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| Subtopic |
Key Concepts |
Video Focus Areas |
| 1.1 |
Types of Errors &
Significant Figures |
True error, absolute error,
relative error, percentage error. Significant figures rules. |
| 1.2 |
Accuracy, Precision & Error
Propagation |
Accuracy vs. Precision. Error
propagation in arithmetic operations (addition, subtraction, multiplication,
division). |
| 1.3 |
Applications in Chemical
Engineering |
Error analysis in reactor
design, process control, and data reconciliation. |
| Subtopic |
Key Concepts |
Video Focus Areas |
| 2.1 |
Basic Properties & Roots of
Equations |
Relations between roots and
coefficients. Descartes' Rule of Signs. Synthetic division of a polynomial. |
| 2.2 |
Bracketing Methods: Bisection
& False Position |
Bisection method (interval
halving). Regula Falsi (False Position) method using secant interpolation. |
| 2.3 |
Secant Method |
Open method (no bracketing
required). Uses two initial guesses. |
| 2.4 |
Newton-Raphson Method |
Open method using derivative.
Quadratic convergence. |
| 2.5 |
Newton-Raphson for Non-Linear
Equations in Two Variables |
Extension of N-R for systems of
non-linear equations. |
| 2.6 |
Convergence of Iterative Methods |
Rate of convergence: Linear,
Super-linear, Quadratic. Convergence criteria. |
| Subtopic |
Key Concepts |
Video Focus Areas |
| 3.1 |
Mathematical Background of
Matrices |
Matrix operations: Addition,
multiplication, transpose, determinant, inverse. |
| 3.2 |
Gauss Elimination Method |
Forward elimination and back
substitution. Augmented matrix. Pivoting (partial and full) to avoid division
by zero. |
| 3.3 |
Gauss-Jordan Elimination Method |
Gauss-Jordan method (elimination
on both sides). Finding the inverse of a matrix. |
| 3.4 |
Jacobi & Gauss-Seidel
Iterative Methods |
Jacobi method (using previous
iteration values). Gauss-Seidel method (using updated values immediately).
Convergence condition (diagonally dominant). |
| 3.5 |
Eigen Value Problem |
Finding eigenvalues and
eigenvectors. Power method (for the largest eigenvalue). |
| Subtopic |
Key Concepts |
Video Focus Areas |
| 4.1 |
Method of Least Squares |
Principle of least squares:
Minimize the sum of squared errors. |
| 4.2 |
Fitting a Straight Line &
Polynomial |
Fitting y = a + bx. Fitting y =
a + bx + cx². Normal equations. |
| 4.3 |
Fitting Non-Linear Functions |
Linearization of non-linear
functions: Geometric (y = ab^x), Exponential (y = ae^(bx)), Hyperbola (y = a
+ b/x). |
| 4.4 |
Fitting Trigonometric Functions |
Fitting periodic data using
sine/cosine functions. |
| Subtopic |
Key Concepts |
Video Focus Areas |
| 5.1 |
Finite Differences: Forward,
Backward & Divided Differences |
Forward difference table.
Backward difference table. Divided difference table. |
| 5.2 |
Newton's Interpolation Formulas |
Newton's Forward Interpolation
formula. Newton's Backward Interpolation formula. Newton's Divided Difference
formula. |
| 5.3 |
Lagrange Interpolation Formula |
Lagrange interpolation (for
unequally spaced data). Inverse Interpolation. |
| Subtopic |
Key Concepts |
Video Focus Areas |
| 6.1 |
Numerical Differentiation
Formulas |
Differentiation at equal
intervals (forward, backward, central difference). Differentiation at unequal
intervals. |
| 6.2 |
Trapezoidal Rule |
Trapezoidal rule for numerical
integration: ∫f(x)dx ≈ (h/2)[f(x₀) + f(xₙ) + 2Σf(xᵢ)]. |
| 6.3 |
Simpson's 1/3 & 3/8 Rules |
Simpson's 1/3 rule (parabolic
approximation). Simpson's 3/8 rule. |
| Subtopic |
Key Concepts |
Video Focus Areas |
| 7.1 |
Taylor's Series & Euler's
Method |
Taylor's series expansion for
ODEs. Euler's method: y_{n+1} = y_n + h f(x_n, y_n). Local truncation error
(O(h²)). |
| 7.2 |
Modifications & Improvements
in Euler's Method |
Modified Euler (Heun's) method:
Predictor-corrector. Improved Euler (midpoint method). |
| 7.3 |
Runge-Kutta 2nd Order Methods |
RK-2 methods (Heun's, Midpoint,
Ralston). Use of multiple slopes for better accuracy. |
| 7.4 |
Runge-Kutta 4th Order Method
(RK4) |
RK4: The classic 4th-order
method. Use of 4 slopes to get a more accurate estimate. |
| 7.5 |
Milne's Predictor-Corrector
Method |
Multi-step method (uses previous
points). Milne's predictor and corrector formulas. |
| 7.6 |
Boundary Value Problems |
Converting BVP to IVP using the
shooting method. Finite difference method for BVP. |