Calculus, Laplace Transform and Numerical Techniques (AAS203C) — Notes & Syllabus

Notes1

Engineering Mathematics I & II — Complete Notes

Part A (Maths I): differential calculus, Leibnitz and Taylor, partial differentiation, Jacobians, matrices and eigenvalues, beta/gamma functions, multiple integrals, vector calculus. Part B (Maths II): differential equations, Laplace transforms, Fourier series and complex analysis. Worked examples throughout.

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Syllabus1

Syllabus — Calculus, Laplace Transform and Numerical Techniques (AAS203C)

Official AKTU syllabus, effective from the academic session 2026-27 (AICTE model curriculum / NEP 2020).

Course objectives

Develop the ability to solve differential equations using series methods and special functions applicable to engineering problems. 2. Introduce Laplace and Fourier transform techniques for analyzing differential equations and signal-processing applications. 3. Build a strong foundation in complex variable theory and its applications in engineering and applied sciences. 4. Understand the concepts of sequences and series, convergence tests, and their role in mathematical modeling. 5. Enable students to apply advanced mathematical techniques to solve real-world problems in computer science, electronics, communication, and engineering.

Course content

Determination of indicial roots, cases of indicial roots: Distinct roots, equal roots, roots differ by an integer and not differ by an integer. Applications: Fluid mechanics, quantum mechanical wave equation, Heat transfer, Solving electromagnetic field equation. Laplace Transform: Definition of Laplace Transform, Existence Conditions, Linearity Property, Laplace Transform of some standard functions, I and II Shifting theorems (without proof), Change of Scale Property, Laplace transform of derivatives and integrals, Periodic Functions, Unit Step Function, Inverse Laplace Transform, Partial Fraction Method, Convolution theorem (without proof), Application of Laplace transform to solve ordinary differential equation. Applications : Transfer Function Determination, Circuit Equations, Step Response. Fourier Series& Fourier Transform: Periodic function, even and odd function, Dirichlet’s Condition for the Existence of a Fourier Series, Fourier series expansion of a function in the interval c to c+2l , Change of Interval, Half Range Series: Half range sine and cosine series in the interval -l to l, Parseval’s Identity, Fourier Integral theorem, Fourier transform, Inverse Fourier Transform, Finite Fourier Sine and Cosine transform. Applications : Computer Graphics, Cyber Security, Digital Image Processing, Machine Learning Feature Extraction, Image compression. Complex Analysis: Functions of Complex Variables, Limit Continuity and Differentiability, Analytic Function, Cauchy-Riemann Equations (Cartesian and Polar Form), Harmonic Function, Milne- Thomson Method, Complex integrals, Cauchy- Integral theorem (without proof), Cauchy integral formula (without proof), Taylor’s series and Laurent’s series, Singularities, Residue of a function, Cauchy Residue Theorem, Evaluating the Real definite integrals. Applications : Network Theory, Electromagnetic Waves, Transmission line. Unit V -Sequence and Series: Definition of a Sequence, Monotonic Sequence, Convergence and Divergent Sequence, Cauchy Sequence, Necessary

Notes1

Engineering Mathematics I & II — Complete Notes

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Free preview

engineering-mathematics-i-and-ii-complete-notes.pdf · 45 KB

Syllabus1

Syllabus — Calculus, Laplace Transform and Numerical Techniques (AAS203C)

Official AKTU syllabus, effective from the academic session 2026-27 (AICTE model curriculum / NEP 2020).

Course objectives

Develop the ability to solve differential equations using series methods and special functions applicable to engineering problems. 2. Introduce Laplace and Fourier transform techniques for analyzing differential equations and signal-processing applications. 3. Build a strong foundation in complex variable theory and its applications in engineering and applied sciences. 4. Understand the concepts of sequences and series, convergence tests, and their role in mathematical modeling. 5. Enable students to apply advanced mathematical techniques to solve real-world problems in computer science, electronics, communication, and engineering.

Course content

Determination of indicial roots, cases of indicial roots: Distinct roots, equal roots, roots differ by an integer and not differ by an integer. Applications: Fluid mechanics, quantum mechanical wave equation, Heat transfer, Solving electromagnetic field equation. Laplace Transform: Definition of Laplace Transform, Existence Conditions, Linearity Property, Laplace Transform of some standard functions, I and II Shifting theorems (without proof), Change of Scale Property, Laplace transform of derivatives and integrals, Periodic Functions, Unit Step Function, Inverse Laplace Transform, Partial Fraction Method, Convolution theorem (without proof), Application of Laplace transform to solve ordinary differential equation. Applications : Transfer Function Determination, Circuit Equations, Step Response. Fourier Series& Fourier Transform: Periodic function, even and odd function, Dirichlet’s Condition for the Existence of a Fourier Series, Fourier series expansion of a function in the interval c to c+2l , Change of Interval, Half Range Series: Half range sine and cosine series in the interval -l to l, Parseval’s Identity, Fourier Integral theorem, Fourier transform, Inverse Fourier Transform, Finite Fourier Sine and Cosine transform. Applications : Computer Graphics, Cyber Security, Digital Image Processing, Machine Learning Feature Extraction, Image compression. Complex Analysis: Functions of Complex Variables, Limit Continuity and Differentiability, Analytic Function, Cauchy-Riemann Equations (Cartesian and Polar Form), Harmonic Function, Milne- Thomson Method, Complex integrals, Cauchy- Integral theorem (without proof), Cauchy integral formula (without proof), Taylor’s series and Laurent’s series, Singularities, Residue of a function, Cauchy Residue Theorem, Evaluating the Real definite integrals. Applications : Network Theory, Electromagnetic Waves, Transmission line. Unit V -Sequence and Series: Definition of a Sequence, Monotonic Sequence, Convergence and Divergent Sequence, Cauchy Sequence, Necessary

📘Notes1

Engineering Mathematics I & II — Complete Notes

📋Syllabus1

Syllabus — Calculus, Laplace Transform and Numerical Techniques (AAS203C)

Course objectives

Course content

📘Notes1

Engineering Mathematics I & II — Complete Notes

📋Syllabus1

Syllabus — Calculus, Laplace Transform and Numerical Techniques (AAS203C)

Course objectives

Course content

Notes1

Syllabus1

Notes1

Syllabus1