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Table of Contents


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  1. Quick Review of Concepts from Calc 1.

    Maxs, Mins Inflection points
    Using maple to enter functions, take derivatives, make univariate plots and manipulate expressions.

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  2. 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 anti-commutatitivity 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...

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  3. 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.

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  4. 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.

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  5. 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...

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  6. 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.

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  7. Surface Area

    Examples of Surface Areas
    Computations with Maple.
    Surface Area of the Intersection of two Cylinders
    Another example with maple.

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  8. 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.

Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Tue Nov 11 09:56:09 EST 1997