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**Unformatted text preview: **18.02 lVlultivariable Calculus (Spring 2009): Lecture 1
Vectors. Dot product. February 3 Reading Material: From Simmons: 17.3 and 18.2 Today: introduction. Vectors. Dot Product. Vectors and Coordinates. 2 Introduction To translate in modern language what Galileo Galilei was saying in the 10th century, we can cer-
tainly describe Calculus as the language of nature. With this language one can compute precisely
completely different objects: from the orbit of a planet to the flow over the wings of an aircraft, from
the amount of yellow paint needed to paint part of the State building to the best approximation to
the data point of an experiment in biology. But Calculus is also the basic foundations for much more sophisticated branches of i'in-itliematics:
analytic number theory, probability, ai'lalysis, differential geometry etc. For some of you this class
will be about learning the tools that will allow you to understand and connnunicate your findings
in Biology, Engineering, Computer Science and so on; for others it will be the beginning of the road
that leads to abstract research in rnatl'lernatics. For all I hope it will be, if not fun, at least a bit
interesting and inspiring. Many of you are coming from Math 18.01. This class is based on single variable calculus. You
studied functions f(:r) depending on a single variable 3:. For these functions you learned about
graphs, tangent lines, integrals, areas of subgraphs. I am sure that each of these abstract concept
has been connect to a concrete concept in nature. On the other hand Math 18.02 is based on multi variable calculus. Here you will study functions
of two (or more) variables; f(.1:,y). For these functions we will deﬁne partial derivatives, tangent
planes, multiple integrals, volumes. ‘Ne will also encounter objects which mathematical definition
may be completely new to you, but not their physical meaning. One example are vectors and vector
ﬁelds. in this class almost no proofs will be given. Sometimes I will give an idea of how a physical concept
that you are familiar with takes a particular mathematical formula. In general I will not present the
sophisticated mathematics behind a certain formula, but rather I will outline the basic ideas that
guided the mathematicians or physicists who came up with those formulas. A more rigorous version of this class is 18.022. 3 Vectors To understand what a vector represents it is better to first deﬁne what a scalar is. o A scalar is a number. Often we denote a number using roman letters like :r, y, 2 etc. if this
number represents a variable, or a, b,c etc. if the number is ﬁxed from the beginning. 0 A vector is a, directed line segment, it hst a length {which is e scalar) and a. direction given
by on urrow. (ﬁgure of a vector “> A Note: for us two vectors with the same length and direction ure equal, we do not care about
their position in space: (figure of two equal vectors A and B:) a ’l/ pf.— 13” Vector Notation 0 fl is a general vector
- ti. is a unit vector (length : 1) - U is the zero vector : the vector of length zero Note: In books often bold is used: A, u. —>
n We also use the notation CD to denote the vector given by the arrow starting at C and ending at D. C
[4/
#7
Vector Arithmetic A C I Addition: A -i- B (ﬁgure relative to the sum of two vectors A and B:) *7 .5.»
-—33
A f} e h. C? o Scalar multiplication: oil. If c 2 Oqthen Oil. is a vector in the direction of fl with length L:
times the length of A. If c < 0 then oil is a vector in the reverse direction of A with length ic|
times the length of A. (figure relative to the scalar product of a scalar c and a vector 11:) 1g (.22 3.91 )7 6A - Subtraction: A — E = fl-i- (—1)B (ﬁgure relative to the subtraction of two vectors A and B1) ..:7
ﬂ ?‘ a? 0 Length of xi: We denote = length of if (this is a scalar!). Note: '8'. = = unit vector in the direction of .4. 4 Dot Product (aka Scalar Product) Dot Product of E = fl |§| cos 6, where 6‘ : angle between E and (ﬁgure relative to the dot product of two vectors A and B2) 2 ﬂ 0
.n“) P) Intuition: It tells you how much A and B point in the same direction. Handy Facts will (cost) 051‘; = ll 1) o .4 - B" : 0 if and only if ( c?» o (A - 11):”: =colnponentz of A in {1. direction. (figure relative to component of A along ’I 4
(/1 WM.) 21
0 Distribution Law: A-(B+ a.) = (A-B)+(.AT.C7) 5 Vectors and Coordinates o (ﬁgure of a coordinate system in the (my) mi‘iublesz) A3 (1.5) ((1,1)) =Vector from origin to point (a, b) (2D picture) c (figure of a coordinate system in the (.1; y, .2) variables:) (a, b,c) :vector from origin to point (a, b,c) (3D picture) ) either L 13" 01' either = 0 or E 0‘ (cos ﬁ/‘Z -_ -. CG. 5c) I Here we are making a so called abuse of notation by using the same symbol b. c)
to describe both the point (o,b,c) in space and the vector starting at the origin of
the coordinate system and ending at the point itself. I 'i,_*j,1.': 2 unit vectors along :5, y, 3 directions I As a consequence A A ﬂ
(o, b,c) = oi + bj + ck. (5.1) u Pythagorean Theorem: |(o.,b,c)| = «(12—1-52 + c2. (5.2) o If C : ((Lhag) and D = (in . ()9) are two points on a coordinate plane, then t—y CD :(i)1,bg)—((J,1‘(},2)=(b]— ebb-3 — (Lo). (ﬁgure of the vector CD on a coordinate system in the (:r, y) variables:) :3 carnal) ——-‘.>
c. D Note: There is an obvious generalization in 3D! Dot Product in coordinate notation If fl = (m, (Lg): E = ([)1,bg). then A - B : o1b1+ (1.252. (5.3)
Proof. It follows directly from the distribution law and Handy Facts: - E = ((1.1% + - (byi + bgfj) : {11be ' "i- (Oilbg + (Lgbﬂt + (Lgbgj - = (“bl + (TL-3'52. Dot Product in 3D coordinate notation If = (ri],(12,ri3).§ : (b1, {1212);1), then A = [Mb] + (Jigbg + (L3i)3. (5.4) CH Exercise 1. Give-:1. ttw'ee points C : (1,1,1),D = (2,3,4) and E : (-1, 3,2), write the coordinates
A —a a —>
of the vectors A = CD and B = CE and ﬁnd the angle 19 between them. (ﬁgure of u. coordinate system in the (11:, 95.2) *umniobtes with vectors 1 = C—lj = and B :
CE: (132)) [Mile 1304‘ Wu. golﬁn‘ts 5, C W, m‘ wan/mug Eu W m
Tau 164.06% I ) Solution: We ﬁrst write f1 = (2 ~ 1,3 — 1.4— 1) = (1,2,3) and, E = (4—1,?) — 1,2 —— 1) : (3,2,1). We now emrtbine the two (equivalent) deﬁnitio'rts for seam-r product to obtain the “identity: B = COS-0 I alb1+ {£21}; + agbg. From here we derive a form/LL50, for 0: In our case a n. of It follows that Study Guide 1. Work on. the following questions: 0 Prove the fb-‘Hnula (5.1) by using elementary geometry and the deﬁnition of sum 0f vectors. I By ﬁrst noticing that for wample = ILLI, prove by using the Pythagorean Theorem. a Prom: by using the HUME my'ument presented to prove (5.3). ...

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