The class of 2024
Assignment Schedule and Announcements
Placement Test: Friday, August 23, 7-10:30 pm (online). It will be open in Canvas. Read the syllabus about the submission of the placement test. Students who are not in the official section should email me their UF ID as soon as possible to get access to the test via Canvas. However, the placement test will also be posted here at 7 pm and you may submit your work by email if Canvas does not work for you. Make sure that your email is sent before the due time (late submissions will not be accepted). Make sure that you refresh this page to view its latest version.
Placement Test with solution Fall 2024
Test schedule: By default, a test day is Thursday and official time E1-E2 (7-9 pm), unless posted otherwise.
Test 1: Week 4 (Sections 1-5), Tuesday, September 10, 7-9 pm, online (via Canvas).
Test 1 with solutions
Test 2: Week 5 (L7-12), Tuesday, September 24, 7-9 pm, online (via Canvas).
Test 2 with solutions
Test 3, Week 8 (L13-18), Tuesday, October 15, 7-9 pm, online (via Canvas)
Test 3 with solutions
Test 4, Week 10, Tuesday, 7-9 pm, online (via Canvas)
Test 4 with solutions
Test 5, Week 12 (November 5, in person, Florida gym 230, 8:20-10:10 pm),
Test 5 with solutions
Test 6, Week 15, Tuesday, December 3, 7-10 pm (via Canvas),
Test 6 with solutions
Final Exam, Final week (Chapter 5), Final exam with solutions
Take-home Final with solutions
Office hours:
Tuesday, 3:00-3:50 pm LIT 464; and, only on the test day, 4:10-5:00 pm (in the classroom)
Friday, LIT 205, 6 pm
You may also stop by LIT 205 at 6 pm on MW. I can answer 1-2 questions IF grad students have no questions
Solution Manual: All homework problems have solutions in the solution manuals posted below.
The UF teaching evaluation period for this semester is November 24 – December 7. It is done online. You have to log in Evaluation Page using your Gatorlink user name and password. Please do not forget to do so!
Textbook: S.V. Shabanov, Concepts in Calculus III
Table of Content
Chapter 1: Vectors and the Space Geometry
Chapter 2: Vector Functions
Chapter 3: Differentiation of Multi-variable Functions
Chapter 4: Multiple Integrals
Chapter 5: Vector Calculus
Acknowledgments
Solution Manual
Additional Solution Manual
Lecture Topics and Homework Assignments
Chapter 1: Vectors and the Space Geometry
Week 1 (August 22- 23)
Lecture 1: Section 1: Distance function in space. Distance axioms. Straight lines and straight line segments. Perpendicular line segments. The angle between two line segments. Planes in space. Areas and volumes. Rectangular coordinates. Points in space as ordered triple of numbers. Coordinate planes.
Week 2 (August 28 – September 1)
Lecture 2: Distance formula. Translations and rotations of a coordinate system. Algebraic description of point sets in space. A sphere.
Recommended Study Problems: 1.1, 1.2, 1.3, 1.4, 1.5, 1.6
Homework, 1.11: 2, 4, 5, 6, 7, 8, 9, 13, 14, 17, 18, 19, 21, 23, 26, 27
Lecture 3: Section 2: Vectors and oriented segments. Vectors as ordered triple of numbers. Position vector of a point. Components of a vector. Equality of two vectors. Computation of components of a vector with given initial and terminal points. Norm (or length) of a vector. Zero vector. Multiplication of a vector by a number. Parallel vectors (algebraic criterion for vectors to be parallel). Unit vector. Addition of vectors (rules of vector algebra). Geometrical interpretation of the rules of vector algebra. The parallelogram rule.
Recommended Study Problems: 2.1, 2.2, 2.3, 2.4, 2.5
Homework, 2.5: 1-11, 13, 16, 17, 18-21, 23, 24
Lecture 4: Section 3: The dot product and its algebraic properties. Geometrical significance of the dot product. The angle between two vectors. Criterion for two vectors to be orthogonal. Orthogonal decomposition of a vector. Scalar and vector projections of a vector onto another vector. Cauchy-Schwarz and triangle inequalities. Directional angles. Basis vectors. Some application of the dot product: Equilibrium of a point object under the action of several forces and the work done by a constant force.
