MATH 2934.270 - Differential and Integral Calculus III - Summer 2014
MTWRF 10:30 a.m. - 12:05 p.m., 212 PHSC

Instructor: Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu

Office Hours: M 1:30-2:30, W 1:30-2:30, F 9:30-10:30.

Math Center: You can get help in the Math Center located at 209 PHSC, open every workday from 9:15 a.m. to 1:15 p.m.

Course catalog description: Prerequisite: 2924 with grade of C or better. Duplicates one hour of 2433 and three hours of 2443. Vectors and vector functions, functions of several variables, partial differentiation and gradients, multiple integration, line and surface integrals, Green-Stokes-Gauss theorems. (F, Sp, Su)

Text: J. Stewart, Calculus, 7th edition, Brooks/Cole, 2012. The course will cover major parts of chapters 12-16.

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Homework:

• Homework 1 (problems given on June 30, July 1, 2), due July 7 (Monday).
• Homework 2 (problems given on July 3, 7), due July 9 (Wednesday).
• Homework 3 (problems given on July 8, 9), due July 11 (Friday).
• Homework 4 (problems given on July 10, 11), due July 15 (Tuesday).
• Homework 5 (problems given on July 14, 15), due July 17 (Thursday).
• Homework 6 (problems given on July 16, 17), due July 22 (Tuesday).
• Homework 7 (problems given on July 21, 22), due July 24 (Thursday).
• Homework 8 (problems given on July 23, 24), due July 29 (Tuesday).
• Homework 9 (problems given on July 25, 28), due July 30 (Wednesday).
• Homework 10 (problems given on July 29, 30, 31), due August 4 (Monday).
• Homework 11 (problems given on August 1, 4, 6), due August 8 (Friday).
• Homework 12 (problems given on August 7, 8), due August 12 (Tuesday).
• Homework 13 (problems given on August 11, 12), due August 14 (Thursday).

Content of the lectures:

• Lecture 1 (Mon, Jun 30): Three dimensional coordinates systems what is a dimension?; coordinates; the space Rⁿ; Cartesian (adjective from the family name of Rene Descartes) coordinates in three dimensions, right-hand rule; two halfs of R, four quadrants of R², eight octants of R³; projection of a point onto a coordinate plane and onto a coordinate axis; Cartesian product of sets, R³ as R×R×R; equations of objects in R³; distance formula in R³; equation of a sphere in R³ [Sec. 12.1]
  Vectors: displacement vector, initial and terminal points; what does u=v mean?; addition of vectors ("the triangle rule" and "the parallelogram rule"), "scalar"="number", multiplication of a vector by a scalar, difference of vectors (defined through the other two operations); components of a vector, position vector, magnitude (length) of a vector, vector addition and multiplication of a vector by a scalar in components; fundamental properties of vectors [pages 815-819 of Sec. 12.2]
  Homework: Exercises 12.1 / 7, 9, 12, 16, 22, 28, 38; 12.2 / 4, 8, 16.
  FFT: Exercises 12.1 / 5 (hint), 13 (hint), 21 (hint), 27 (hint), 33 (hint), 37 (hint), 41 (hint); 12.2 / 3 (hint), 7, 13 (hint).
  Remark: The FFT ("Food For Thought") exercises are to be solved like regular homework problems, but do not have to be turned in.
• Lecture 2 (Tue, Jul 1): Vectors (cont.): standard basis vectors i=⟨1,0,0⟩, j=⟨0,1,0⟩, k=⟨0,0,1⟩; a=⟨a₁,a₂,a₃⟩=a₁i+a₂j+a₃k; the unit vector in the direction of a≠0 is a/|a| [pages 820-821 of Sec. 12.2]
  The dot product: definition of dot product (scalar product, inner product); elementary properties of dot product; angle θ between vectors, a⋅b=|a||b|cosθ expressing cosθ through a⋅b; perpendicular (orthogonal) vectors, condition for orthogonality expressed through the dot product; direction angles and direction cosines, vector projection proj_ab of b onto a, scalar projection comp_ab of b onto a, geometric meanings and formal expressions for proj_ab and comp_ab [pages 824-829 of Sec. 12.3]
  The cross product: matrices; determinants of square matrices of size 2×2 and 3×3; definition of a cross product (vector product) [pages 832-833 of Sec. 12.4]
  Homework: Exercises 12.2 / 20, 24, 28, 42(a,b); 12.3 / 11 (hint), 19 (hint), 27 (hint), 38, 39, 54.
  FFT: Exercises 12.3 / 26, 45 (hint), 47 (hint), 55 (hint).
• Lecture 3 (Wed, Jul 2): The cross product (cont.): basic facts about the cross product: (a×b)⊥a, (a×b)⊥b, |a×b|=|a||b|sinθ, direction of a×b (the right hand rule), condition for a to be perpendicular to b (a×b=0), geometric interpretation of |a×b| as the area of a parallelogram; further properties of the cross product; triple product a⋅(b×c), geometric interpretation of the triple product as (plus or minus) the volume of a parallelopiped, properties of the triple product [pages 834-837 of Sec. 12.4]
  Equations of lines and planes: lines: vector equation, parameter, parametric equations, direction numbers, symmetric equations [pages 840-842 of Sec. 12.5]
  Homework: Exercises 12.4 / 12, 14, 20, 31 (hint), 33, 45 (hint), 48, 49 (hint); 12.5 / 10, 17, 20, 22.
  FFT: Exercises 12.4 / 9, 11, 13 (hint), 19 (hint), 37, 38, 53 (hint); 12.5 / 1, 7 (hint), 13 (hint), 19 (hint).
  The complete Homework 1 is due on July 7 (Monday).
• Lecture 4 (Thu, Jul 3): Equations of lines and planes (cont.): planes: normal vector, vector, scalar, and linear equations of a plane; parallel planes, angle between planes; distance from a point to a line, distance from a point to a plane [pages 843-847 of Sec. 12.5]
  Homework: Exercises 12.5 / 26, 28, 30, 34, 37, 57, 64, 71.
  FFT: Exercises 12.5 / 51 (hint), 63 (hint), 67, 68, 75 (hint).
• Lecture 5 (Mon, Jul 7): Vector functions and space curves: component functions, vector functions, limits and continuity of vector functions, space curves, parametric equations of a space curve, curves obtained as intersections of surfaces, examples [Sec. 13.1]
  Derivatives and integrals of vector functions: derivative of a vector function, tangent vector to a space curve at a point; differentiation rules [pages 871-874 of Sec. 13.2]
  Homework: Exercises 13.1 / 2, 4, 18, 28, 30, 42, 48 (hint: see Exercise 13.1/47); 13.2 / 8, 16, 18, 21.
  FFT: Exercises 13.1 / 21-26 (hint), 27 (hint), 41 (hint); 13.2 / 3 (hint), 15 (hint), 19 (hint).
  The complete Homework 2 is due on July 9 (Wednesday).
• Lecture 6 (Tue, Jul 8): Derivatives and integrals of vector functions (cont.): a vector function determines a single space curve, a space curve corresponds to infinitely many vector functions (differing by the choice of parameterization); higher-order derivatives; unit tangent vector to a space curve at a point; examples (|r(t)|=const implies that r'(t)⋅r(t)=0, d|r(t)|/dt=r'(t)⋅r(t)/|r(t)|; definite integrals of vector functions [pages 874-875 of Sec. 13.2]
  Arc length and curvature: derivation of the expression for arc length by representing it as a limit of Riemann sums [pages 877-879 of Sec. 13.3]
  Homework: Exercises 13.2 / 23, 34, 36, 51, 55; 13.3 / 3 (hint), additional problem.
  FFT: Exercises 13.2 / 25 (hint), 53 (hint); 13.4 / 22 (hint, compare this exercise with 13.2/53).
• Lecture 7 (Wed, Jul 9): Functions of several variables: functions of two variables, independent variables, dependent variable, domain, range; graph, level curves; examples; functions of three or more variables [Sec. 14.1]
  Limits and continuity: limits of functions of one and several variables, examples; continuity of functions of one and several variables; proving discontinuity of a function at a point by finding different values when approaching the point along different paths, examples [skip Definition 1, read Examples 1-3 of Sec. 14.2]
  Homework: Exercises 14.1 / 14, 18, 20, 47 (hint); 14.2 / 5 (see Example 5), 9 (hint), 15 (see Example 3 and consider, e.g., the path x=0 and the path y=2x²), 17 (use polar coordinates), 19, 41.
