MTC-1011 :
Multivarable Calculus
Unit 1: VECTOR SPACES
1.1 Introduction.
1.2 Definitions and Examples of a Vector Space.
1.3 Subspace.
1.4 Linear Dependence and Independence
1.5 Basis and Dimension
1.6 Vector Space as a Direct Sum of Subspaces
1.7 Null Space and Range Space
Unit 2: INNER PRODUCT SPACES
2.1 Introduction.
2.2 Definitions, Properties and Examples.
2.3 Length (Norm), Distance in Inner product space.
2.4 Angle and Orthogonality in Inner product space
2.5 Orthonormal basis and Orthonormal projection
2.6 Gram-Schmidt Process of Orthogonalization
(1) Limit and Continuity of Multivariable functions: . . . [06]
1.1 Functions of several variables, graphs and level curves of function of two variables .
1.2 Limit and continuity in higher dimensions .
(2) Partial Derivatives: . . . [04]
2.1 Definition and examples .
2.2 Second order partial derivative, the mixex derivative theorem .
2.3 Partial derivatives of higher order.
(3) Differentiability: . . . [12]
3.1 Differentiability, the increment theorem for functions of two variables(without proof).
3.2 Chain rules for composite function.
3.3 Directional derivatives, gradient vectors.
3.4 Tangent planes, normal lines and differentials.
(4) Extreme Values: . . . [10]
4.1 Extreme values, First derivative test and Second derivative test for local extreme values.
4.2 Lagrange’s multipliers method for finding extreme values of constraint function (One Constraint)
4.3 Taylors Formula for two variables.
(5) Multiple Integrals: . . . [16]
5.1 Double Integral over rectangles, Fubini’s theorem for calculating double integrals (Without proof)
5.2 Double integrals in polar form.
5.3 Triple integral in cylindrical and spherical coordinates.
5.4 Triple integral in cylindrical and spherical coordinates.
5.5 Substitution in multiple integrals, Application to are and volumes.
Text book: Prepared by the BOS Mathematics, Savitribai Phule Pune University, Pune.
Recommended Book: Thomas’ Calculus’ 11th Edition, G.B. Thomas. Revised by Maurice
D. Weir, Joel Hass and Frank R. Giordano. Pearson Education 2012. Articles: 14.1 to 14.10,
15.1,15.3,15.4,15.6,15.7
Reference Books:
(1) Basic Multivariable Calculus, J.E. Marsden, A.J. Tromba, A. Weinstein. Springer Verlag (Indian Edition).
(2) Shanti Narayan, R.K. Mittal, A Text-book of Vector Calculus, S. Chand and Company.
(3) D.V.Widder, Advanced Calculus 2 (2 nd Edition)
(4) T.M. Apostol. Calculus Vol.2 (2 nd Edition), John Wiley, Newyork (1967).
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