University:
Amherst College
Course:
MATH 211 | Multivariable Calculus
Academic year:

2009
Views:

409

Pages:

16
Author:

Lauren Larson

Tuesday September 8, 2009 MA13 - Multivariable Calculus Fall 2009 Text Multivariable Calculus 6rd Edition Math 12 September 11, 2009 1 / 51 Tuesday September 8, 2009 Overview/Evaluation Elementary vector calculus; introduction to partial derivatives; extrema of functions of several variables; multiple integrals in two and three dimensions; line integrals in the plane; Green’s theorem. 5% Group Projects 15% Homework 50% Three in-class exams 30% Final Exam Math 12 September 11, 2009 2 / 51 Tuesday September 8, 2009 Curving Grades: Final grades from a previous course Math 12 September 11, 2009 3 / 51 Tuesday September 8, 2009 Three Dimensions What separates multivariable from single variable calculus is the addition of a third dimension. You have a 3-dimensional coordinate axes labeled x, y, z which are broken up into 8 octants 3 coordinate planes In 2-dim you have 1 points 2 lines Two distinct lines either intersect in a point or are parallel. In 3dim you have 1 points 2 lines 3 planes Two distinct planes either intersect in a line or are parallel. Two distinct lines intersect in a point, are parallel, or are skew. Math 12 September 11, 2009 5 / 51 Tuesday September 8, 2009 The distance between two points You can have distance formula formula q |P1 P2 | = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 Benjamin Hutz (Amherst College) Math 12 September 11, 2009 6 / 51 Tuesday September 8, 2009 The distance between two points You can have distance formula formula q |P1 P2 | = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 Proof. Rectangular box with P1 and P2 in opposite corners and label the in-between corners as A, B. So you have from Pythagorean Theorem that |P1 P2 |2 = |P1 B|2 + |P2 B|2 |P1 B|2 = |P1 A|2 + |AB|2 Combining these we get |P1 P2 |2 = |P1 A|2 + |AB|2 + |P2 B|2 = |x2 − x1 |2 + |y2 − y1 |2 + |z2 − z1 |2 Math 12 September 11, 2009 6 / 51 Tuesday September 8, 2009 A problem Problem Find an equation for describing the set of all points equidistant from (−1, 2, 3) and (3, 0, −1). (What common geometric object is the set?) Math 12 September 11, 2009 7 / 51 Tuesday September 8, 2009 A problem Problem Find an equation for describing the set of all points equidistant from (−1, 2, 3) and (3, 0, −1). (What common geometric object is the set?) Proof. We need q q (x + 1)2 + (y − 2)2 + (z − 3)2 = (x − 3)2 + y 2 + (z + 1)2 which is 2x − y − 2z + 1 = 0. This is a plane. Benjamin Hutz (Amherst College) Math 12 September 11, 2009 7 / 51 Tuesday September 8, 2009 Another example Problem Find an equation of the sphere with center (2, −6, 4) and radius 5. Describe its intersection with each of the coordinate planes. Math 12 September 11, 2009 8 / 51 Tuesday September 8, 2009 Another example Problem Find an equation of the sphere with center (2, −6, 4) and radius 5. Describe its intersection with each of the coordinate planes. Proof. The equation is (x − 2)2 + (y + 6)2 + (z − 4)2 = 25. The intersection with the xy -plane(z = 0) is (x − 2)2 + (y + 6)2 = 9 a circle of radius 3 centered at (2, −6). The intersection with the xz-plane (y = 0 is (x − 2)2 + (z − 4)2 = −9. In other words, there intersection is empty. 2 4 The intersection with the yz-plane(x √ = 0) is (y + 6) + (z − 4) = 21. In other words, a circle of radius 21 centered at (−6, 4). Math 12 September 11, 2009 8 / 51 Tuesday September 8, 2009 Vectors Definition A vector has both a magnitude a direction. A scalar has only a magnitude. For example, 20mph is a scalar, but 20mph due east is a vector. Some Terminology: Initial Point Terminal Point Components Magnitude or length → − We will denote a vector as a = ha1 , a2 , a3 i where a1 is the number of units to move in the x direction, etc. Math 12 September 11, 2009 9 / 51 Tuesday September 8, 2009 Magnitude and Unit vectors Definition → − The magnitude of a vector a is the distance from its initial point to its terminal point and is denoted |a|. (The initial point is assumed to be the origin unless otherwise specified). Example |h3, 4i| = √ 33 + 42 = 5. Definition A unit vector is a vector with magnitude 1 and is generally thought of as a “direction”. Example √1 3 h1, 1, 1i. Math 12 September 11, 2009 10 / 51 Tuesday September 8, 2009 Parallel in 3-dimensions Problem How do we tell if two lines are parallel? Math 12 September 11, 2009 11 / 51 Tuesday September 8, 2009 Parallel in 3-dimensions Problem How do we tell if two lines are parallel? Proof. In two dimensions we have y = mx + b. If m1 = m2 then they are parallel, in other words if hm1, 1i = c hm2, 1i for some constant c. Then we can easily generalize to 3-dimensions with vectors: two lines are parallel if their direction vectors are equal up to scaling. Math 12 September 11, 2009 11 / 51 Tuesday September 8, 2009 Vector Addition and Scalar Multiplication Addition of vectors ha1 , a2 i + hb1 , b2 i = ha1 + b1 , a2 + b2 i. (triangle and parallelogram laws(commutativity)) scalar multiplication c ha1 , a2 i = hca1 , ca2 i. There are at least two ways to define vector multiplication (dot product, cross product) which we will discuss later. Math 12 September 11, 2009 12 / 51 Tuesday September 8, 2009 Alternate Notation We define î = h1, 0, 0i ĵ = h0, 1, 0i k̂ = h0, 0, 1i and so we can write ha1 , a2 , a3 i = a1 î + a2 ĵ + a3 k̂. Math 12 September 11, 2009 13 / 51

Multivariable Calculus Lecture Slides Part 1

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