  FFT: Exercises 14.1 / 19 (hint), 21, 25 (hint), 26, 32 (hint), 36, 39-42, 59(hint)-64; 14.2 / 13 (hint), 21 (hint).
  The complete Homework 3 is due on July 11 (Friday).
• Lecture 8 (Thu, Jul 10): Limits and continuity (cont.): more on the meaning continuity, using polar coordinates to understand the discontinuity of the function from Example 1 (a digression on trigonometry: everything follows from the formulas sin(α+β)=sin(α)cos(β)+cos(α)sin(β), cos(α+β)=cos(α)cos(β)−sin(α)sin(β)); extending or modifying a function to make it continuous (not always possible!); continuity of rational functions where the denominator is non-zero, continuity of compositions of continuous functions, examples [read Examples 5-9 of Sec. 14.2]
  Partial derivatives: definition of partial derivatives for functions of two variables; practical rules for finding partial derivatives, examples; geometric meaning of the partial derivatives of a function of two variables [pages 924-929 of Sec. 14.3]
  Homework: Exercises 14.2 / 30, 32, 36, 42 (no graphing); 14.3 / 16, 24, 26, 34, 40, 41.
  FFT: Exercises 14.2 / 25 (hint), 33; 14.3 / 10, 73 (hint), 74.
• Lecture 9 (Fri, Jul 11): Partial derivatives (cont.): more on the geometric meaning of partial derivatives of a function of two variables; definition of partial derivatives for functions of n variables; higher derivatives; Clairaut's Theorem; ordinary and partial differential equations; implicit differentiation [pages 929-933 of Sec. 14.3]
  Tangent planes and linear approximations: equation of the tangent plane to a surface defined as a graph of a function, linear approximation (tangent plane approximation) of a function at a point [pages 939-941 of Sec. 14.4]
  Remark: Reviewing Sec. 2.6 and pages 183-184 of Sec. 2.9 will be useful.
  Homework: Exercises 14.3 / 30, 50, 52 (hint), 53, 64, 71, 78(a), 93 (hint); 14.4 / 6, 11 (only find the linearization, hint), 42.
  FFT: Exercise 14.3 / 59.
  The complete Homework 4 is due on July 15 (Tuesday).
• Lecture 10 (Mon, Jul 14): Tangent planes and linear approximations (cont.): a function ƒ(x,y) is said to be differentiable if it has a tangent plane (for an example of a function that has partial derivatives but no tangent plane see Exercise 14.4/46); the continuity of the partial derivatives of a function implies the differentiability of the function (Theorem 8); differentiability of a function at a point implies continuity of the function at that point; using the linear approximation to approximate the increment of the value of a function due to small changes of the values of the arguments [pages 941-943 of Sec. 14.4; skip Equations 5, 6, and Definition 7 on page 942]
  The chain rule: Case 1 for a function of two variables and for a function of several variables (skip the proof of the chain rule) [pages 948-950 of Sec. 14.5]
  Homework: Exercises 14.4 / 19, 21 (hint); 14.5 / 2, 4, 35 (hint); additional problem.
  FFT: Exercise 14.4 / 46.
• Lecture 11 (Tue, Jul 15): The chain rule (cont.): Case 2; the general case of the chain rule for functions of several variables; implicit differentiation of a function of one variable and a function of more than one variables, examples [pages 950-954 of Sec. 14.5; reviewing pages 148-152 of Sec. 2.5 may be useful]
  Directional derivatives and the gradient vector: directional derivative of a function of two variables; expressing D_uƒ(x₀,y₀) in terms of the partial derivatives of ƒ and the components of the unit vector u; definition of the gradient vector ∇ƒ(x₀,y₀); expressing the directional derivative as a scalar product of a unit vector and the gradient vector: D_uƒ(x₀,y₀)=∇ƒ(x₀,y₀)⋅u [pages 957-962 of Sec. 14.6]
  Homework: Exercises 14.5 / 10, 14, 16, 23, 47 (hint), 49; 14.6 / 5, 8, 13 (v is not a unit vector!), 20.
  FFT: Exercises 14.5 / 17 (hint), 41; 14.6 / 19 (hint).
  The complete Homework 5 is due on July 17 (Thursday).
• Lecture 12 (Wed, Jul 16): Directional derivatives and the gradient vector (cont.): maximizing the directional derivative; significance of the gradient vector - the tangent lines to level curves at (x₀,y₀) are perpendicular to ∇ƒ(x₀,y₀); path ⟨x(t),y(t)⟩ of steepest descent/ascent on the graph z=ƒ(x,y) of a function ƒ(x,y) - the tangent vector ⟨x'(t),y'(t)⟩ at each point of the path is proportional to the gradient vector ∇ƒ(x(t),y(t)); directional derivatives of functions of three and more variables; tangent planes to level surfaces of a function F(r): the normal vector n to the surface at a point r₁ on the surface can be chosen to be equal to the gradient vector ∇F(r₁) [pages 962-966 of Sec. 14.6]
  Homework: Exercises 14.6 / 23 (hint), 37(a,b,d), 49, 56.
  FFT: Exercise 14.6 / 33 (hint).
• Lecture 13 (Thu, Jul 17): Maximum and minimum values: local minima and maxima, global (absolute) minima and maxima, if a differentiable function has an extremum (i.e., a minimum or a maximum) at (a,b), then ƒ_x(a,b)=0 and ƒ_y(a,b)=0 (Theorem 2), critical (stationary) points of a function, examples: (1) ƒ(x,y)=x²+y²: (0,0) is a critical point and a local (and global) mimimum; (2) ƒ(x,y)=x²−y²: (0,0) is a critical point but is not a local extremum; (3) ƒ(x,y)=|x|: all points of the form (0,y) are critical points and they are all local (and global) minima; (4) ƒ(x,y)=|x|+y: all points of the form (0,y) are critical points but none of them is a local extremum; second derivatives test, examples [pages 970-974 of Sec. 14.7]
  Homework: Exercises 14.7 / 2, 11, 43 (hint), additional problem.
  FFT: Exercises 14.7 / 1 (hint), 3, (hint), 13 (hint), 19, 55.
  The complete Homework 6 is due on July 22 (Tuesday).
• Lecture 14 (Fri, Jul 18): Exam 1 [on the material from Sections 12.1-12.5, 13.1, 13.2, 14.1-14.6 covered in Lectures 1-12]
