CalcIII_LectureWorkbook2017_new.pdf - MATH 25000 Calculus III Lecture Notes Dr Amanda Harsy 1 Contents 1 Syllabus and Schedule 7 2 Sample Gateway 19 3

CalcIII_LectureWorkbook2017_new.pdf - MATH 25000 Calculus...

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MATH 25000: Calculus III Lecture NotesDr. Amanda HarsyOctober 17, 20171
Contents1Syllabus and Schedule72Sample Gateway193Parametric Equations213.1Calculus with Parametric Equations. . . . . . . . . . . . . . . . . . . . . . .223.2Arc Length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .233.3ICE 1 -Parametric Equations. . . . . . . . . . . . . . . . . . . . . . . . . .2543-Space274.1Cartesian Coordintates in 3 Space. . . . . . . . . . . . . . . . . . . . . . . .274.2Linear Equations and Traces. . . . . . . . . . . . . . . . . . . . . . . . . . .284.3Parametric Equations in 3 Dimensions. . . . . . . . . . . . . . . . . . . . .295Vectors315.1Introduction to Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .315.2Displacement Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .335.3ICE 2 – Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .355.4Dot Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .375.4.1Projections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .385.5Cross Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .405.5.1Torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .425.6ICE 3 –Dot and Cross Products. . . . . . . . . . . . . . . . . . . . . . . . .436Vector Valued Functions457Lines and Planes497.1Lines In Space!. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .497.2Planes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .507.3Tangent Lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .517.4ICE 4 – Vector Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . .538Curvature558.1Definitions of Curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . .558.2Components of Acceleration. . . . . . . . . . . . . . . . . . . . . . . . . . .568.3ICE – Curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .599Surfaces619.1Review of Conic Sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . .619.1.1Parabolas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .619.1.2Ellipses and Hyperbolas. . . . . . . . . . . . . . . . . . . . . . . . .629.1.3Optional Topic: Translations and Rotations. . . . . . . . . . . . . .662
9.2Graphing Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .699.3General Forms of Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . .729.4ICE 6 – Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7310 Multivariate Functions7510.1 Functions of Two Variables. . . . . . . . . . . . . . . . . . . . . . . . . . .7510.2 Level Curves and Contour Maps. . . . . . . . . . . . . . . . . . . . . . . . .7610.3 ICE 7 –Level Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7911 Partial Derivatives8311.1 Higher Partial Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . .8411.2 Thinking about Partial Derivatives. . . . . . . . . . . . . . . . . . . . . . .8511.2.1 Partial Derivatives Numerically. . . . . . . . . . . . . . . . . . . . .8511.2.2 Partial Derivatives Verbally. . . . . . . . . . . . . . . . . . . . . . .8511.2.3 Partial Derivatives Graphically. . . . . . . . . . . . . . . . . . . . .8611.3 Visualizing Partial Derivatives with Play-Doh. . . . . . . . . . . . . . . . .8811.4 Ice 8— Partial Derivative Review. . . . . . . . . . . . . . . . . . . . . . . .9912 Polar Coordinates10112.1 Defining Polar Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . .10112.2ICE 9 – Polar Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . .10312.3 Polar Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10412.4 Graphing Polar Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . .10412.4.1 Circles In Polar Coordinates. . . . . . . . . . . . . . . . . . . . . . .10612.4.2 More Complex Curves In Polar Coordinates. . . . . . . . . . . . . .10612.5 ICE 10 – Polar Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10712.6 Derivatives and Tangents Lines in Polar Curves. . . . . . . . . . . . . . . .10812.7 Calculating Polar Areas. . . . . . . . . . . . . . . . . . . . . . . . . . . . .10912.8 Arc Length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11112.9 ICE 11 – Polar Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . .11313Multivariate Limits and Continuity11513.1 Continuity of Multivariate Functions. . . . . . . . . . . . . . . . . . . . . .11513.2 Evaluating Multivariate Limits. . . . . . . . . . . . . . . . . . . . . . . . .11614 Linear Approximations of the Derivative11914.1 Tangent Plane Revisited. . . . . . . . . . . . . . . . . . . . . . . . . . . . .11914.2 The Differential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12015 Differentiability of Multivariate Functions1213
16 The Chain Rule12316.1 Implicit Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12516.2ICE 12– Chain Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12717 Directional Derivatives and Gradient12917.1 Gradient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13017.2 Directional Derivatives Definition. . . . . . . . . . . . . . . . . . . . . . . .13118 Optimization13518.1 Max and Mins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13518.2 Critical Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13518.3 Using the Contour Map to Classify Critical Points. . . . . . . . . . . . . . .13818.4ICE 13– Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13919 Lagrange Multipliers14119.1 Optimizing with Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . .14119.2 2 Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14219.3ICE 14– Lagrange Multipliers. . . . . . . . . . . . . . . . . . . . . . . . . .14720Multivariate Integration14920.1 Double Integrals over Rectangular Regions. . . . . . . . . . . . . . . . . . .14920.2 Iterated Integrals over General Regions. . . . . . . . . . . . . . . . . . . . .15120.2.1General (non-rectangular) Regions of Integration. . . . . . . . . . .15120.2.2 Iterated Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . .15120.3ICE 15– Iterated Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . .15320.4Integration over Polar Regions. . . . . . . . . . . . . . . . . . . . . . . . .15521 Applications of Multivariate Integration15721.1 Density and Mass (Optional Topic). . . . . . . . . . . . . . . . . . . . . . .15721.2 Moments of Inertia/ Rotational Inertia (Optional Topic). . . . . . . . . . .16021.3ICE 16 – Mass and Density (Optional Topic). . . . . . . . . . . . . . . . .16321.4 Surface Area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16521.5ICE 17– Surface Area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16722Triple Integrals16922.1 Triple Integrals in Cartesian Coordinates. . . . . . . . . . . . . . . . . . . .16922.2ICE 18– Triple Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . .17322.3Triple Integrals Using Cylindrical and Spherical Coordinates. . . . . . . . .17522.3.1 Cylindrical Coordinates. . . . . . . . . . . . . . . . . . . . . . . . .175

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