept of directional derivatives in this section. We will also see how to compute them and see a couple of nice facts pertaining to directional derivatives.

Applications of Partial Derivatives

Tangent Planes and Linear Approximations We’ll take a look at tangent planes to surfaces in this section as well as an application of tangent planes.

Gradient Vector, Tangent Planes and Normal Lines In this section we’ll see how the gradient vector can be used to find tangent planes and normal lines to a surface.

Relative Minimums and Maximums Here we will see how to identify relative minimums and maximums.

Absolute Minimums and Maximums We will find absolute minimums and maximums of a function over a given region.

Lagrange Multipliers In this section we’ll see how to use Lagrange Multipliers to find the absolute extrema for a function subject to a given constraint.

Multiple Integrals

Double Integrals We will define the double integral in this section.

Iterated Integrals In this section we will start looking at how we actually compute double integrals.

Double Integrals over General Regions Here we will look at the most general double integral.

Double Integrals in Polar Coordinates In this section we will take a look at evaluating double integrals using polar coordinates.

Triple Integrals Here we will define the triple integral as well as how we evaluate them.

Triple Integrals in Cylindrical Coordinates We will evaluate triple integrals using cylindrical coordinates in this section.

Triple Integrals in Spherical Coordinates In this section we will evaluate triple integrals using spherical coordinates.

Change of Variables In this section we will look at change of variables for double and triple integrals.

Surface Area Here we look at the one real application of double integrals that we’re going to look at in this material.

Area and Volume Revisited We summarize the area and volume formulas from this chapter.

Line Integrals

Vector Fields In this section we introduce the concept of a vector field.

Line Integrals Part I Here we will start looking at line integrals. In particular we will look at line integrals with respect to arc length.

Line Integrals Part II We will continue looking at line integrals in this section. Here we will be looking at line integrals with respect to x, y, and/or z.

Line Integrals of Vector Fields Here we will look at a third type of line integrals, line integrals of vector fields.

Fundamental Theorem for Line Integrals In this section we will look at a version of the fundamental theorem of calculus for line integrals of vector fields.

Conservative Vector Fields Here we will take a somewhat detailed look at conservative vector fields and how to find potential functions.

Green’s Theorem We will give Green’s Theorem in this section as well as an interesting application of Green’s Theorem.

Curl and Divergence In this section we will introduce the concepts of the curl and the divergence of a vector field. We will also give two vector forms of Green’s Theorem.

Surface Integrals

Parametric Surfaces In this section we will take a look at the basics of representing a surface with parametric equations. We will also take a look at a couple of applications.

Surface Integrals Here we will introduce the topic of surface integrals. We will be working with surface integrals of functions in this section.

Surface Integrals of Vector Fields We will look at surface integrals of vector fields in this section.

Stokes’ Theorem We will look at Stokes’ Theorem in this section.

Divergence Theorem Here we will take a look at the Divergence Theorem.