• Lecture 15 (Mon, Jul 21): Maximum and minimum values (cont.): global (absolute) maxima/minima, definition of a boundary point of a set, definition of a closed set, definition of a bounded set, extreme value theorem for continuous functions of two variables on a closed bounded set, algorithm for finding the absolute minima and maxima of a function on a region D (don't forget the values on the boundary of the region D!) [pages 975-976 of Sec. 14.7]
  Double integral over rectangles: areas and single integrals in one-dimensional Calculus, definition of area as a limit of Riemann sums [page 998 of Sec. 15.1]
  Homework: Exercises 14.7 / 30, 41 (hint), 51 (hint).
  FFT: Chapter 14 Concept Check on page 991 (skip questions 9, 10, and 19); Chapter 14 True-False Quiz on pages 991-992 (the answers are given on page A123).
• Lecture 16 (Tue, Jul 22): Double integral over rectangles (cont.): volumes and double integrals; double Riemann sums; double integral over a rectangle; volumes under graphs of functions of two variables over rectangles, midpoint rule for double integrals; average value; linearity of double integrals, monotonicity of double integrals [Sec. 15.1]
  Iterated integrals: the concept of an iterated integral, Fubini's theorem, examples [Sec. 15.2]
  Homework: Exercises 15.1 / 12, 14, 18; 15.2 / 2, 9 (hint), 11, 17 (hint), 22.
  FFT: Exercises 15.2 / 27 (hint), 35 (hint), 37.
  The complete Homework 7 is due on July 24 (Thursday).
• Lecture 17 (Wed, Jul 23): Double integrals over general regions: definition of a double integral over a general region D (by continuing the function on a rectangle R containing D and integrating over the rectangle); type I regions, integrals over a type I region; type II regions, integrals over a type II region; computing integrals over general regions by subdividing the region D into non-overlapping regions D₁, ..., D_k of type I or type II, and using the property that ∫∫_DƒdA=∫∫_D1ƒdA+⋅⋅⋅∫∫_DkƒdA [pages 1012-1016 of Sec. 15.3]
  Homework: Exercises 15.3 / 9, 14, 17 (hint), 44, 46, 48, 56 (in Exercise only express the double integral as a sum of double integrals over regions of type I or type II , do not evaluate the integrals, but be completely specific about the limits of integration!).
  FFT: Exercises 15.3 / 11, 12, 15, 47 (hint).
• Lecture 18 (Thu, Jul 24): Double integrals over general regions (cont.): properties of double integrals, examples [pages 1017-1019 of Sec. 15.3]
  Double integrals in polar coordinates: polar coordinates; geometric meaning of the curves r=const and θ=const; polar rectangles; subdividing a polar rectangle into smaller polar rectangles and constructing a Riemann sum [pages 1021-1022 of Sec. 15.4]
  Homework: Exercises 15.3 / 25 (hint), 51, 58, 62 (hint), 64.
  FFT: Exercises 15.3 / 37, 43, 45, 53.
  The complete Homework 8 is due on July 29 (Tuesday).
• Lecture 19 (Fri, Jul 25): Double integrals in polar coordinates (cont.): derivation of the expression for the area ΔA_ij≈x_i-1(Δx_i)(Δθ_j) of an infinitesimally small polar rectangle; area element in polar coordinates: dA=dxdy; change to polar coordinates in double integrals (two things to remember: to include the factor r in the expression for the area element and to change x and y to r and θ in the integrand); example: calculating the area of a circle using different methods: (1) by a single integral (as in Calculus I), (2) by a double integral in Cartesian coordinates, (3) by a double integral in polar coordinates, with a coordinate system centered at the center of the circle [pages 1022-1025 of Sec. 15.4]
  Triple integrals: remarks on using single integrals to compute the total mass of a thin wire of length L of linear density λ(x) and using double integrals to compute the total mass of a thin sheet shaped as a 2-dimensional region D with a given area density σ(x,y); defining a triple integral over a box as the limit of a triple Riemann sum; defining a triple integral over an arbitrary bounded region; Fubini's Theorem for triple integrals [pages 1027-1028 of Sec. 15.5; page 1041 of Sec. 15.7]
  Homework: Exercises 15.4 / 2, 4, 11 (hint), 13 (hint), 25 (only set up the integral in polar coordinates, do not evaluate it; hint), 30, 31 (in Exercise 31 only set up the integral in polar coordinates, do not evaluate it), additional problem.
  FFT: Exercises 15.4 / 6, 29, 39 (hint), 40.
• Lecture 20 (Mon, Jul 28): Triple integrals: type-1, type-2, and type-3 regions in R³; computing triple integrals over a type-1 region, over a type-2 region, and over a type-3 region; examples (Exercises 15.7/33 and 15.7/35) [pages 1042-1045 of Sec. 15.7]
  Homework: Exercises 15.7 / 10, 13 (hint), 19 (hint), 34, 36, 55(a).
  FFT: Exercises 15.7 / 23 (hint), 27 (hint), 33 (a detailed solution), 35 (hint).
  The complete Homework 9 is due on July 30 (Wednesday).
• Lecture 21 (Tue, Jul 29): Triple integrals (cont.): more examples.
  Triple integrals in cylindrical coordinates: cylindrical coordinates in R³; computing triple integrals in cylindrical coordinates [Sec. 15.8]
  Homework: Exercises 15.8 / 8 (see Example 2), 10, 16, 21 (hint), 25(a), 29 (see Example 4).
  FFT: Exercises 15.8 / 3, 5, 6, 7 (see Example 3), 9 (hint).
• Lecture 22 (Wed, Jul 30): Triple integrals in cylindrical coordinates (cont.): more examples.
  Triple integrals in spherical coordinates: spherical coordinates, volume element in spherical coordinates, examples [Sec. 15.9]
  Homework: Exercises 15.9 / 4, 8, 10, 14, 20, 28, 40.
  FFT: Exercises 15.9 / 1, 5 (hint), 6, 7, 9, 13, 15, 17 (hint), 19.
• Lecture 23 (Thu, Jul 31): Triple integrals in spherical coordinates (cont.): more examples on using spherical coordinates; examples of computing triple integrals by using the particular form of the integrand or the region of integration in order to reduce the integral to a two- or one-dimensional integrals: (1) computing the average value of the x coordinate in the part of the ball of radius R (centered at the origin) in the first quadrant by using a triple integral; (2) computing the same quantity as in (1) by "slicing" the (1/8)-ball with many densely-spaced parallel planes (all of them perpendicular to the x-axis) and finding the limit of the Riemann sum by using a triple integral; computing the volume of a ball of radius R in the n-dimensional space (see the handout).
