# Schedule of Lectures for Mat214

### Lecture 1. *Tues. Jan 23*

• Distribution of the syllabus.
• VAX accounts are required by next week. The use of: help, mail or pine, maple and lynx on the vax.
• A first quick over-all look at maple and the windows interface.

### Lecture 2. *Thrs. Jan 25*

• 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.
• Orthogonal Projections
Maple exercises
Cornell lectures available: Vectors, Vector Geometry, The Dot Product

### Lecture 3. *Tues. Jan 30*

• Algebraic definition of the Cross Product: 1-2-3 rule, determinant, cross products for the basis vectors i,j,k.
• Simple properties: Distributivity, anticommutativity, linearity under scalar multiplication.
• Geometric property of the cross product.
• The triple scalar product: cross and dot = volume of the parallelepiped.
• Equations of lines in 3D.

### Lecture 4 *Thrs. Feb. 1*

• 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
Assignment set 1: p655/5,10,13,18,21,24,29,32,37,49,56
p663-665/9,14,19,24,27,28,29,30,31,34,39,47,50,60
p673-674/4,7,12,14,16,19,21,24,31,36,45,52,63,64,65,66

### Lecture 5 *Tues. February 6*

Review exercises in preparation for Exam1 this Thrs.
Recall only Maple and Chap10 are included. Don't forget to bring a calculator (able to handle sines and cosines) 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});

### Exam 1 *Thrs. February 8*

In class Exam on Maple and the Geometry and Algebra of 3D euclidean space.

### Lecture 6: *Tues. February 13*

Vector functions: Limits, derivatives and integrals of vector valued functions.
Cornell lectures available: Kinematics with vector Calculus.

### Lecture 7: *Thrs. February 15*

Unit Tangent, Unit Normal, curvature. Tangential and Normal components of acceleration.
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 8: *Tues. February 20*

Functions of several variables, Level Curves, plot3d in Maple, Limits and Continuity, partial derivatives.

Assignment set 3: p816/1,4,5,8,10,11,13,14,16,19,20,21,22,23.

### Lecture 9: *Thrs. February 22*

Tangent planes, differentiability.
Assignment set 4: p762-4/31,42,48. p772-4/3,4,5,12,21,22,26,28,29,33,44.

### Lecture 10: *Tues. February 27*

Partial derivatives and Chain rules.

### Lecture 12: *Thrs. March 14*

Maxima and minima of functions of several variables. Lagrange Multipliers.
Assignment set 5: p779-80/4,7,8,11,12,15,18,19,31,32. p791-2/10,14,18,32,38,52,53. p802-5/35,37,47,48,50. p813-14/13,14,20,23,40.
Available: maxs and mins.. (under construction)

### Lecture 13: *Tues. March 19*

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.
Assignment set 6: p829-30/9,24,29,36(use maple),38,41. p836-8/3,4,14,15,18,30, 37, 44,50,53.

### Lecture 14: *Thrs. March 21*

Properties of double integrals. Changing the variables to polar coordinates. Polar rectangles. Element of area in polar coordinates. Change of variables formula. Examples.

### Lecture 15: *Tues. March 26*

Review exercises and preparation for Exam3. Sections included in the exam are: 12.6, 12.7, 12.8, 13.1, 13.2, 13.3.

Exam 3.

### Lecture 17: *Tues. April 2*

Introduction to vector analysis. Definition of a vector field, divergence and curl

### Lecture 19: *Tues. April 9*

Fundamental theorem for line integrals, Conservative vector fields, scalar potentials, conservation of energy, introduction to Green's theorem on the plane.

### Lecture 20: *Thrs. April 11*

Positively oriented, closed, piecewise smooth, simple curves. Statement and proof of Green's Theorem, Extension to regions with wholes, Computation of area with line integrals, Other forms of Green's Theorem.
Assignment set 7: p914-916/4,5,7,13,16,21,26,36,39,40,44,47,48. p922-24/4,6,7,9,13,21,26,29,30,36,37,39. p931-33/1,4,6,9,10,13,20,30,31,34.

### Lecture 21: *Tues. April 16*

General parametric surfaces, Surface integrals, The element of surface area, Orientable surfaces, Stokes' theorem.

### Lecture 22: *Thrs. April 18*

Introduction to geometric algebra, bivectors, the wedge product, differential forms, the exterior derivative, Statement of the General Fundamental Theorem of Calculus.

### Lecture 23: *Tues. April 23*

Review problems from chapter 14. Preparation for Exam4 on Thursday April 25.

Exam4.

### Lecture 25: *Tues. April 30*

Final Grades, Evaluations, Explanation of how the Make-up on Thrs May 2nd can change your final letter grade in the course.

### Lecture 26: *Thrs. May 2*

Make-up Exam5. All is included.

HINT: Equations of planes, cross product, tangent plane to a surface, total differentials, length of a parametric curve, double integrals, double int in polar coordinates, Lagrange Multipliers, Green's theorem, line integrals, curl and div.