Engineering Mathematics M1
Calculus and Vector Analysis
Limits
M1Basic Limit Definition
\[ \lim_{x \to a} f(x) = L \]
If for every ε > 0, there exists δ > 0 such that 0 < |x - a| < δ implies |f(x) - L| < ε.
Limit Laws
\[ \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \]
\[ \lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \]
\[ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, \quad \text{if } \lim_{x \to a} g(x) \neq 0 \]
Important Limits
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]
\[ \lim_{x \to 0} \frac{1 - \cos x}{x} = 0 \]
\[ \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e \]
Continuity
M1Definition of Continuity
\[ f \text{ is continuous at } a \iff \lim_{x \to a} f(x) = f(a) \]
Intermediate Value Theorem
\[ \text{If } f \text{ is continuous on } [a,b] \text{ and } k \text{ is between } f(a) \text{ and } f(b), \]
\[ \text{then } \exists c \in (a,b) \text{ such that } f(c) = k \]
Differentiation
M1Derivative Definition
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
Basic Rules
\[ \frac{d}{dx}(c) = 0 \]
\[ \frac{d}{dx}(x^n) = nx^{n-1} \]
\[ \frac{d}{dx}(e^x) = e^x \]
\[ \frac{d}{dx}(\ln x) = \frac{1}{x} \]
\[ \frac{d}{dx}(\sin x) = \cos x \]
\[ \frac{d}{dx}(\cos x) = -\sin x \]
Product & Quotient Rules
\[ (fg)' = f'g + fg' \]
\[ \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2} \]
Chain Rule
\[ \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \]
Mean Value Theorems
M1Rolle's Theorem
\[ \text{If } f \text{ is continuous on } [a,b], \text{ differentiable on } (a,b), \text{ and } f(a) = f(b), \]
\[ \text{then } \exists c \in (a,b) \text{ such that } f'(c) = 0 \]
Mean Value Theorem (MVT)
\[ \text{If } f \text{ is continuous on } [a,b] \text{ and differentiable on } (a,b), \]
\[ \text{then } \exists c \in (a,b) \text{ such that } f'(c) = \frac{f(b) - f(a)}{b - a} \]
Cauchy's Mean Value Theorem
\[ \frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(c)}{g'(c)} \]
Taylor's & Maclaurin's Series
M1Taylor Series Expansion
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n \]
Maclaurin Series (a=0)
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \]
Common Expansions
\[ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \]
\[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \]
\[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \]
\[ \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \]
Integration
M1Basic Integrals
\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1 \]
\[ \int e^x dx = e^x + C \]
\[ \int \frac{1}{x} dx = \ln|x| + C \]
\[ \int \sin x dx = -\cos x + C \]
\[ \int \cos x dx = \sin x + C \]
Integration by Parts
\[ \int u dv = uv - \int v du \]
Substitution Rule
\[ \int f(g(x))g'(x) dx = \int f(u) du, \quad u = g(x) \]
Definite Integrals
M1Fundamental Theorem of Calculus
\[ \int_a^b f(x) dx = F(b) - F(a), \quad \text{where } F'(x) = f(x) \]
Properties
\[ \int_a^b f(x) dx = -\int_b^a f(x) dx \]
\[ \int_a^b [f(x) + g(x)] dx = \int_a^b f(x) dx + \int_a^b g(x) dx \]
\[ \int_a^b cf(x) dx = c \int_a^b f(x) dx \]
Partial Derivatives
M1Definition
\[ \frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h,y) - f(x,y)}{h} \]
Clairaut's Theorem
\[ f_{xy} = f_{yx} \quad \text{if both are continuous} \]