  Homework: Click here for the assigned homework.
  FFT: Chapter 15 Concept Check (page 1073; skip questions 4(b,d), 8(b,d), and 10), Chapter 15 True-False Quiz (pages 1073-1074; skip question 8).
The complete Homework 10 is due on August 4 (Monday).
• Lecture 24 (Fri, Aug 1 ): Vector fields: vector fields on subsets of R² and R³; plots of vector fields; gradient (conservative, potential) vector fields; potential function of a conservative vector field; examples (Newton's law of gravity, Coulomb's law of interaction between stationary electric charges); an example of a vector field that is not conservative: F(x,y)=−yi+xj (to show that it is not conservative, we assumed that it is conservative, i.e., that F=Pi+Qj=∇ƒ=ƒ_xi+ƒ_xj, and using Clairaut's Theorem (Sec. 14.3) to come to a contradiction because (ƒ_x)_y=P_y=(−y)_y=−1 is different from (ƒ_y)_x=Q_x=(x)_x=1 [Sec. 16.1]
  Line integrals: motivation: mass of a thin wire with a known shape and linear density; line integral (with respect to arc length) of a function along a curve C in R² and R³; piecewise-smooth curves; velocity v(t)=r'(t) and speed v(t)=|v(t)|; expressing the arc length as ds=v(t)dt; parameterizing a straight line as r(t)=(1−t)r₀+tr₁; line integrals with respect to x, y, and z; work expressed as a sum of line integrals to x, y, and z; a line integral of vector fields as a sum of line integrals w.r.t. x, y, and z [Sec. 16.2]
  Homework: Exercises 16.1 / 10, 12, 14, 24; 16.2 / 11 (hint), 14, 21 (hint), 33 (see Example 3, hint), 39 (hint).
  FFT: Exercises 16.1 / 1, 3, 5 (hint), 11 (hint), 13, 15-18 (hint), 29(hint)-32; 16.2 / 17 (hint), 18, 35(a), 52.
• Lecture 25 (Mon, Aug 4) Line integrals (cont.): unit tangent vector T(t)=r'(t)/|r'(t)|; expressing work as an integral of F(r)⋅dr, as an integral of F⋅T with respect to the arc length, as an integral of F(r(t))⋅r'(t) with respect to the parameter t; behavior of integrals with respect to arc length and with respect to x and y when changing the direction of traversing the curve C [pages 1092, 1094, 1095 of Sec. 16.2]
  The Fundamental Theorem for line integrals: the Fundamental Theorem for line integrals (with proof using the Chain Rule and the Fundamental Theorem of Calculus); independence of path - definition; vanishing of the line integral over any closed path is a necessary and sufficient condition for path independence (Theorem 3) [pages 1099-1101 of Sec. 16.3]
  Homework: Exercises 16.2 / 50; 16.3 / 2, 9, 29 (hint), 30.
• Lecture 26 (Tue, Aug 5): Exam 2 [on the material from Sections 14.7, 15.1-15.4, 15.7-15.9 covered in Lectures 13, 15-23]
• Lecture 27 (Wed, Aug 6): The Fundamental Theorem for line integrals (cont.): definitions of open region and connected region, path independence of a vector field over an open connected region implies that the vector field is conservative (Theorem 4); for a conservative vector field F(x,y)=P(x,y)i+Q(x,y)j, the equality ∂P/∂y=∂Q/∂x holds (Theorem 5); simple curve, simply-connected region, necessary and sufficient condition for a vector field in a simply-connected region in R² to be conservative (Theorem 6); an example: the vector field F(x,y)=(e^y+2x)i+(xe^y+y)j (defined in R² which is open and simply-connected) is conservative: F(x,y)=∇ƒ(x,y) for ƒ(x,y)=xe^y+x²+y²/2 (check that F(x,y) satisfies P_y=Q_x); practical recipe for finding the function ƒ(x,y): integrate the condition P=ƒ_x with respect to x to find ƒ(x,y) up to an arbitrary function φ(y), and then impose the condition Q=ƒ_y to determine φ(y) [alternatively, integrate the condition Q=ƒ_y with respect to y to findƒ(x,y) up to an arbitrary function ψ(x), and then impose the condition P=ƒ_x to determine ψ(y)]; this practical recipe would not work if at least one of the conditions in Theorem 6 is violated - as an example, consider the vector field F(x,y)=(e^y+2x)i+(xe^y+y+x²sin(y))j which does not satisfy P_y=Q_x, so there is no function ƒ(x,y) such that F(x,y)=∇ƒ(x,y) (check that the practical recipe will lead to a non-solvable equation) [pages 1101-1105 of Section 16.3]
  Food for thought: The vector field F(x,y)=(−yi+xj)/(x²+x²) satisfies P_y=Q_x but is non-conservative because integral of it over the unit circle in counterclockwise direction is 2π (this can be checked by a direct calculation) - therefore, there exists a closed curve C the integral of F over which is non-zero, which by Theorem 3 implies that the integral of F is not path-independent, so by the Fundamental Theorem for line integrals F is not conservative; these facts do not contradict Theorem 6 because F is not defined at (0,0), so that the region D of F is not simply-connected [see Exercise 16.3/35 (hint)]
  Homework: Exercises 16.3 / 5, 11 (hint), 18, 20, 28, 32, 34, additional problem.
  FFT: Exercises 16.3 / 1, 21, 22, 25 (hint).
  The complete Homework 11 is due on August 8 (Friday).
• Lecture 28 (Thu, Aug 7): The Fundamental Theorem for line integrals (cont.): conservation of energy [pages 1105, 1106 of Section 16.3]
  Green's Theorem: orientation of the boundary of a planar region; Green's Theorem; proof for a simple region (i.e., region that is both type-I and type-II); proof that the theorem works for regions that are not type-I and type-II, and for regions with holes; examples of using Green's Theorem for practical calculations; derivation of formulas for areas of planar regions; an example when the contour can be deformed to some extent but the value of the integral of a vector field does not change (because P_y=Q_x) - Example 5 (recall Exercise 16.3/35 considered in detail in Lecture 24) [pages 1110-1113 of Section 16.4]
