Multivariable calculus (Spring 2017)

Office hour: Wednesday 3-4PM(B3 & B4), 4-5PM(B1-B2)

Instructor (A 404, Main building)

Teaching Assistants:

1. Mr. Shashikant Ghanwat (A 427)

2. Mr. Onkar Kale (A 427)

3. Mr. Uday Jagdale (A 427)

4. Ms. Sneha Jondhale (A 427)

5. Dr. Rohit Holkar (A 464)

6. Dr. Venkat Krishna Kishore (A 405)

Lectures

1. 03/01 Pythagoras's Theorem, Definition of vectors, R^n, dot product. [001.pdf]

2. 04/01 Cauchy-Schwarz Inequality, Angle between two vectors, Functions of several variables, Some examples. [001.pdf]

3. 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]

4. 11/01 Repeated limit, Continuity, Partial derivatives. [003.pdf, page 61 and 62 of E-COPY]

5. 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]

6. 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]

   1. 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]

   2. 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]

   3. 31/01 Directional derivative, Chain rule. [008.pdf]

   4. 01/02 Jacobian matrix, General form of chain rule using Jacobian matrix. [008.pdf]

   5. 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]

   6. 08/02 Taylor's theorem for functions of several variables, Local extrema of functions of several variables, Hessian matrix. [010.pdf]

   7. 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]

   8. 15/02 Definition of definite and indefinite matrices. Statements of Inverse Function Theorem and Implicit Function Theorem. Discussion with an example. [011.pdf]

7. 28/02 Expression of derivative of an implicitly given function. Statement of a constrained optimization problem. [012.pdf]

• 1. 01/03 Derivation of Lagrange's undetermined multiplier method for solving constrained optimization problem, illustration with an example. [012.pdf]

  2. 07/03 Illustration of Lagrange's undetermined multiplier method with two more examples. [013.pdf]

  3. 08/03 Line integral of a scalar field. [014.pdf]

  4. 14/03 Illustration of line integral of a scalar function with two examples. Line integral of a vector field. [015.pdf]

  5. 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]

  6. 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]

  7. 22/03 Sufficient conditions for integrability. integration on regular region. [016.pdf]

  8. 29/03 Example of double integration on a non-regular region. Differentiation under the sign of integral. [017.pdf]

  9. 04/04 Green's Theorem, path independent integrals. [018.pdf]

  10. 05/04 Change of variables in double integral. [018.pdf]

  11. 11/04 Area formula of a differentiable surface. [019.pdf]

  12. 12/04 Surface integral of Scalar and Vector field over a smooth surface. [019.pdf]

8. 18/04 Example of calculation of surface integrals. [020.pdf]

9. 19/04 General discussion on integrations.

Exams

• 1. 20/01 Quiz-1: Topics: Limit

  2. 03/02 Quiz-2: Topics: Continuity and differentiability

  3. 22/02 Midsem: Topics: Lecture 1-14 except the topics on local extrema.

  4. 10/03 Quiz-3: Topics: Directional derivative, Chain rule, and Taylor's theorem.

  5. 24/03 Quiz-4: Topics: Constrained and unconstrained extrema.

  6. 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).

  7. Duration 4:15-4:30 PM, Venue Madhava Hall.

  8. 07/04 Quiz-5: Line integral for scalar or vector fields.

  9. 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

Feedback: 4/5