## Lecture 1. *Tues. Sept. 3*- Distribution of the syllabus.
- VAX accounts are required by next week. The use of: help, mail or pine, maple and lynx on the vax. Graphical web browsers are available in the library and in terminal rooms around campus.
- A first quick over-all look at maple and the windows interface.
## Lecture 2. *Thrs. Sept. 5*- Introduction to the Algebra and Geometry of Euclidean Space.
- Right handed 3D coordinate systems.
- Addition of vectors and scalar multiplication.
- Inner product and length.
- Distance between two points and angle between two vectors.
Maple exercises Cornell lectures available: Vectors, Vector Geometry, The Dot Product ## Lecture 3. *Tues. Sep. 10*- Orthogonal Projections
- Geometric property of the cross product.
- Algebraic definition of the Cross Product: 1-2-3 rule, determinant, cross products for the basis vectors i,j,k.
Maple Proofs
## Lecture 4. *Thrs. Sep. 12*- More properties of Cross Products. Examples.
Maple proof that cross products are not associative - Lines in 3D: velocity vector, position vector, change of clocks.
p655-7/4,5,13,14,17,18,19,20,24,26,27,29,33,34,36,37,38,40,44,48,51,56, 57,58,59. p663-5/5,6,7,11,12,16,18,22,23,26,28,29,30,31,39,40,41,43,44,48. ## Lecture 4 *Tues. Sept. 17*- Equations of planes in 3D.
- A plane through the origin perpendicular to a given vector N.
- A plane through an arbitrary point perpendicular to N.
- A plane through 3 points.
- Intersections of lines and planes.
- Intersection of two lines.
- Intersection of two planes.
- Intersection of a line and a plane.
- Plotting lines and planes with Maple.
Cornell lectures available: The Cross Product, Applications of the cross product: planes, volumes The plane through 3 points The plane containing two lines Example: 1 Example: 2 Example: 3 Two planes and one point Two planes, angle, line.. ## Lecture 5 *Thrs. Sept. 19*The distance from a point to a lineThe distance from a point to a plane ## Lecture 6 *Tues. Sept. 24*Review exercises in preparation for Exam1 this Thrs.Recall only Maple and Chap10 are included. Don't forget to bring a number 2 pencil and your SUNYA ID.
- Find the line of intersection of the planes:
y - x - z = 1 x + y - z = -1 - Find the angle that
makes with (-1,1,1), (1,0,1), (-2,2,1).**v**= 2 i + j - Do the vectors:
i + 2k -j, i + j - k, 3i + j lie on a plane? - Compute the area of the triangle PQR where:
P(0,1,0), Q(2,1,0), R(1,0,1) - Find the symmetric equations of the line through the origin
perpendicular to the plane:
z - x - y = 5 - Consider the line:
x = -3t - 1 y = 2t - 2 z = t - 1 and the plane: x + y + z = 3. Do the line and the plane intersect? You may be able to look at both the line and the plane with maple:`> line:=plot3d([-3*t-1,2*t-2,t-1], t=-5..5,s=0..1,grid=[2,2]):``> plane:=plot3d(3-x-y,x=-3..3, y=-3..3,orientation=[103,110]):``> with(plots): display3d({line,plane});`
## Exam 1: *Thrs. Sept 26 13*In class Exam on Maple and the Geometry and Algebra of 3D euclidean space.## Lecture 7: *Tues. Oct. 1*Vector functions:[Limits], [derivatives] and integrals of vector valued functions.Cornell lectures available: Kinematics with vector Calculus. ## Lecture 8: *Thrs. Oct. 3*[Unit Tangent, Unit Normal], [Arc length and curvature]. Tangential and Normal components of acceleration. [Formulas for plane curves].Cornell Lectures available: [Tangent vector and curvature],[Normal, Twist and Binormal],[Tangential and Normal components of acceleration] Assignment set 2: p700-1/2,5,8,10,12,15,20,25,45,48 p708-10/10,17,20,26,32,33. ## Lecture 9: *Tues. Oct. 8*[Functions of several variables, Level Curves, plot3d in Maple], [Limits and Continuity.][Exercises of limits of functions of several variables] Assignment set 3: p816/1,4,5,8,10,11,13,14,16,19,20,21,22,23. ## Lecture 10: *Thrs. Oct. 10*Partial DerivativesExamples with Maple Tangent planes Examples with Maple Assignment set 4: p762-4/31,42,48. p772-4/3,4,5,12,21,22,26,28,29,33,44. ## Lecture 11: *Tues. Oct. 15*DifferentiabilityChain Rules. Examples of the Chain Rule with Maple Using the Chain Rule to compute rates ## Lecture 12: *Thrs. Oct. 17*Directional derivativesThe Gradient. ## Lecture 13: *Tues. Oct. 22*Review exercises for Exam2 this Thursday.
Included are: ## Exam2: *Thrs. Oct. 24*In class Exam.## Lecture 14: *Tues. Oct. 29*Maxs, Mins and Saddle pointsExamples with maple. Lagrange Multipliers. More on Lagrange Multipliers. ## Lecture 15: *Thrs. Oct. 31*Multipe Integrals. Definition of the double integral of f(x,y) over a rectangular domain D. Fubini's theorem. Volumes and Areas computed with double integrals. Integration over non-rectangular domains.## Lecture 16: *Tues. Nov. 5*Properties of double integrals. Changing the variables to polar coordinates. Polar rectangles. Element of area in polar coordinates.## Lecture 17: *Thrs. Nov. 7*Change of variables formula. Examples.## Lecture 18: *Tues. Nov. 12*Surface Integrals. Examples## Lecture 19: *Thrs. Nov. 14*EXAM 3 in class.## Lecture 20: *Tues. Nov. 19*Vector Fields. Definition. Divergence and Curl. Physical interpretation. Examples. Computations with maple (diverge and curl commands).## Lecture 21: *Thrs. Nov. 21*Line IntegralsDefinition for scalar functions. Example of a computation from the definition. Formula for computing line integrals in terms of a parametrization for the curve. Line integrals of vector fields. The physical concept of Work. Formula involving the unit tangent vector. |

Carlos Rodriguez <carlos@math.albany.edu> Last modified: Fri Nov 22 10:50:37 EST 1996