  Reading assignment (optional): Read the sketch of the proof of Theorem 16.3.6 on page 1113.
  Homework: Exercises 16.4 / 1, 7 (hint), 13 (clockwise=negative!), 19.
• Lecture 29 (Fri, Aug 8): Parametric surfaces and their areas: different ways to represent surfaces in R³ studied so far: defined implicitly by an equation Φ(x,y,z)=0, or defined as a graph of a function of two variables, z=ƒ(x,y); parametric curves and parametric surfaces in R³ defined by a function r(u,v); examples - non-vertical and vertical planes, a sphere; grid curves in a parameterized surface; equations of the tangent plane to a surface defined parametrically, ∇Φ(r₀)⋅(r-r₀)=0, or as a graph, z=ƒ_x(x₀,y₀)(x-x₀)+ƒ_y(x₀,y₀)(y-y₀); the vectors r_u(u₀,v₀) and r_v(u₀,v₀) are tangent to the surface r(u,v) at the point r(u₀,v₀); the vector r_u(u₀,v₀)×r_v(u₀,v₀) is normal to the tangent plane of the surface r(u,v) at the point r(u₀,v₀); the equation of the tangent plane is r_u(u₀,v₀)×r_v(u₀,v₀)⋅(r-r₀)=0; |a×b| as the area of the parallelogram spanned by the vectors a and b; the area ΔS_ij of a surface element is ΔS_ij≈|r_u(u_i,v_j)×r_v(u_i,v_j)|ΔA_ij, where ΔA_ij=(Δu_i)(Δv_j); area of a parametric surface as a limit of Riemann sums, i.e., as an integral ∫∫_D|r_u(u_,v)×r_v(u,v)|dA where dA=dudv [Sec. 16.6]
  Reading assignment 1: Surfaces of revolution [page 1127 of Sec. 16.6]
  Reading assignment 2: Surface area of a graph of a function [page 1130 of Sec. 16.6]
  Food for thought (optional): Look at Equation 4 on page 926 and Figure 1 on page 927, and reread "Interpretations of partial derivatives" on pages 927-928 of Sec. 14.3; look at Fig. 12 on page 1127 and Equations 4 and 5 on pages 1127-1128, and think about the connection with the definition and interpretation of partial derivatives from Sec. 14.3 that you just read.
  Homework: Exercises 16.6 / 2, 3 (see Example 3, hint), 19 (hint), 26 (hint), 30 (skip the graphing), 34, 49 (hint), 51 (use property 15.3.11 on page 1018 to give bounds on A(S)), 59(a) (for the equation of an ellipsoid see Table 1 on page 854, hint).
  FFT: Exercises 16.6 / 4 (cylinder with elliptical cross-section), 6 (paraboloid), 13-18.
  The complete Homework 12 is due on August 12 (Tuesday).
• Lecture 30 (Mon, Aug 11): Surface integrals: integral of a scalar function over a surface Σ defined parametrically by r(u,v), (u,v)∈D - do not forget the area scaling factor |r_u×r_v|, so that ∫∫_ΣƒdS=∫∫_Dƒ(r(u,v))|r_u(u,v)×r_v(u,v)|dA, dA=dudv; orientable surfaces; an example of a non-orientable surface - the Möbius strip; unit normal vector n to an orientable parameterized surface: n is equal to (plus or minus) r_u×r_v/|r_u×r_v|; the positive orientation of a closed orientable surface is given by the outward unit normal vector to the surface; surface integrals of vector fields; flux of a vector field; examples from physics: fluid flow through a surface, the flux of the electric field E; "natural" orientation of a parametric surface S defined by a vector equation r(u,v) - given by the unit normal vector n=r_u×r_v/|r_u×r_v| (Equation 6 on page 1139); alternative expressions for surface integrals ∫∫_SF⋅dS=∫∫_SF⋅ndS=∫∫_SF(r(u,v))⋅n(r(u,v))|r_u(u,v)×r_v(u,v)|dA=±∫∫_SF(r(u,v))⋅r_u(u,v)×r_v(u,v)dA [Sec. 16.7]
  Homework: Exercises 16.7 / 9 (hint), 23 (hint), 39 (hint).
  FFT: Exercise 16.7 / 38.
• Lecture 31 (Tue, Aug 12): Curl and divergence: dot and cross product of vectors in R³; methods for memorizing the expression for a×b, cyclic permutations; the gradient operator as a vector: ∇=i∂_x+j∂_y+k∂_z; definition of curl(F)=∇×F of a vector field F(x,y,z) in R³; curl(grad(f))=0 for any function f:R³→R; definition of divergence div(F)=∇⋅F of a vector field F; div(curl(F))=0 for any vector field F(x,y,z), condition for conservativeness in terms of curl; Laplacian of a scalar function; an example: proving the vector identity ∇⋅(ƒF)=∇ƒ⋅F+ƒ∇⋅F (Exercise 16.5/25); Maxwell's equations [pages 1115-1119 of Sec. 16.5]
  Homework: Exercises 16.5 / 3, 16, 19 (hint), 21 (hint, "irrotational" means "with zero curl"), 28, 30(a,b), 31 (hint).
  FFT: Exercises 16.5 / 9, 11 (hint), 12, 13 (hint); Chapter 16 Concept Check (page 1160; skip questions 14-16), Chapter 16 True-False Quiz (page 1060; skip question 11).
  The complete Homework 13 is due on August 14 (Thursday).
  Please take a couple of minutes to fill out your evaluation of the course at http://eval.ou.edu!
• Lecture 32 (Wed, Aug 13): Curl and divergence (cont.): heat and wave equations use the Laplacian; Laplace and the early history of black holes (see also in the FAQ of the Hubble telescope).
  Stokes' Theorem and the Divergence Theorem: orientation of the boundary of an oriented surface consistent with the orientation of the surface; Stokes' Theorem (skip the proof); circulation of a vector field over a simple closed curve (i.e., the integral ∫_CF⋅dr); the Divergence Theorem (skip the proof); examples; the FTC for line integrals, Stokes Theorem, and the Divergence Theorem as particular cases of a general theorem that can be written as ∫_Σdω=∫_∂Σω; physical meaning of the curl of a vector field [Sec. 16.8, Sec. 16.9, the Summary table on page 1159]
  Homework: Exercises 16.8 / 1 (hint), 16, 20 (hint: use Exercise 16.5/26); 16.9 / 7 (hint), 25 (hint), 27, 29. [Not to be turned in!]
  FFT: Exercises 16.8 / 7 (hint); questions 14-16 of Chapter 16 Concept Check (page 1160); question 11 of Chapter 16 True-False Quiz (page 1060).
• Lecture 33 (Thu, Aug 14) Stokes' Theorem and the Divergence Theorem: physical interpretation of div(F); derivation of Coulomb's law from the equation div(E)=ρ/ε₀.
  Final remarks: last words of wisdom and advice about math, science, and life in general.
• Lecture 34 (Fri, Aug 15): Final Exam [cumulative]

