Basic InformationCourse OutlineSyllabus ReviewVectors and Arithmetic (§12.2)Multiplying Vectors: Dot product (§12.3)Math 241, Spring 2014Jayadev S. AthreyaSpring 2014Jayadev S. Athreya Math 241, Spring 2014Basic InformationCourse OutlineSyllabus ReviewVectors and Arithmetic (§12.2)Multiplying Vectors: Dot product (§12.3)WelcomeWelcome to Math 241:Multivariable CalculusJayadev S. Athreya Math 241, Spring 2014Basic InformationCourse OutlineSyllabus ReviewVectors and Arithmetic (§12.2)Multiplying Vectors: Dot product (§12.3)Instructor InformationOffice 371 Altgeld HallOffice phone 244-0225Email [email protected] http://math.illinois.edu/~jathreya/241Office Hours Monday 2:00–3:00, Wednesday 11:00–12:00,Friday 1:00–2:00, and by appointment (subject tochange in the first few weeks).TA Room M-T 4-9pm, W-Thu, 4-8pmJayadev S. Athreya Math 241, Spring 2014Basic InformationCourse OutlineSyllabus ReviewVectors and Arithmetic (§12.2)Multiplying Vectors: Dot product (§12.3)Calculus: from single to multiple variablesMultiple VariablesOne-variable calculusIn one variable calculus, we study functions of a single realvariable:y = f (x)We take limits, derivatives, and integrals.Jayadev S. Athreya Math 241, Spring 2014Basic InformationCourse OutlineSyllabus ReviewVectors and Arithmetic (§12.2)Multiplying Vectors: Dot product (§12.3)Calculus: from single to multiple variablesMultiple VariablesDerivativesf0(x), Slope of a tangent line; instantaneous velocity; rate ofchange.Jayadev S. Athreya Math 241, Spring 2014Basic InformationCourse OutlineSyllabus ReviewVectors and Arithmetic (§12.2)Multiplying Vectors: Dot product (§12.3)Calculus: from single to multiple variablesMultiple VariablesIntegrals(Signed) area under a curve,Rbaf (x)dx:Z3/20x2dxxf (x)1322 31943x2Can be used to take averages1baRbaf (x)dxJayadev S. Athreya Math 241, Spring 2014Basic InformationCourse OutlineSyllabus ReviewVectors and Arithmetic (§12.2)Multiplying Vectors: Dot product (§12.3)Calculus: from single to multiple variablesMultiple VariablesFundamental Theorem of CalculusRate of change of area under the curve of a function is given bythe function itself:Jayadev S. Athreya Math 241, Spring 2014Basic InformationCourse OutlineSyllabus ReviewVectors and Arithmetic (§12.2)Multiplying Vectors: Dot product (§12.3)Calculus: from single to multiple variablesMultiple VariablesEvaluating IntegralsF0(x)=f (x),Zbaf (x)dx = F (b) F (a)Jayadev S. Athreya Math 241, Spring 2014Basic InformationCourse OutlineSyllabus ReviewVectors and Arithmetic (§12.2)Multiplying Vectors: Dot product (§12.3)Calculus: from single to multiple variablesMultiple VariablesReal-world problemsWe don’t often encounter quantities which are functions of onereal variable:Temperature Depends on time, location - and just location is 3variables: latitude, longtitude, elevation.Motion of a particle in space Need to describe position,velocity, each are 3 variables- so 6 dimensionalsystem.Jayadev S. Athreya Math 241, Spring 2014Basic InformationCourse OutlineSyllabus ReviewVectors and Arithmetic (§12.2)Multiplying Vectors: Dot product (§12.3)Calculus: from single to multiple variablesMultiple Variablesn-dimensional space RnNumber line R := {x : x 2 R}Plane R2:= {( x1, x2):x1, x22 R}Space R3= {(x1, x2, x3):x1, x2, x32 R}In general Rn= {(x1, x2,...,xn):xi2 R, 1 i n}Jayadev S. Athreya Math 241, Spring 2014Basic InformationCourse OutlineSyllabus ReviewVectors and Arithmetic (§12.2)Multiplying Vectors: Dot product (§12.3)Calculus: from single to multiple variablesMultiple VariablesNumber Line R x yd(x, y)Figure: The number line. d(x,y) = |x - y|.Jayadev S. Athreya Math 241, Spring 2014Basic InformationCourse OutlineSyllabus ReviewVectors and Arithmetic (§12.2)Multiplying Vectors: Dot product (§12.3)Calculus: from single to multiple variablesMultiple VariablesThe plane R2Points have two components: x =(x1, x2), y =(y1, y2)x1y2x2y1··xyd(x, y)Figure: d(x, y)=p(y1 x1)2+(y2 x2)2. What famous theorem isthis?Jayadev S. Athreya Math 241, Spring 2014Basic InformationCourse OutlineSyllabus ReviewVectors and Arithmetic (§12.2)Multiplying Vectors: Dot product (§12.3)Calculus: from single to multiple variablesMultiple VariablesThree-space R3··xyd(x, y)Figure: Three-space.d(x, y)=p(y1 x1)2+(y2 x2)2+(y3 x3)2.Jayadev S. Athreya Math 241, Spring 2014Basic InformationCourse OutlineSyllabus ReviewVectors and Arithmetic (§12.2)Multiplying Vectors: Dot product (§12.3)Calculus: from single to multiple variablesMultiple VariablesEquation of a SphereA sphere S(p, R) of radius R > 0 centered at p =(p1, p2, p3) isthe set of points in R3at distance exactly R from p. That isS(p, R):={(x1, x2, x3):(x1p1)2+(x2p2)2+(x3p3)2= R2}Jayadev S. Athreya Math 241, Spring 2014Basic InformationCourse OutlineSyllabus ReviewVectors and Arithmetic (§12.2)Multiplying Vectors: Dot product (§12.3)Calculus: from single to multiple variablesMultiple VariablesDistances and LimitsThe most important concept of calculus is the limit:Informally, we saylimx!af (x)=Lif no matter how close (✏ ) you want f (x) to get to L, you can finda small neighborhood (of size ) so that x being within of aguarantees 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 canapproach a point in many different ways.Jayadev S. Athreya Math 241, Spring 2014Basic InformationCourse OutlineSyllabus ReviewVectors and Arithmetic (§12.2)Multiplying Vectors: Dot product (§12.3)Calculus: from single to multiple variablesMultiple VariablesDistance in Rnx =(x1,...,xn), y =(y1,...,yn),d(x, y)=q(x1 y1)2+ ...+(xn yn)2.Jayadev S. Athreya Math 241, Spring 2014Basic InformationCourse OutlineSyllabus ReviewVectors and Arithmetic (§12.2)Multiplying Vectors: Dot product (§12.3)Calculus: from single to multiple variablesMultiple VariablesTechniquesMost techniques we learn will work in any dimension n, someare specific to 2 and 3 dimensions.Jayadev S. Athreya Math 241, Spring 2014Basic InformationCourse OutlineSyllabus ReviewVectors and Arithmetic (§12.2)Multiplying Vectors: Dot product (§12.3)Course PoliciesTextbooksWe will cover Chapters 12-16 ofStewart James Stewart, Calculus: Early Transcendentals,7th edition, with Enhanced WebAssign.Edition Please note that this course uses the 7th editionrather than the 6th.WebAssign You will also need WebAssign access to do thehomework. If you have
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