UMA001 MATHEMATICS-I

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**Successive
Differentiation: **Higher order derivatives, n^{th} derivatives of
standard functions, nth derivatives of
rational functions, Leibnitz theorem.

**Applications of
Derivatives:** Mean value theorems and their geometrical interpretation,
Cartesian graphing using first and second order derivatives, Asymptotes and
dominant terms, Graphing of polar curves, Polar equations for conic sections.

**Sequences and Series**: Introduction
to sequences and Infinite series, Tests for convergence/divergence: Limit
comparison test, Ratio test, Root test, Cauchy integral test, Cauchy
condensation test. Alternating series, Absolute convergence and conditional
convergence.

**Series Expansions**: Power series,
Taylor series, Convergence of Taylor series, Error estimates, Term by term
differentiation and integration, Multiplication and division process in power
series.

**Partial Differentiation**: Functions
of several variables, Limits and continuity, Chain rule, Change of variables,
Partial differentiation of implicit functions, Taylor series of two variables,
Directional derivatives and its properties, Maxima and minima by using second
order derivatives.

**Multiple Integrals:**
Change of order of integration, Change of variables, Applications of multiple
integrals to areas and volumes.

**
Vector Calculus**: Differentiation and integration of vector valued
functions, velocity, acceleration, tangent, principle normal and binormal
vectors, Curvature, Torsion and TNB frame. Scalar and vector fields, Gradient,
Divergence and Curl. Line integrals, Work, Circulation and Flux. Green’s
theorem in Plane, Gauss-divergence and Stoke’s theorem (without proof).

*Text Books *

*1.
**Thomas, G.B. and Finney, R.L., Calculus and
Analytic Geometry, Pearson Education (2007) 9 ^{th} ed.*

*2.
**Stewart James, Essential Calculus; Thomson
Publishers (2007) 6 ^{th} ed.*

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*Reference Books*

*1.
**Wider David V, Advanced Calculus: Early
Trancedentals, Cengage Learning (2007).*

*2.
**Apostol Tom M, Calculus, Vol I and II John
Wiley (2003).*

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