Attendance: You are required to attend class on those days when an examination is being given; attendance during other class periods is also expected. You are fully responsible for the material covered in each class, whether or not you attend. Make-ups for missed exams will be given only if there is a compelling reason for the absence, which I know about beforehand and can document independently of your testimony (for example, via a note or a phone call from a doctor or a parent).

Homework: It is absolutely essential to solve a large number of problems on a regular basis!
Homework will be assigned regularly and will be posted on the this web-site. The homework will be due at the start of class on the due date. Each homework will consist of several problems, of which some pseudo-randomly chosen problems will be graded. Your lowest homework grade will be dropped. Giving just an answer to a problem is not worthy any credit - you have to write a complete solution which gives your step-by-step reasoning and is written in grammatically correct English. Although good exposition takes time and effort, writing your thoughts carefully will greatly increase your understanding and retention of the material.
You are allowed (and encouraged) to work in small groups. However, each of you will need to prepare individual solutions written in your own words - this is the only way to achieve real understanding!
Your homework should have your name clearly written on it, and should be stapled; the problems should be in the same order in which they are given in the assignment. No late homework will be accepted!

Exams: There will be two in-class midterms and a comprehensive in-class final.
Tentative dates for the midterms are July 18 (Friday) and August 5 (Tuesday).
The final will be given during the regular class time on August 15 (Friday).
All tests must be taken at the scheduled times, except in extraordinary circumstances.
Please do not arrange travel plans that prevent you from taking the exams at the scheduled times.

