CALCULUS IN THREE DIMENSIONS (21259)
SPRING 2015
Instructor:
Dr. Dana Mihai, 8128 Wean Hall
[email protected]
Lecture:
Lecture 1: MWF, DH A302, 12:301:20 pm
Lecture 2: MWF, BH A51, 9:3010:20 am
PAPER HOMEWORK 1
21259: Calculus in Three Dimensions, Spring 2015
Due: Wednesday, January 21st, in lecture
1. In R3 we will encounter the following type of objects: points (0dimensional), curves (1d
Learning goals for section 14.2 (Limits and Continuity):
At the end of this lecture you should be able to:
 nd the limit of a function of two variables if it exists, by either plugging
in the coordin
Review 12.2: Vectors in R2
Geometrically, a vector can be represented by an arrow with a specied
direction and length (magnitude). Notation: !, or v
v
Algebraically, a vector in R2 can be represented
1. You are using a jet pilot dogght simulator. Your monitor tells you that two missiles
have honed in on your plane along the directions (3,5,2) and (1,3,2). In what direction
should you turn to have
(PG 13: for some reference to violence and use of weapons)
1. A superhero stands on the top of a building and wants to shoot a laser gun at his evil
nemesis, who is standing on the corner of the stree
13.1: Vector Functions and Space Curves
Vector Valued Function: rule that assigns to each element in its
domain a vector in its range:
v : R ! Rn
v(t) = hv1 (t); :; vn (t)i
so v(t) is a vector with n
Cylinders
0 Let C be a planar curve (generating curve)
0 Lettr be a line not in the plane of the curve (axis of the cylinder) (note: we only
consider axes parallel to the coordinate axes)
0 Cylinder:
13.3: Arc Length and Curvature
Review questions:
 what is the formula for the length of the curve with vector equation
r(t), a t b
 what is the formula for the arclength s(t) of a curve with vector
12.1: Threedimensional coordinate system
R3 : f(x; y; z) j x, y, z are real numbersg
Read: "R three: the set of all ordered triples (x; y; z) such that x,
y and z are real numbers"
Geometric represe
Review 12.1:
R2
R2 : f(x; y) j x and y are real numbersg
Read: "R two: the set of all ordered pairs (x; y) such that x and
y are real numbers"
In R2 it makes no sense to talk about a plane, because
14.2: Limits and Continuity
Review questions:
 What does
lim
(x;y)!(a;b)
f (x; y) = L
mean? How can you show that such a limit doesn exist? How can you show
t
that a limit exists? (methods)
 True or
14.5: The Chain Rule
Review questions:
 State the Chain Rule (write formulas) for the case z = f (x; y) and x
and y are functions of one variable, t. What if x and y are functions of two
variables, u
Learning goals for section 14.3: Partial Derivatives
At the end of this section you should be able to:
 nd rst partial derivatives of given functions (evaluate them at given
points): 11, 12, 1540, 4
Sowhowé BLUE. Bum
NAME:
SECTION:
I have not received any help during this exam from unauthorized sources and the work '
submitted in this exam is entirely my own.
Signature: '
MATH 21259, CALCULUS I
134 "3 Arctength formula
Arclength Formula for a curve y = x) or 70) in R2:
0 Partition [a,b] into n subintervals:
a=to<t1<.<ln=b
o The arciength fort e [1}, n+1] is:
Li = IAl'il = Il'(li+1) r(t:)i,
Learning goals for section 14.6: Directional Derivative and
the Gradient Vector
At the end of this section you should be able to:
 calculate and evaluate gradients of functions of several variables

Learning goals for section 14.5: The Chain Rule
At the end of this section you should be able to:
 use the chain rule and tree diagrams to calculate and evaluate total and
partial derivatives of func
Learning goals for section 12.5 (Equations of Lines and Planes):
At the end of this lecture you should be able to:
 nd vector equations, parametric equations, symmetric equations of
lines in R3 based
Learning goals for section 12.6 (Cylinders and Quadrics):
At the end of this lecture you should be able to:
 identify a given surface as a plane, cylinder or quadric
 distinguish between a curve in
Learning goals for section 13.2 (Vector Functions and Space
Curves):
At the end of this lecture you should be able to:
 sketch plane curves and derivatives of vector function: 38
 nd intersections
Learning goals for section 13.3: Arc Length and Curvature
At the end of this lecture you should be able to:
 nd lengths of curves by using the arc length formula: 16, 11, 12, 15
 reparameterize cur
Learning goals for section 12.4 (The Cross Product):
At the end of this lecture you should:
 know how to calculate the cross product of two vectors given on components: 17, 1718
 determine whether
Carnegie Mellon University
Department of Mathematical Sciences
21259, Fall 2017
Homework 2, Problem 6 Hint
6. Find the unit tangent vector T and the unit normal vector N for the vector function
r (t)