Recommended Study Problems: 3.1, 3.2, 3.3
Homework, 3.8: 1-5, 9-17, 25, 26, 28, 30
Week 3 (September 3-6)
Lecture 5: Section 4: The determinant of a square matrix. The cross product of two vectors. Algebraic properties of the cross product. The bac-cab rule for the double cross product. The Jacobi identity. Non-commutativity and non-associativity of the cross product. Geometrical significance of the cross product. The areas of a parallelogram and a triangle. The right hand rule. Criterion for two vectors to be parallel. Applications of the cross product: the torque. Equilibrium of an extended rigid object.
Recommended Study Problems: 4.1, 4.2, 4.3, 4.4, 4.6
Homework, 4.6: 1-11, 15-17, 19, 21, 26, 27, 30
Lecture 6: Section 5: The triple product. Geometrical significance of the triple product. Volume of a parallelepiped. Criterion for three vectors to be coplanar. Applications of the triple product. The distance between two sets of points in space. Criteria for two lines being parallel, or intersecting, or skew. The distance between two skewed lines.
Recommended Study Problems: 5.2, 5.3, 5.4,
Homework, 5.5: 1-11, 13, 15, 18.
Week 4 (September 9-13)
Lecture 7: Section 6: Planes in space. A geometrical description of a plane. Normal to a plane. An equation of a plane that has a given normal and goes through a given point. Parallel planes. Angle between two planes. Distance between a plane and a point.
Recommended Study Problems: 6.1, 6.2, 6.3, 6.4
Homework, 6.5: 1-3, 7-9, 11-16, 21-23
Lecture 8: Section 7: Lines in space. Geometrical description of a line. Vector, parametric, and symmetric equations of a line through a given point and parallel to a given vector. Distance between a point and a line. Intersection of a line with another line, a plane, and a sphere. Relative positions of planes and lines in space.
Recommended Study Problems: 7.2, 7.3
Homework, 7.5: 2, 3, 5-7, 9-11, 14, 17, 18, 21, 23, 24, 26
Lecture 9: Section 9: Quadric cylinders. Rotations and translations of the coordinate system in a plane. Classification theorem for quadric cylinders. Quadric surfaces. Point sets in space defined by a general quadratic equation. Rotations and translations of the coordinate system in space. Standard form of the quadratic equation. Classification theorem for quadric surfaces. Graphing quadric surfaces by means of sections by coordinate planes. Shifted quadric surfaces.
Recommended Study Problems: 9.1, 9.2, 9.3, 9.4
Homework, 9.6: 1-15, 18, 19, 24, 25, 28, 31
Week 5 (September 16-20)
Chapter 2: Vector Functions
Lecture 10: Section 10: Curves in space and vector functions. Definition of a vector function.
Domain and range of a vector function. Limits and continuity of a vector function. Graph of a continuous
vector function. Continuous vector functions and curves in space. Parametric curves. Simple parametric
curve. A curve as a point set in space.
Recommended study problems: 10.1, 10.2, 10.3, 10.4, 10.5, 10.6
Homework, 10.5: 3, 4, 11, 12, 14, 15, 17, 18, 22, 23, 27, 30, 34, 38, 42-44
Lecture 11, Section 11: Differentiation of vector functions. Definition of the derivative.
Basic rules of differentiation. Geometrical significance of the derivative. Tangent line
and a unit tangent vector of a curve. Normal plane. Smooth parametric curves.
Smooth curves as a point set in space. Verification whether or not a given space curve is smooth.
Recommended Study Problems: 11.1, 11.2, 11.3
Homework, 11.5: 4-6, 8, 11, 12, 15, 16, 19, 21, 23, 26, 31, 32
Lecture 12, Section 12: Integration of vector functions. Definite integral of a vector function.
Indefinite integral of a vector function. Fundamental Theorem of calculus for vector functions.
Application to motion in space. Reconstruction of the trajectory from Newton’s equations of motion.
Motion under a constant force. Trajectory of the motion under a constant gravitational force.
Recommended Study Problems: 12.1, 12.2, 12.3
Homework, 12.4: 3-9, 13, 18, 21, 22
Week 6 (September 23-27)
Lecture 13, Section 13: Arc length of a curve.
Arc length of a smooth curve via the Riemann integral. Reparameterization of a space curve. Natural parameterization
of a space curve.
