B.Tech 1st Year Notes (Mathematics – I & Physics)
■ Engineering Mathematics – I
• Limits, Continuity & Differentiability
Limit: lim(x→a) f(x) = L if left-hand limit = right-hand limit = L.
Continuity: f(x) is continuous at x=a if lim(x→a-)f(x) = lim(x→a+)f(x) = f(a).
Differentiability: Function must be continuous + derivative exists.
Common results: lim(x→0) sinx/x = 1, lim(x→0) (1-cosx)/x² = 1/2.
Derivative rule: d(x^n)/dx = n*x^(n-1).
• Partial Differentiation
If z=f(x,y), then ∂z/∂x and ∂z/∂y are partial derivatives.
Chain Rule Example: z=x²+y², x=r cosθ, y=r sinθ → ∂z/∂r=2r, ∂z/∂θ=0.
• Matrices & Determinants
Types: Square, Diagonal, Identity, Singular, Orthogonal.
Properties: Interchanging rows → sign changes; if two rows equal → det=0.
Inverse (Adjoint method): A^-1 = Adj(A)/|A|.
Rank: Number of non-zero rows in echelon form.
• Differential Equations
First order linear DE: dy/dx + Py = Q. Solution: y·e^(∫Pdx) = ∫Qe^(∫Pdx)dx + C.
Second order: y'' + ay' + by=0. Solve using auxiliary equation: m²+am+b=0.
• Vector Calculus
Gradient: ∇f = ∂f/∂x i + ∂f/∂y j + ∂f/∂z k.
Divergence: ∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z.
Curl: ∇×F = | i j k; ∂/∂x ∂/∂y ∂/∂z; P Q R |.
■ Engineering Physics
• Wave Optics
Interference: Fringe width β = λD/d.
Diffraction: Minima condition a sinθ = nλ.
Polarization: Malus Law I=I0 cos²θ.
• Lasers & Optical Fibers
Laser: Light Amplification by Stimulated Emission of Radiation.
Properties: Coherent, monochromatic, intense, directional.
Optical Fiber principle: Total internal reflection.
■ Engineering Mathematics – I
• Limits, Continuity & Differentiability
Limit: lim(x→a) f(x) = L if left-hand limit = right-hand limit = L.
Continuity: f(x) is continuous at x=a if lim(x→a-)f(x) = lim(x→a+)f(x) = f(a).
Differentiability: Function must be continuous + derivative exists.
Common results: lim(x→0) sinx/x = 1, lim(x→0) (1-cosx)/x² = 1/2.
Derivative rule: d(x^n)/dx = n*x^(n-1).
• Partial Differentiation
If z=f(x,y), then ∂z/∂x and ∂z/∂y are partial derivatives.
Chain Rule Example: z=x²+y², x=r cosθ, y=r sinθ → ∂z/∂r=2r, ∂z/∂θ=0.
• Matrices & Determinants
Types: Square, Diagonal, Identity, Singular, Orthogonal.
Properties: Interchanging rows → sign changes; if two rows equal → det=0.
Inverse (Adjoint method): A^-1 = Adj(A)/|A|.
Rank: Number of non-zero rows in echelon form.
• Differential Equations
First order linear DE: dy/dx + Py = Q. Solution: y·e^(∫Pdx) = ∫Qe^(∫Pdx)dx + C.
Second order: y'' + ay' + by=0. Solve using auxiliary equation: m²+am+b=0.
• Vector Calculus
Gradient: ∇f = ∂f/∂x i + ∂f/∂y j + ∂f/∂z k.
Divergence: ∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z.
Curl: ∇×F = | i j k; ∂/∂x ∂/∂y ∂/∂z; P Q R |.
■ Engineering Physics
• Wave Optics
Interference: Fringe width β = λD/d.
Diffraction: Minima condition a sinθ = nλ.
Polarization: Malus Law I=I0 cos²θ.
• Lasers & Optical Fibers
Laser: Light Amplification by Stimulated Emission of Radiation.
Properties: Coherent, monochromatic, intense, directional.
Optical Fiber principle: Total internal reflection.
