

Quick Review of Concepts from Calc 1.
 Maxs, Mins Inflection points
 Using maple to enter functions, take derivatives, make
univariate plots and manipulate expressions.

Introduction to the Algebra and Geometry of Euclidean Space
 Vectors
 Introduction to the concept of vector. Magnitud,
direction, addition.
 Vector Geometry
 Cartesian and spherical coordinate systems. Describing, surfaces,
lines, points with vectors.

Working with vectors in Maple
 Using maple to compute addition of vectors, magnitudes, angles.
The plane in the wind problem is here...
 The Dot Product
 Introducing the inner product. Scalar and vector projections.
 The Cross Product
 Definition. Cross products of the i,j,k basis vectors. Examples.

Properties of Cross products
 Maple proofs of the distributivity and anticommutatitivity
properties of cross products.

Cross products are NOT associative.
 Maple proof that cross products are not associative.

Applications of the cross product: planes, volumes
 Triple products. The volume generated by 3 vectors. Projected Area.

Lines with Maple
 Position vector plus t times the velocity vector: Howto with maple.

The plane through 3 points
 The equation of the plane containing 3 given points. The maple
procedure P3points for computing it is here...

The plane containing two lines
 The equation of the plane containing two given lines. The maple
procedure interlines for finding the point of intersection
of two lines in 3D is here...

The distance from a point to a line
 How far away is this point from that line?
The maple proc d2line is here...

The distance from a point to a plane
 How far is that point from this plane ?
The maple proc p2plane is here...

Plane containing two lines: Example1
 Given two lines in symmetric form, maple is used to find
the plane that contains them. A picture of the plane with the
two lines is here...

Example: angle of diagonals
 Simple Maple proof that when the diagonals of a rectangle
intersect at right angles then the rectangle is a square.

Example: bisecting the angle between u and v
 Length of u times v plus length of v times u does it!
The proof with maple is here...

Two planes and one point
 The equation of the plane that contains the line of intersection
of two other planes and a given point.

Two planes, angle, line..
 Finding the angle between two planes and the line of
intersection in symmetric form.
 A few review exercises
 Seven problems on lines, planes, angles, innerprods etc...

The Real Thing: Geometric Algebra of Euclidean Space

The true geometric product of two vectors
 Forget cross products: THEY ARE THE WRONG PRODUCT!
Unlearn them by discovering the dual to a bivector.
Vector Functions

Limits and Continuity
 Introducing vector functions. Definition of Limit and
continuity.

Derivatives
 The derivative of a vector function. Maple tricks.

Kinematics with vector Calculus.
 Projectile motion with vectors.

Unit Tangent, Unit Normal
 What they are what they mean and howto with maple.

Arc length and curvature
 Definitions, formulas and the very useful
maple proc curvature is here...

Formulas for plane curves
 When the vector function stays on a plane the formulas
simplify and when y=f(x) the calc1 results are recovered.

Tangent vector and curvature
 Again... but now with 'gif' formulas... Examples
for the circle, ellipses.

Normal, Twist and Binormal
 The tangent the normal and the binormal define a
turning twisting frame at each point...

Tangential and Normal components of acceleration
 They are useful in mechanics.
Functions of Several Variables

What is a function of several variables?
 Functions of several variables, Level Curves,
plot3d in Maple.

Limits and Continuity
 Definitions of limit and continuity.

Computing limits with maple
 Exercises of limits of functions of several variables.

Partial Derivatives
 Definition of partial derivatives.

Computing partials with Maple
 Maple tricks and notations.

Tangent planes
 The equation of the tangent plane to a surface at a point.

Tangent planes with Maple
 Two problems with solutions and pictures are here...

Differentiability
 The differential of a function of several variables.

The Chain Rule
 Examples of chain rules with maple.

Rates of change
 Using the Chain Rule to compute rates of change.

Directional derivatives and the gradient
 What is a directional derivative? The definition of
the gradient is also here...

Computing directional derivatives
 Examples of directional derivatives from their definition
and from the grad formula. The fast climber is here...

Surfaces and the Gradient
 The gradient is always perpendicular to the level surfaces.
Examples.

Some review exercies
 Review exercises and their solutions with maple.

Max, Mins and Saddle points
 Definition of local and global maxs and mins.

The test of second partials
 After you know that the partial derivatives are zero
you need to check for the 'sign' of the second derivatives..
Here is how and why.

Finding Extrema with Maple
 Two problems with solutions.

Lagrange Multipliers
 The method of Lagrange multipliers.

More on Lagrange Multipliers.
 Max of f subject to constraints...
Integration of Functions of Several Variables

Double Integrals and Iterated Integrals
 Double integrals over rectangles are defined here. The
procedure ApproxInt2d for computing approximating
Riemann sums for integrals in 2D is here.

Computing Volumes: Examples
 Computation of volumes of solids defined by the space
in between a rectangle and a surface. An easy integral
where Mighty Maple fails is here...

Double Integrals over General Regions
 If the domain of integration is NOT a rectangle
a simple trick can be used to define the integral. Examples
of computations of double integrals over regions with
boundaries given in different forms.
 Properties of Double Integrals
 Double integrals inherit most of the properties of sums:
Linearity, monotonicity,... etc.

Transformation to Polar Coordinates
 Polar rectangles. The element of area in different coordinate
systems.
Surface Area

Examples of Surface Areas
 Computations with Maple.

Surface Area of the Intersection of two Cylinders
 Another example with maple.
Vector Calculus

Definition of a Line Integral
 A line integral is just an integral...

Definition for Scalar Functions
 (CL) Definition, Example and Formula.

Definition for Vector Fields
 (CL) The concept of Work is here.

Computing Line Integrals with Maple
 Three problems with solutions using Maple.
The cool lint (and even cooler Lint) maple procedure
for finding the line integral
of a vector field along a path is here.