Recommended Study Problems: 13.1
Homework, 13.4: 4, 5, 7-9, 11, 14, 17, 20, 22, 23, 24
Lecture 14, Section 14: Curvature of a smooth curve. Evaluation of curvature
via the derivatives of a vector function that traces out a smooth space curve.
Curvature of a planar curve. Curvature of the graph.
Geometrical significance of the curvature. Curvature radius. Osculating plane.
Osculating circle.
Recommended Study Problems:14.1, 14.2, 14.3, 14.4, 14.5
Homework, 14.3:3, 5, 7, 9-11, 12, 14, 15-17, 20, 24, 27, 30, 33, 35.
Lecture 15, Section 15: Applications to mechanics. Normal and tangential accelerations. Applications to geometry. Unit normal and binormal vectors to a space curve. Torsion. Frenet-Serret equations. Shape of a space curve. Curves with constant curvature and torsion.
Recommended Study Problems:15.1, 15.2, 15.3, 15.4, 15.5
Homework, 15.4:3, 8, 10-14, 17, 18, 22
Week 7 (September 30- October 4)
Chapter 3: Differentiation of Multi-Variable Functions
Lecture 16, Sections 16, 17: Functions of several variables. Domain. Range. Graph of a function
of two variables. Level sets. Contour map of a function of two variables. Level surfaces
of a function of three variables. Limit points of a subset of a Euclidean space.
Definition of the limit of a function of several variables. Continuity of a function of several variables.
Homework, 16.6: 4, 8, 10, 11, 12, 15, 17, 18, 22, 24, 27, 33-36, 37, 41, 45
Lecture 17, Sections 17, 18: Basic limit laws. Squeeze principle.
Properties of continuous functions (continuity of the sums, products, and ratios of continuous
functions). Continuity of a composition. Continuity of polynomial and rational functions.
Continuous extension to a limit point. A general strategy to study limits. Continuity argument.
Composition rule. Limits along curves. Criterion for non-existence of a multivariable limit.
Repeated limits. Limits along straight lines and power curves.
Limits at infinity and infinite limits.
Recommended Study Problems: 17.1
Homework, 17.7: 1-3, 6-8, 10, 11, 15, 16, 17, 21, 23-25
Recommended Study Problems: 18.1, 18.2, 18.3, 18.4,
Homework, 18.7: 1-3, 5, 6, 7, 10, 13, 15, 16, 19, 22, 24, 26, 27, 30
Lecture 18, Sections 19, 20: Partial derivatives. Definition of partial derivatives of a function of several variables at a point.
Geometrical significance of the partial derivatives as the rates of change of the function at
a point along the coordinate axes. Partial derivatives as functions. Basic rules of differentiation. Higher-order partial derivatives. Mixed derivatives. Clairaut’s theorem. Reconstruction of a function from its partial derivatives. Integrability conditions. Partial differential equations. A solution of a partial differential equation. Example: Wave equation.
Homework, 19.4: 1-3, 6,7, 8-14, 22, 23, 25, 28
Recommended Study Problems: 20.1, 20.2
Homework, 20.4: 1, 3, 6, 7, 9, 12, 21-24, 26-28, 30, 33, 34, 39, 40-42, 44
Week 8 (October 7-11)
Lecture 19, Section 21: A good linear approximation to a single-variable function and differentiability of the function. A good linear approximation to a function of several variables. Differentiability of a multi-variable function. Continuity and the existence of partial derivatives of a differentiable function. Relations between differentiability, continuity, and the existence of partial derivatives for multi-variable functions. Linearization of a differentiable function at a point. Tangent plane to the graph of a function of two variables. Tangent plane approximation.
Recommended Study Problem: 21.1
Homework, 21.6: 1-11, 14, 17, 19, 21-28.
Lecture 20, Section 22: Chain rules. Composition of functions of several variables. Simple chain rule for the rate of change of a function of two variables along a curve. Differentiability and the chain rule. General chain rule for functions of several variables. Functions defined implicitly. Implicit differentiation. Implicit function theorem.
Recommended Study Problem: 22.1
Homework: 1-6, 11, 12, 18, 20, 23, 25, 28, 31, 33, 35-41.
Lecture 21, Section 23: The differential and Taylor polynomials. Differential and its geometrical significance. Linearization of a function and its differential. Application: Error analysis. Accuracy of the linear approximation. Higher-order differentials of multi-variable functions and Taylor polynomials.
