Basic Information Course Outline Syllabus Review Vectors and Arithmetic 12 2 Multiplying Vectors Dot product 12 3 Math 241 Spring 2014 Jayadev S Athreya Spring 2014 Jayadev S Athreya Math 241 Spring 2014 Basic Information Course Outline Syllabus Review Vectors and Arithmetic 12 2 Multiplying Vectors Dot product 12 3 Welcome Welcome to Math 241 Multivariable Calculus Jayadev S Athreya Math 241 Spring 2014 Basic Information Course Outline Syllabus Review Vectors and Arithmetic 12 2 Multiplying Vectors Dot product 12 3 Instructor Information Office 371 Altgeld Hall Office phone 244 0225 Email jathreya illinois edu Website http math illinois edu jathreya 241 Office Hours Monday 2 00 3 00 Wednesday 11 00 12 00 Friday 1 00 2 00 and by appointment subject to change in the first few weeks TA Room M T 4 9pm W Thu 4 8pm Jayadev S Athreya Math 241 Spring 2014 Basic Information Course Outline Syllabus Review Vectors and Arithmetic 12 2 Multiplying Vectors Dot product 12 3 Calculus from single to multiple variables Multiple Variables One variable calculus In one variable calculus we study functions of a single real variable y f x We take limits derivatives and integrals Jayadev S Athreya Math 241 Spring 2014 Basic Information Course Outline Syllabus Review Vectors and Arithmetic 12 2 Multiplying Vectors Dot product 12 3 Calculus from single to multiple variables Multiple Variables Derivatives f 0 x Slope of a tangent line instantaneous velocity rate of change Jayadev S Athreya Math 241 Spring 2014 Basic Information Course Outline Syllabus Review Vectors and Arithmetic 12 2 Multiplying Vectors Dot product 12 3 Calculus from single to multiple variables Multiple Variables Integrals Signed area under a curve f x Rb a f x dx x2 3 9 4 1 Z 3 2 2 x dx 0 1 3 2 2 Can be used to take averages 1 Jayadev S Athreya 3 b a Rb a x f x dx Math 241 Spring 2014 Basic Information Course Outline Syllabus Review Vectors and Arithmetic 12 2 Multiplying Vectors Dot product 12 3 Calculus from single to multiple variables Multiple Variables Fundamental Theorem of Calculus Rate of change of area under the curve of a function is given by the function itself Jayadev S Athreya Math 241 Spring 2014 Basic Information Course Outline Syllabus Review Vectors and Arithmetic 12 2 Multiplying Vectors Dot product 12 3 Calculus from single to multiple variables Multiple Variables Evaluating Integrals F 0 x f x Z b a f x dx F b Jayadev S Athreya F a Math 241 Spring 2014 Basic Information Course Outline Syllabus Review Vectors and Arithmetic 12 2 Multiplying Vectors Dot product 12 3 Calculus from single to multiple variables Multiple Variables Real world problems We don t often encounter quantities which are functions of one real variable Temperature Depends on time location and just location is 3 variables latitude longtitude elevation Motion of a particle in space Need to describe position velocity each are 3 variables so 6 dimensional system Jayadev S Athreya Math 241 Spring 2014 Basic Information Course Outline Syllabus Review Vectors and Arithmetic 12 2 Multiplying Vectors Dot product 12 3 Calculus from single to multiple variables Multiple Variables n dimensional space Rn Number line R x x 2 R Plane R2 x1 x2 x1 x2 2 R Space R3 x1 x2 x3 x1 x2 x3 2 R In general Rn x1 x2 xn xi 2 R 1 i n Jayadev S Athreya Math 241 Spring 2014 Basic Information Course Outline Syllabus Review Vectors and Arithmetic 12 2 Multiplying Vectors Dot product 12 3 Calculus from single to multiple variables Multiple Variables Number Line R d x y x y Figure The number line d x y x y Jayadev S Athreya Math 241 Spring 2014 Basic Information Course Outline Syllabus Review Vectors and Arithmetic 12 2 Multiplying Vectors Dot product 12 3 Calculus from single to multiple variables Multiple Variables The plane R2 Points have two components x x1 x2 y y1 y2 y2 x2 x y x d x1 p Figure d x y y1 this x1 2 y2 Jayadev S Athreya y y1 x2 2 What famous theorem is Math 241 Spring 2014 Basic Information Course Outline Syllabus Review Vectors and Arithmetic 12 2 Multiplying Vectors Dot product 12 3 Calculus from single to multiple variables Multiple Variables Three space R3 x Figure Three space p d x y y1 x1 2 y2 d x y x2 2 y3 Jayadev S Athreya y x3 2 Math 241 Spring 2014 Basic Information Course Outline Syllabus Review Vectors and Arithmetic 12 2 Multiplying Vectors Dot product 12 3 Calculus from single to multiple variables Multiple Variables Equation of a Sphere A sphere S p R of radius R 0 centered at p p1 p2 p3 is the set of points in R3 at distance exactly R from p That is S p R x1 x2 x3 x1 p1 2 x2 p2 2 x3 p3 2 R 2 Jayadev S Athreya Math 241 Spring 2014 Basic Information Course Outline Syllabus Review Vectors and Arithmetic 12 2 Multiplying Vectors Dot product 12 3 Calculus from single to multiple variables Multiple Variables Distances and Limits The most important concept of calculus is the limit Informally we say lim f x L x a if no matter how close you want f x to get to L you can find a small neighborhood of size so that x being within of a guarantees that f x is within of L All this relies on is a notion of distance We have to be careful in higher dimensions because you can approach a point in many different ways Jayadev S Athreya Math 241 Spring 2014 Basic Information Course Outline Syllabus Review Vectors and Arithmetic 12 2 Multiplying Vectors Dot product 12 3 Calculus from single to multiple variables Multiple Variables Distance in Rn x x1 xn y y1 yn q d x y x1 y1 2 xn Jayadev S Athreya Math 241 Spring 2014 yn 2 Basic Information Course Outline Syllabus Review Vectors and Arithmetic 12 2 Multiplying Vectors Dot product 12 3 Calculus from single to multiple variables Multiple Variables Techniques Most techniques we learn will work in any dimension n some are specific to 2 and 3 dimensions Jayadev S Athreya Math 241 Spring 2014 Basic Information Course Outline Syllabus Review Vectors and Arithmetic 12 2 Multiplying Vectors Dot product 12 3 Course Policies Textbooks We will cover Chapters 12 16 of Stewart James Stewart Calculus Early Transcendentals 7th edition with Enhanced WebAssign Edition Please note that this course uses the 7th edition rather than the 6th WebAssign You will also need WebAssign access to do the homework If you have the standard text and WebAssign package from Math 220 221 or 231 for last year then you already have everything you need for this course Jayadev S Athreya Math 241 Spring 2014 Basic Information Course Outline Syllabus Review Vectors
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