# Schedule of Lectures for Mat214 Fall 1996

### 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.
Using maple to find max, min and inflection points.

### 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.
Working with vectors in Maple
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*

Assignment1: p637-8/4,7,12,13,16,17,22,27,29,30,31,32,37,39,41,42,46,47, 49,50,51
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.
Maple exercises
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 line
The 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
v= 2 i + j
makes with (-1,1,1), (1,0,1), (-2,2,1).
• 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});

Solutions

### 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 Derivatives
Examples 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*

Differentiability
Chain Rules.
Examples of the Chain Rule with Maple
Using the Chain Rule to compute rates

### Lecture 12: *Thrs. Oct. 17*

Directional derivatives

### Lecture 13: *Tues. Oct. 22*

Review exercises for Exam2 this Thursday.

Included are:
Chap11: 1,2,3-,4,5-
Chap12: 1,2,3,4,5,6

In class Exam.

### Lecture 14: *Tues. Oct. 29*

Examples 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

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 Integrals
Definition 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>