Recommended Study Problem: 23.1, 23.2
Homework: 1-4, 8-11, 13-15, 18-20, 23, 27-29, 31-33, 38, 39.
Week 9 (October 14-18)
Lecture 22, Section 24: Directional derivative and its significance. The gradient. The relation between the gradient of a differentiable function at a point and the directional derivative of the function at that point. Conditions under which level sets of a function of two variables are smooth curves. Conditions under which level sets of a function three variables are smooth surfaces. Geometrical properties of the gradient. Tangent plane and normal line to a level surface at a point. Tangent and normal lines to a level curve.
Recommended Study Problems: 24.1, 24.2, 24.3
Homework: 1, 2, 4, 5, 6, 11, 14, 15, 16, 23-26, 30, 31, 34, 35, 41, 42, 44, 45, 46, 52
Lecture 23, Section 25: Minimum and maximum values of a multi-variable function. Absolute and local maxima and minima. Critical points. Saddle point. Second-derivative test for functions of two variables.
Recommended Study Problems: 25.1, 25.2, 25.3, 25.4, 25.5
Homework, 25.6: 1, 2, 6-16, 28.
Week 10 (October 21-25)
Lecture 24, Section 26: Second-derivative test for functions of more than two variables. Use of Taylor polynomials to study the local behavior of a function near a critical point when the second derivative test is inconclusive. Extreme values of a function on a set. The extreme value theorem. The intermediate value theorem.
Recommended Study Problems: 26.1, 26.2
Homework, 26.6: 1-5, 13-19, 26-30.
Lecture 25, Section 27: Extreme values of a function subject to constraints. The method of Lagrange multipliers. The case of one constraint for functions of several variables. The case of two and more constraints for multi-variable functions. Local extreme values of a function subject to a constraint.
Recommended Study Problems: 27.1, 27.2
Homework, 27.6: 1, 3-6, 8, 9, 11, 12, 25, 26, 28, 31, 32
Week 11 (October 28-November 1)
Chapter 4: Multiple integrals
Lecture 26, Section 28: The volume problem and intuitive idea of the double integral. Rectangular partition of a bounded region. Upper and lower sums for a bounded function. Definition of the double integral of a bounded function over a bounded closed region. Integrability and continuity. An example of a non-integrable function. Riemann sums. Convergence of Riemann sums for an integrable function. Approximation of a double integral by a Riemann sum.
Homework, 28.6: 2, 3, 6, 10, 11, 12, 13, 17, 18
Lecture 27, Sections 29 and 30: Properties of the double integral: linearity, area of a planar region, additivity, positivity, upper and lower bounds, integral mean value theorem, independence of partition (theorem). Iterated double integrals. Double integrals over a rectangle. Fubini’s theorem. Factorization of the double integral.
Recommended Study Problem: 30.1
Homework, 29.1: 4-8, 10, 12, 15; 30.3: 1-10, 16, 17, 21
Lecture 28, Section 31: Double integrals over general regions. Simple regions. Iterated integrals over horizontally and vertically simple regions. Reversing the order of integration in an iterated double integral. Use of symmetries of the integrand to evaluate the double integral.
Recommended Study Problems: 31.1, 31.2
Homework, 31.6: 1-17, 21, 22, 24-26, 29-35, 39-42
Week 12 (November 2-8)
Lecture 29, Section 32: Double integrals in polar coordinates. Polar coordinates in integration. Partition of a region by coordinate curves of polar coordinates. Transformation of the area element. Jacobian of polar coordinates. Use of polar coordinates to evaluate double integrals.
Recommended Study Problem: 32.1
Homework, 32.5: 1-4, 9-11, 17-20, 27-29, 34-39
Lecture 30, Section 33: Change of variables in double integrals. Transformations. Jacobian of a transformation. Change of variables. Coordinate curves. The area transformation under a change of variables. Use of change of variables to simplify the region of integration.
Recommended Study Problem: 33.1
Homework, 33.5: 7-9, 11-14, 19-23.
Lecture 31, Section 34: Triple integrals. Definition of a triple integral. Reduction of a triple integral to an iterated integral. Fubini’s theorem. Use of symmetry.
Recommended Study Problems: 34.1, 34.2
Homework, 34.6: 1-4, 7, 8, 12, 13, 18, 21, 22-24, 32
Lecture 32, Section 35: Triple integral in spherical coordinates. Spherical coordinates. Coordinate surfaces of spherical coordinates. Jacobian of spherical coordinates. Triple integral in spherical coordinates. Use of spherical coordinates to evaluate triple integrals.