Grading: Your grade will be determined by your performance on the following coursework:

Useful links: the academic calendar, the OU enrollment web-site.

Policy on W/I Grades : From July 5 to July 25 (for undergraduate students), respectively from July 5 to July 16 (for graduate students), you can withdraw from the course with an automatic "W". Dropping after July 26 requires a petition to the Dean. (Such petitions are not often granted. Furthermore, even if the petition is granted, I will give you a grade of "Withdrawn Failing" if you are indeed failing at the time of your petition.) Please check the dates in the academic calendar!

The grade of "I" (Incomplete) is not intended to serve as a benign substitute for the grade of "F". I only give the "I" grade if a student has completed the majority of the work in the course (for example everything except the final exam), the coursework cannot be completed because of compelling and verifiable problems beyond the student's control, and the student expresses a clear intention of making up the missed work as soon as possible.

Academic Misconduct: All cases of suspected academic misconduct will be referred to the Dean of the College of Arts and Sciences for prosecution under the University's Academic Misconduct Code. The penalties can be quite severe. Don't do it!
For details on the University's policies concerning academic integrity see the Student's Guide to Academic Integrity at the Academic Integrity web-site. For information on your rights to appeal charges of academic misconduct consult the Academic Misconduct Code. Students are also bound by the provisions of the OU Student Code.

Students With Disabilities: The University of Oklahoma is committed to providing reasonable accommodation for all students with disabilities. Students with disabilities who require accommodations in this course are requested to speak with the instructor as early in the semester as possible. Students with disabilities must be registered with the Office of Disability Services prior to receiving accommodations in this course. The Office of Disability Services is located in Goddard Health Center, Suite 166: phone 405-325-3852 or TDD only 405-325-4173.

Good to know:

• The greek_alphabet.
• Some useful notations.