# Home

#### Course Notes

Links to the class notes also appear in the schedule. If you use Firefox, the links will take you directly to the relevant section for the day; otherwise, use the table of contents to navigate to the relevant section.

#### Schedule

This schedule will be updated as necessary throughout the semester. Section numbers refer to the text Calculus: Early Transcendentals, James Stewart; Brooks/Cole, 8th edition, 2016.

 Week 1 (8/21–8/25)    Introduction to Vectors M No Class: Eclipse Day W Topic 1.1: Vectors & 3-Space, Introduction (sec 12.1, 12.2) Cartesian co-ordinates and vectors in component form. The position vector $\boldsymbol{r}$. Activity / Key Slides / Class Notes F Topic 1.2: Working with Vectors (sec 12.2) Magnitude of vectors. Vector addition and scalar multiplication. Unit vectors. $\boldsymbol{\hat{\imath}}$, $\boldsymbol{\hat{\jmath}}$, $\boldsymbol{\hat{k}}$ basis vectors. Activity / Key Slides Week 2 (8/28–9/1)    Dot and Cross Products M Topic 1.3: Dot and Cross Products, Introduction (sec 12.3, 12.4) Definitions and computation. Activity / Key Slides W Topic 1.4: Geometry of the Dot and Cross Products (sec 12.3, 12.4) The relationships of the dot and cross products to angles, area, and orthogonality. Activity / Key Slides F Topic 1.5: Projections & Work (sec 12.3) Applications of the dot product: projections and work. Draft Activity/Class Notes Slides Week 3 (9/4–9/8)    Lines & Planes; Introduction to Vector Functions M No Class: Labor Day W Topic 1.6: Lines & Planes (sec 12.5) Applications of vectors and vector operations: equations of lines and planes. Activity / Key Slides F Topic 1.7: Vector Functions & Curves, Introduction (sec 13.1) The position vector and vector-valued functions. Curves in the plane and in 3-space. Common Parameterizations for Some Important Curves Activity / Key Slides Week 4 (9/11–9/15)    Derivatives of Vector Functions M Topic 1.8: Derivatives of Vector Functions – Computation & Geometry (sec 13.2, 13.3) The tangent vector $\boldsymbol{r}'(t)$, the unit tangent vector $\boldsymbol{\hat{T}}$, and the vector and scalar line elements $d\boldsymbol{r}$ and $ds$. Activity / Key Slides / Class Notes W Topic 1.9: Applications of Vector Derivatives: Motion on Curves (sec 13.3, 13.4) Velocity, speed, and acceleration; length of curves & the arc length function. Activity / Key Slides F [Topic 1.10: Applications of Vector Derivatives: Line Elements $d\boldsymbol{r}$ and $ds$, and Arc Length][1.10N] (sec 13.3) Bridge Book: More details on the vector line element $d\boldsymbol{r}$ and its magnitude $ds$. Slides: Topic 1 Review Class Notes Week 5 (9/18–9/22)    Multivariate Functions & Partial Derivatives M Topic 2.1: Multivariate Functions, Introduction (sec 14.1) Domain and range. Graphs, traces, level curves, and contour maps. Links to online and downloadable graphing apps. Activity / Key Slides W Topic 2.1: Multivariate Functions, Introduction (continued) (sec 14.1) Domain and range. Graphs, traces, level curves, and contour maps. Links to online and downloadable graphing apps. Activity / Key Slides F Topic 2.2: Partial Derivatives. (sec 14.3) Derivatives of functions of two or more variables. Activity / Key Slides Week 6 (9/25–9/29)    TBA & Exam 1 M Topic 2.3: Differentiability & Tangent Planes (sec 14.4) Higher-dimensional analogues of tangent lines. Class Notes Slides W Exam 1: Take-Home (Due at the beginning of class on Friday) You must come to class to pick up the take-home exam. Exam Information F Exam 1: In-Class During class, in our regular classroom. Exam Information Week 7 (10/2–10/6)    Differentials; Directional Derivatives; the Gradient; Chain Rules M Topic 2.4: Applications of Differentiability: the Differential $df$ (sec 14.4) Differentials of functions. Activity / Key Slides W Topic 2.5: Directional Derivatives & the Gradient (sec 14.6) More derivatives of functions of two or more variables. The gradient of a function $\boldsymbol{\nabla}f$. Activity / Key Slides F Topic 2.6: Chain Rules (sec 14.5, 14.6) Chain rules for multivariate functions. Class Notes Slides Week 8 (10/9–10/13)    Optimization; Introduction to Double Integrals M Topic 2.7: Local Extrema & the Second Derivative Test (sec 14.7) Application of partial derivatives and the gradient: Finding maxima and minima of functions of two variables. Critical points, local