Recommended Study Problem: 35.1
Homework, 35.6: 4-13, 18-25, 30, 31
Week 13 (November 11-15)
Lecture 33, Section 35: Triple integral in cylindrical coordinates. Cylindrical coordinates. Coordinate surfaces of cylindrical coordinates. Partition of a region by coordinate surfaces of cylindrical coordinates. Transformation of the volume element. Jacobian of cylindrical coordinates. Triple integral in cylindrical coordinates.
Recommended Study Problem: 35.1
Homework, 35.6: 4-13, 18-25, 30, 31
Lecture 34, Section 36: Change of variables in triple integrals. Jacobian. Change of variables. Coordinate surfaces. The volume transformation under a change of variables. Triple integrals in curvilinear coordinates. Use of change of variables to simplify the region of integration.
Recommended Study Problem: 36.1
Homework, 36.3: 1-3, 6-8, 13, 14, 15
Lecture 35, Sections 38: Line integral of a function along a smooth curve.
Homework, 38.3: 1-5, 9, 18, 19.
Lecture 36, Sections 38: Surface area of a smooth surface defined by the graph of a function with continuous partial derivatives. Parametric surfaces.
Homework, 38.3:1-5, 9, 18, 19.
Lecture 37, Sections 39: Surface integral of a function over a smooth surface defined by the graph of a function with continuous partial derivatives.
Homework, 39.5: 1-10
Week 14 (November 18 – 22)
Chapter 5: Vector Calculus
Lecture 38, Section 41: Vector fields. Flow lines of a vector field. Line integral of a vector field. Conservative vector field.
Recommended Study Problem: 41.1.
Homework, 41.6: 1-4, 9-11, 16, 17-25.
Lecture 39, Section 42: Fundamental theorem for line integrals. Path-independence property of a vector field. Curl of a vector field. Classification of planar regions. Connected and simply connected regions. Test for a vector field to be conservative.
Recommended Study Problems: 42.1, 42.2
Homework, 42.6: 1-5, 9-11, 17-19.
Lecture 40, Section 43: Green’s theorem. Applications of Green’s theorem to evaluate double integrals and line integrals over planar curves. Area of a region bounded by a smooth closed parametric curve.
Recommended Study Problems: 43.1, 43.2, 43.3
Homework, 43.5: 3-6, 9 12, 13, 16, 17, 18, 21, 24
Lecture 41, Section 44: Orientable surfaces. Orientation of a surface. Flux of a vector field across an oriented surface. Evaluation of the flux across a surface represented by a graph.
Homework, 44.5: 1-5, 8-13, 15, 18;
Fall Break (Thanksgiving): No classes, November 25-29
Week 15 (December 2 – 4)
Lecture 42, Section 46: Another vector form of Green’s theorem. Divergence of a vector field. Divergence theorem. Volume of a solid region as a surface integral. Use of the divergence theorem to evaluate the flux of a vector field. Deformation of the surface in the flux integral. Geometrical significance of the divergence of a vector field.
Recommended Study Problem: 46.1
Homework, 46.6: 2, 6, 7, 13, 14, 18-26, 32, 33.
Lecture 43, Section 45: Vector form of Green’s theorem. Relation between a line integral of a vector field and a flux of the curl of the vector field (Stokes’ theorem). Use of Stokes’ theorem to evaluate line integrals along a closed curve. Geometrical significance of the curl of a vector field. Deformation of the curve in the line integral.
Recommended Study Problem: 45.1
Homework, 45.7: 1, 4-7, 9, 10, 14, 17, 19.
TO BE UPDATED
Section 40: Applications of multiple integrals: Moments of inertia and the center of mass of extended objects.
Recommended Study Problem: 40.1, 40.2, 40.3
Homework, 40.3: 2-4, 7, 9, 11, 18, 22, 23, 25, 32, 34
Section 37: Improper multiple integrals. Exhaustion of an integration region. Improper multiple integrals over bounded regions. Improper multiple integrals over unbounded regions. Absolute and conditional convergence. Absolute integrability tests.
Recommended Study Problem: 37.1
Homework, 37.6: 1-3, 6, 7, 11-14, 16, 27, 28