Chain Rule for Partial Derivatives
\[ \frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt} \]
Vector Calculus Basics
M1Gradient
\[ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \]
Divergence
\[ \nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \]
Curl
\[ \nabla \times \mathbf{F} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
F_x & F_y & F_z
\end{vmatrix} \]
Engineering Mathematics M2
Linear Algebra and Differential Equations
Matrices
M2Matrix Multiplication
\[ (AB)_{ij} = \sum_{k=1}^n A_{ik} B_{kj} \]
Determinant of 2×2
\[ \det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc \]
Inverse of 2×2
\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
Transpose Properties
\[ (A^T)^T = A \]
\[ (AB)^T = B^T A^T \]
Eigenvalues & Eigenvectors
M2Characteristic Equation
\[ \det(A - \lambda I) = 0 \]
Eigenvalue Definition
\[ A\mathbf{v} = \lambda\mathbf{v} \]
Trace and Determinant
\[ \text{tr}(A) = \sum_{i=1}^n \lambda_i \]
\[ \det(A) = \prod_{i=1}^n \lambda_i \]
System of Linear Equations
M2Matrix Form
\[ A\mathbf{x} = \mathbf{b} \]
Cramer's Rule (2×2)
\[ x = \frac{\det\begin{pmatrix} b_1 & a_{12} \\ b_2 & a_{22} \end{pmatrix}}{\det(A)}, \quad y = \frac{\det\begin{pmatrix} a_{11} & b_1 \\ a_{21} & b_2 \end{pmatrix}}{\det(A)} \]
Gaussian Elimination
\[ \begin{bmatrix} a_{11} & a_{12} & | & b_1 \\ a_{21} & a_{22} & | & b_2 \end{bmatrix} \rightarrow \begin{bmatrix} 1 & a_{12}' & | & b_1' \\ 0 & 1 & | & b_2' \end{bmatrix} \]
Differential Equations
M2First Order Linear
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
\[ \text{Solution: } y = e^{-\int P dx} \left( \int Q e^{\int P dx} dx + C \right) \]
Second Order Constant Coefficients
\[ ay'' + by' + cy = 0 \]
\[ \text{Characteristic: } ar^2 + br + c = 0 \]
Homogeneous Solution Cases
\[ \text{Real distinct roots: } y = C_1 e^{r_1 x} + C_2 e^{r_2 x} \]
\[ \text{Real repeated root: } y = (C_1 + C_2 x) e^{r x} \]
\[ \text{Complex roots: } y = e^{\alpha x} (C_1 \cos \beta x + C_2 \sin \beta x) \]
Laplace Transforms
M2Definition
\[ \mathcal{L}\{f(t)\} = F(s) = \int_0^\infty e^{-st} f(t) dt \]
Common Transforms
\[ \mathcal{L}\{1\} = \frac{1}{s} \]
\[ \mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}} \]
\[ \mathcal{L}\{e^{at}\} = \frac{1}{s-a} \]
\[ \mathcal{L}\{\sin(kt)\} = \frac{k}{s^2 + k^2} \]
\[ \mathcal{L}\{\cos(kt)\} = \frac{s}{s^2 + k^2} \]
Derivative Property
\[ \mathcal{L}\{f'(t)\} = sF(s) - f(0) \]
Fourier Series
M2Fourier Series Expansion
\[ f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty \left[ a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right] \]
Coefficients
\[ a_0 = \frac{1}{L} \int_{-L}^L f(x) dx \]
\[ a_n = \frac{1}{L} \int_{-L}^L f(x) \cos\left(\frac{n\pi x}{L}\right) dx \]
\[ b_n = \frac{1}{L} \int_{-L}^L f(x) \sin\left(\frac{n\pi x}{L}\right) dx \]
Numerical Methods
M2Newton-Raphson Method
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
Trapezoidal Rule
\[ \int_a^b f(x) dx \approx \frac{h}{2} \left[ f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right] \]
Simpson's 1/3 Rule
\[ \int_a^b f(x) dx \approx \frac{h}{3} \left[ f(x_0) + 4\sum_{i=1,3,5}^{n-1} f(x_i) + 2\sum_{i=2,4,6}^{n-2} f(x_i) + f(x_n) \right] \]
Euler's Method
\[ y_{n+1} = y_n + h f(x_n, y_n) \]