Applied Mathematics I

Engineering Mathematics - I

First Year • Semester 1 • NEP Syllabus

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📘 Syllabus
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Applied Mathematics I (1AS100BS)

**Course Aim:** To impart the sound knowledge on the principles of Mathematics involving the different application-oriented topics required for all engineering students.

Unit I: Infinite Series (07 Hrs)
Convergence of series, Tests for Convergence (p-series test, Comparison test, Root test, Ratio test, Raabe's test).
Unit II: Differential Calculus (07 Hrs)
Successive Differentiation, Leibnitz's Theorem, Taylor's series and Maclaurin's series.
Unit III: Partial Differentiation (07 Hrs)
Partial derivatives, change of variables, Euler's theorem on homogeneous function.
Unit IV: Application of Partial Differentiation (08 Hrs)
Maxima and minima for the function of two variables, Maxima and minima for function of several connected independent variables (Lagrange's Multiplier).
Unit V: Differential Equation of First Order (08 Hrs)
Solutions of differential equations by using Variable Separable form, Homogeneous DE, Exact DE, Non-Exact DE, Linear DE.
Unit VI: Applications of Differential equations (08 Hrs)
Application of Differential equations of first order and first degree to the problems on Electrical Circuits, Solution of Differential equations of First order and Higher degree.
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📝 Unit-wise Notes
Unit 1: Infinite Series

Infinite Series

Convergence: A series converges if the sum of its terms approaches a finite limit.
Tests: D'Alembert's Ratio Test, Cauchy's Root Test, and Raabe's Test are used to determine convergence.

Unit 2: Differential Calculus

Differential Calculus

Successive Differentiation: Finding nth order derivatives.
Leibnitz's Theorem: Used to find the nth derivative of the product of two functions.
Expansions: Taylor's and Maclaurin's series expand functions into infinite sums.

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Unit 3: Partial Differentiation

Partial Differentiation

Partial Derivative: Derivative of a function of multiple variables with respect to one variable, keeping others constant.
Euler's Theorem: Relates homogeneous functions to their partial derivatives: x(∂u/∂x) + y(∂u/∂y) = nu.

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Unit 4: Application of Partial Differentiation

Applications of PD

Maxima & Minima: Finding extreme values of functions of two variables (rt - s² > 0).
Lagrange's Multipliers: A method to find local maxima and minima of a function subject to equality constraints.

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Unit 5: First Order DE

First Order Differential Equations

Exact DE: Mdx + Ndy = 0 is exact if ∂M/∂y = ∂N/∂x.
Linear DE: Equation of the form dy/dx + Py = Q. Solved using Integrating Factor (I.F. = e^∫Pdx).

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Unit 6: Applications of DE

Applications of Differential Equations

Electrical Circuits: Modeling L-R and C-R circuits using first-order differential equations.
Kirchhoff's Laws: Used to set up the DEs for current and charge in circuits.

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Unit 1 — Infinite Series (Notes)