Calculus III, by Andrew Incognito

Chapter 1: Vectors

1.1 Distance in Space

In this section we create the distance formula in space and apply it to spheres.

1.2 Vectors in the Plane

In this section we define vectors in two dimensions and study their algebraic and geometric properties.

1.3 Vectors in Space

In this section we define vectors in three dimensions and study their algebraic and geometric properties.

1.4 The Dot Product

In this section we define the dot product and we use it to find the angle between vectors.

1.5 The Cross Product

In this section we define the cross product and we use it to create orthogonal vectors.

1.6 Lines in Space

In this section we describe lines in space analytically.

1.7 Planes in Space

In this section we describe planes in space analytically.

1.8 Cylinders and Quadric Surfaces

In this section we discover cylinders and quadric surfaces in $\R ^3$ .

Chapter 2: Vector Valued Functions

2.1 Space Curves

In this section we create the describe curves in space.

2.2 Calculus of Space Curves

In this section we define limits, derivatives and integrals of vector-valued functions.

2.3 Differentiation Rules

In this section we will derive differentiation rules for vector-valued functions.

2.4 Arc Length

In this section we compute arc length and we define the arc length parameter.

2.5 Curvature

In this section we compute the curvature of a space curve.

2.6 TNB Frames

In this section we determine the unit Normal and Binormal vectors.

Chapter 3: Partial Derivatives

3.1 Functions of Several Variables

In this section we describe functions of two or more variables.

3.2 Limits and Continuity

In this section we compute limits and define continuity.

3.3 Partial Derivatives

In this section we define and compute partial derivatives.

3.4 The Gradient Vector

In this section we compute the gradient vector and directional derivatives.

3.5 Tangent Planes

In this section we determine tangent planes to surfaces.

3.6 Chain Rule

In this section we compute partial derivatives using the chain rule.

3.7 Maxima and Minima

In this section we determine local maxima and minima of a surface.

3.8 Lagrange Multipliers

In this section we use Lagrange multipliers to find absolute maxima and minima.

Chapter 4: Multiple Integrals

4.1 Double Integrals

In this section we define the double integral over a rectangle.

4.2 More Double Integrals

In this section we compute double integrals over various regions.

4.3 Polar Integrals

In this section we compute double integrals using polar coordinates.

4.4 Triple Integrals

In this section we compute triple integrals over various regions.

4.5 Cylindrical and Spherical Coordinates

In this section we compute triple integrals using cylindrical and spherical coordinates.

4.6 Line Integrals and Work

We compute integrals of vector-valued functions along curves.