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Excellent collection of guides/resources for Calculus II - online and e-book (download)INTEGRATION TECHNIQUESIntegration by Parts - Of all the integration techniques covered in this chapter this is probably the one that students are most likely to run into down the road in other classes.Integrals Involving Trig Functions - In this section we look at integrating certain products and quotients of trig functions. Trig Substitutions - Here we will look using substitutions involving trig functions and how they can be used to simplify certain integrals. Partial Fractions - We will use partial fractions to allow us to do integrals involving some rational functions. Integrals Involving Roots - We will take a look at a substitution that can, on occasion, be used with integrals involving roots. Integrals Involving Quadratics - In this section we are going to look at some integrals that involve quadratics. Using Integral Tables - Here we look at using Integral Tables as well as relating new integrals back to integrals that we already know how to do. Integration Strategy - We give a general set of guidelines for determining how to evaluate an integral. Improper Integrals - We will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section. Comparison Test for Improper Integrals - Here we will use the Comparison Test to determine if improper integrals converge or diverge. Approximating Definite Integrals - There are many ways to approximate the value of a definite integral. We will look at three of them in this section. APPLICATIONS OF INTEGRALSArc Length - We'll determine the length of a curve in this section.Surface Area - In this section we'll determine the surface area of a solid of revolution. Center of Mass - Here we will determine the center of mass or centroid of a thin plate. Hydrostatic Pressure and Force - We'll determine the hydrostatic pressure and force on a vertical plate submerged in water. Probability - Here we will look at probability density functions and computing the mean of a probability density function. PARAMETRIC EQUATIONS AND POLAR COORDINATESParametric Equations and Curves - An introduction to parametric equations and parametric curves (i.e. graphs of parametric equations)Tangents with Parametric Equations - Finding tangent lines to parametric curves. Area with Parametric Equations - Finding the area under a parametric curve. Arc Length with Parametric Equations - Determining the length of a parametric curve. Surface Area with Parametric Equations - Here we will determine the surface area of a solid obtained by rotating a parametric curve about an axis. Polar Coordinates - We'll introduce polar coordinates in this section. We'll look at converting between polar coordinates and Cartesian coordinates as well as some basic graphs in polar coordinates. Tangents with Polar Coordinates - Finding tangent lines of polar curves. Area with Polar Coordinates - Finding the area enclosed by a polar curve. Arc Length with Polar Coordinates - Determining the length of a polar curve. Surface Area with Polar Coordinates - Here we will determine the surface area of a solid obtained by rotating a polar curve about an axis. Arc Length and Surface Area Revisited - In this section we will summarize all the arc length and surface area formulas from the last two chapters. SEQUENCES AND SERIESSequences - We will start the chapter off with a brief discussion of sequences. This section will focus on the basic terminology and convergence of sequencesMore on Sequences - Here we will take a quick look about monotonic and bounded sequences. Series The Basics - In this section we will discuss some of the basics of infinite series. Series Convergence/Divergence - Most of this chapter will be about the convergence/divergence of a series so we will give the basic ideas and definitions in this section. Series Special Series - We will look at the Geometric Series, Telescoping Series, and Harmonic Series in this section. Integral Test - Using the Integral Test to determine if a series converges or diverges. Comparison Test/Limit Comparison Test - Using the Comparison Test and Limit Comparison Tests to determine if a series converges or diverges. Alternating Series Test - Using the Alternating Series Test to determine if a series converges or diverges. Absolute Convergence - A brief discussion on absolute convergence and how it differs from convergence. Ratio Test - Using the Ratio Test to determine if a series converges or diverges. Root Test - Using the Root Test to determine if a series converges or diverges. Strategy for Series - A set of general guidelines to use when deciding which test to use. Estimating the Value of a Series - Here we will look at estimating the value of an infinite series. Power Series - An introduction to power series and some of the basic concepts. Power Series and Functions - In this section we will start looking at how to find a power series representation of a function. Taylor Series - Here we will discuss how to find the Taylor/Maclaurin Series for a function. Applications of Series - In this section we will take a quick look at a couple of applications of series. Binomial Series - A brief look at binomial series. VECTORSVectors The Basics - In this section we will introduce some of the basic concepts about vectors.Vector Arithmetic - Here we will give the basic arithmetic operations for vectors. Dot Product - We will discuss the dot product in this section as well as an application or two. Cross Product - In this section we'll discuss the cross product and see a quick application. THREE DIMENSIONAL SPACEThis 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. 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. Curvature - We will determine the curvature of a function in this section. 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. 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. Extremely useful for teachers of Maths/Mathematics working through Calculus II (Math 2414). |
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Excellent collection of guides/resources for Calculus III - online and e-book (download)THREE DIMENSIONAL SPACEThis 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. 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. Curvature - We will determine the curvature of a function in this section. 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. 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 DERIVATIVESLimits - 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 DERIVATIVESTangent 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 INTEGRALSDouble 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 some general double integrals. 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 INTEGRALSVector 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 INTEGRALSParametric 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. Extremely useful for teachers of Maths/Mathematics working through Calculus III (Math 2415). |
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Generate math worksheets for the following -ADDITIONSUBTRACTIONMULTIPLICATIONLONG MULTIPLICATIONDIVISIONLONG DIVISIONDECIMALSDECIMALS - CONVERSION FROM FRACTIONSDECIMAL FRACTIONSDECIMAL FRACTIONS AND CONVERSION FROM FRACTIONSFRACTIONSTIME WORKSHEETSVery useful for teachers of Math/Mathematics. |
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