maxima and minima, and the second derivative test. Activity / Key Slides W Topic 2.8: Method of Lagrange Multipliers (sec 14.8) Application of partial derivatives and the gradient: Finding the critical points of a multivariate function subject to a constraint using the method of Lagrange multipliers. Class Notes Slides F Topic 3.1: Double Integrals, Introduction (sec 15.1, 15.2) Double integrals in Cartesian cooridinates, and the area element $dA$. Activity / Key Slides Week 9 (10/16–10/20)    Double Integrals; Polar Coordinates; Introduction to Triple Integrals M Topic 3.1: Double Integrals, Continued (sec 15.2, 15.4) Integration over general regions, and applications of the double integral (area, volume, mass). Class Notes Slides W Topic 3.2: Double Integrals in Polar Coordinates (sec 10.3, 15.3) The area element $dA$ in polar coordinates. Finding limits of integration. Activity / Key Slides F Topic 3.3: Triple Integrals in Cartesian Coordinates (sec 15.6) The volume element $dV$. Finding limits of integration. Applications of triple integrals: volume and mass. Activity / Key Slides Week 10 (10/23–10/27)    Triple Integrals in Cylindrical & Spherical Coordinates M Topic 3.4: Triple Integrals in Cylindrical Coordinates (sec 15.7) Cylindrical coordinates. The volume element $dV$ in cylindrical coordinates. Finding limits of integration. Activity / Key Slides W Topic 3.5: Triple Integrals in Spherical Coordinates (sec 15.8) Spherical coordinates. The volume element $dV$ in spherical coordinates. Finding limits of integration. Activity / Key Slides F Homework Day In-class work on assignments #22 & #23 (triple integrals in cylindrical and spherical coordinates). Class Notes Week 11 (10/30–11/3)    TBA & Exam 2 M Exam Review Triple Integral Practice W Exam 2: Take-Home (Due at the beginning of class on Friday) You must come to class to pick up the take-home exam. Exam Information F Exam 2: In-Class During class, in our regular classroom. Exam Information Week 12 (11/6–11/10)    Line Integrals M Topic 4.1: Line Elements & Scalar Line Integrals (sec 13.3, 16.2) The scalar line element $ds$ and the vector line element $d\boldsymbol{r}$. Scalar line integrals. Bridge Book: More details on the vector line element $d\boldsymbol{r}$. Common Parameterizations for Some Important CurvesGuide to Setting Up Scalar Line Integrals from a Parameterization Mini-Activity / Key + Example Slides W Topic 4.2: Vector Fields, Introduction (sec 16.1) Vector fields and the geometry of vector line integrals. Activity / Key Slides F Topic 4.3: Vector Line Integrals (sec 16.2) Vector line element $d\boldsymbol{r}$ and vector line integrals.Guide to Setting up Vector Line Integrals from a Parameterization Class Notes Slides Week 13 (11/13–11/17)    Integral Theorems: FTLI & Green’s; Introduction to Surfaces M Topic 4.4: Conservative Vector Fields (sec 16.3) Conservative vector fields and the Fundamental Theorem of Line Integrals. Activity / Key Slides W Topic 4.5: Green’s Theorem (sec 16.4) An integral theorem equating line integrals in the plane to double integrals. Class Notes Slides F Topic 4.6: Surfaces & Surface Elements (sec 16.6) Surfaces, parameterizations, grid curves, and the geometry of the scalar surface element $dS$ and the vector surface element $d\boldsymbol{S}$. Common parameterizations for some important surfaces. Activity / Key Slides 11/20–11/24: Fall Break – No Class Week 14 (11/27–12/1)    Surface Integrals; Divergence Theorem M Topic 4.7: Scalar Surface Integrals (sec 16.7) Integrating a scalar function over a surface. Applications of scalar surface integrals: surface area, mass.Guide to Setting Up Surface Integrals from a Parameterization W Topic 4.8: Vector Surface Integrals (sec 16.7) Integrating a vector field over a surface. (Also called a flux integral.) F Topic 4.9: The Divergence Theorem (sec 16.9)The Divergence Theorem is an integral theorem equating surface integrals to triple integrals. Week 15 (12/4–12/8)    Stokes’ Theorem; Catch-up/Review M Topic 4.10: Stokes’ Theorem (sec 16.8) Stokes’ Theorem is an integral theorem equating line integrals and surface integrals. (Green’s Theorem is a special case of Stokes’ Theorem.) W Open/TBA F Open/TBA Finals Week: 12/11–12/15 M sec 001: Mon 12/11 10am–12pm in our usual classroom. W sec 002: Wed 12/13 10am–12pm in our usual classroom. Exam Information