Course code: MT1213
Course Title: Multivariable Calculus
Credits: 3
Course Coordinators: Anup Biswas and Anindya Goswami
Pre-requisites: Single Variable Calculus
Objectives (goals, type of students for whom useful, outcome, etc): This course is aimed at providing a concise introduction to the calculus of the vector-valued functions of several variables. It is useful for the students who want to study topics like Differential Geometry, Topology, Analysis, Lie groups, Differential Equations, Theory of Relativity, Quantum Mechanics, Mathematical Biology, etc. The goal is to get the students acquainted with the basic notions of partial and directional derivatives, multiple integrals, line and surface integrals of functions of several variables.
Course contents: Euclidean space, Pythagoras' Theorem, Distance between two points, F unctions on Rn , Limit and continuity of functions on Rn , Sufficient conditions for continuity, Partial and directional derivatives, Total derivative, Sufficient conditions for differentiability, Higher order partial derivatives, Sufficient condition for change of order of differentiation, Equation of tangent plane to a differentiable surface, Chain rule, Jacobian Matrix, Taylor's Theorem, Implicit function theorem, Inverse function theorem, Local optima, Lagrange multiplier, Curves, Line integrals, Regular domain, Multiple integrals, Fubini's theorem, Differentiation under the integral sign, Green's theorem, Path independent integral, Change of variables in double integral, Surface integral of Scalar and Vector field over a smooth surface.
Number of lectures: 29 (excluding tutorials), 10 tutorials
Evaluation:
End-sem examination 35%
Mid-sem examination 35%
Quiz 30%
Text-book:
Calculus Vol. II: Multivariable calculus and linear algebra with applications to differential equations and probability by Tom Apostol (John Wiley and Sons, 2005). This book has 3 parts. Part 2 is relevant for this course.
Mathematical Analysis by S. C. Malik and Savita Arora (New Age International P.Ltd., 1992). Only Chapters 15, 16, 17, and 18 are relevant for this course.
Calculus of Several Variables by Serge Lang (Addison-Wesley 1973) E-COPY
Multivariable Calculus by James Stewart Link
Class Schedule: @ NLH, Monday and Thursday 8:45 AM
Tutorial Schedule: Every Tuesday at 10 AM
Office hour: TBA(B3 & B4), TBA(B1-B2)
Instructor Main building 3rd Floor
Teaching Assistants:
1. Mr. Sooraj M (A ?)
2. Mr. Aniket Sen (A ?)
3. Mr. Sandip Mandal (A ?)
4. Mr. Abirranga Datta (A ?)
5. Ms. Nishtha Sharma (A ?)
6. Mr. Mohd. Sharukh (A ?)
Lectures
03/01 Pythagoras's Theorem, Definition of vectors, R^n, dot product. [001.pdf]
04/01 Cauchy-Schwarz Inequality, Angle between two vectors, Functions of several variables, Some examples. [001.pdf]
10/01 Equivalence of norms in a finite dimensional Euclidean space, Definition of limit of a real-valued function of several variables. Limit along a path, non-existence of limit, example. [002.pdf]
11/01 Repeated limit, Continuity, Partial derivatives. [003.pdf, page 61 and 62 of E-COPY]
17/01 Properties of the continuous functions, continuity of components and wrt each variable. A set of sufficient conditions for continuity of a function at a point in a plane using the knowledge of partial derivatives at and about the point. [003.pdf]
18/01 A set of sufficient conditions for continuity of a function defined on Rn. Two examples related to this result. The derivative of a function from a line to a Euclidean space. The derivative of a function from a Euclidean space to a line. [004.pdf]
24/01 Discussion on differentiability condition, differentiability implies continuity, expression of the derivative in terms of the partial derivatives. The equation of tangent plane to a differentiable surface. [005.pdf]
25/01 Differentiability of a vector valued function of several variables and the definition of the derivative. A set of sufficient conditions for differentiability of a function defined on Rn. Example. Partial derivatives of higher order, a set of sufficient conditions for the change of order of differentiation (Young's Theorem). [007.pdf]
31/01 Directional derivative, Chain rule. [008.pdf]
01/02 Jacobian matrix, General form of chain rule using Jacobian matrix. [008.pdf]
07/02 The matrix inverse of the Jacobian matrix of an invertible function is the Jacobian matrix of the inverse function, higher order derivatives of the map t⟼f(a+th) where a,h are from Rn . [009.pdf]
08/02 Taylor's theorem for functions of several variables, Local extrema of functions of several variables, Hessian matrix. [010.pdf]
14/02 Comparison between Jacobian and Hessian matrices, Symmetry of Hessian matrix for C2 function. Some general discussion and recapitulation of definition of limit, differentiability, tangent plane etc. [011.pdf]
15/02 Definition of definite and indefinite matrices. Statements of Inverse Function Theorem and Implicit Function Theorem. Discussion with an example. [011.pdf]
28/02 Expression of derivative of an implicitly given function. Statement of a constrained optimization problem. [012.pdf]
01/03 Derivation of Lagrange's undetermined multiplier method for solving constrained optimization problem, illustration with an example. [012.pdf]
07/03 Illustration of Lagrange's undetermined multiplier method with two more examples. [013.pdf]
08/03 Line integral of a scalar field. [014.pdf]
14/03 Illustration of line integral of a scalar function with two examples. Line integral of a vector field. [015.pdf]
15/03 Illustration of the line integral of a vector field with one example. Region, boundary of a region, regular region. Multiple integral. [015.pdf]
21/03 Integrability of a scalar field, illustration with an example and a non-example, Reduction of double integral into repeated integral. Fubini's Theorem. [016.pdf]
22/03 Sufficient conditions for integrability. integration on regular region. [016.pdf]
29/03 Example of double integration on a non-regular region. Differentiation under the sign of integral. [017.pdf]
04/04 Green's Theorem, path independent integrals. [018.pdf]
05/04 Change of variables in double integral. [018.pdf]
11/04 Area formula of a differentiable surface. [019.pdf]
12/04 Surface integral of Scalar and Vector field over a smooth surface. [019.pdf]
18/04 Example of calculation of surface integrals. [020.pdf]
19/04 General discussion on integrations.
Exams
20/01 Quiz-1: Topics: Limit
03/02 Quiz-2: Topics: Continuity and differentiability
22/02 Midsem: Topics: Lecture 1-14 except the topics on local extrema.
10/03 Quiz-3: Topics: Directional derivative, Chain rule, and Taylor's theorem.
24/03 Quiz-4: Topics: Constrained and unconstrained extrema.
05/04 Extra Quiz: Topics: [Lecture 14 and 15] Inverse Function Theorem and Implicit Function Theorem, Examples, Expression of derivative of an implicitly given function (Those who missed one or more quizzes are eligible only).
Duration 4:15-4:30 PM, Venue Madhava Hall.
07/04 Quiz-5: Line integral for scalar or vector fields.
24/04 Endsem: Full marks 35. Topics: complete syllabus, including the topics covered in midsem which would have at most 10 marks out of 35.
Result
4 Chain Rule - Thanks to "Mathematics Calculus Online Tutorials" by "Harvey Mudd College".
3 Geometrical interpretation of Df(a) - Thanks to Wiki
2 Repeated Limit - Thanks to Encyclopediaofmath.org Note carefully, although it seems that for repeated limits, only the paths along the x and y axis matter, but that is not the case. Indeed the function's values at the axes do not play any role. What matters is the behavior of the function near an axis (neighborhood of the axis).
1 Examples of parametric surfaces - An article from Resonance.