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
- Introduction to the concept of vector. Magnitud,
- 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...
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.
Limits and Continuity
- Introducing vector functions. Definition of Limit and
- 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.
- Definition of partial derivatives.
Computing partials with Maple
- Maple tricks and notations.
- The equation of the tangent plane to a surface at a point.
Tangent planes with Maple
- Two problems with solutions and pictures are here...
- 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.
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.
- 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
Examples of Surface Areas
- Computations with Maple.
Surface Area of the Intersection of two Cylinders
- Another example with maple.
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.