Questions
2 questions in major PSU papers
Difficulty
Medium
Importance
High yield for HPCL/NTPC/ONGC
Overview
Calculus forms the backbone of engineering mathematics, focusing on rates of change, accumulation, and spatial dynamics. Mastering this topic is essential for PSU exams as it provides the analytical tools required for solving complex mechanical and electrical system problems. Aspirants must prioritize fluency in derivative applications and integral transformation techniques to secure marks.
Functions and Partial Derivatives
Partial derivatives allow for the analysis of functions where multiple independent variables interact simultaneously. In PSU exams, these are frequently tested through chain rule applications and total differential equations.
- Total Differential: dz = (∂z/∂x)dx + (∂z/∂y)dy
- Euler's Theorem for Homogeneous Functions: x(∂u/∂x) + y(∂u/∂y) = nu
- Chain Rule: ∂z/∂t = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)
- Schwarz's Theorem: ∂²f/∂x∂y = ∂²f/∂y∂x
Maxima and Minima
This section involves identifying stationary points in multi-variable functions to optimize engineering designs. Candidates must be proficient in checking the Hessian matrix or second-order partial derivatives to classify points.
- Condition for stationary point: ∂f/∂x = 0 and ∂f/∂y = 0
- Hessian determinant: D = (fxx * fyy) - (fxy)²
- Local Maximum: D > 0 and fxx < 0
- Local Minimum: D > 0 and fxx > 0
- Saddle Point: D < 0
Vector Calculus
Vector calculus is crucial for understanding electromagnetic fields and fluid flow, which are staples of PSU technical papers. The focus remains on evaluating line, surface, and volume integrals using fundamental integral theorems.
- Gradient of scalar field: ∇φ
- Divergence of vector field: ∇·F
- Curl of vector field: ∇×F
- Gauss Divergence Theorem: ∭(∇·F)dV = ∬(F·n)dS
- Stokes' Theorem: ∮(F·dr) = ∬(∇×F)·ndS
Definite and Improper Integrals
Definite integrals are used to calculate areas, volumes, and centroids, while improper integrals test convergence limits. Success here depends on identifying standard patterns like Beta and Gamma functions.
- Leibniz Rule for differentiation under integral sign
- Gamma Function: Γ(n) = ∫[0 to ∞] x^(n-1)e^(-x) dx
- Beta Function: B(m,n) = Γ(m)Γ(n)/Γ(m+n)
- Wallis Formula for integrals of trigonometric powers
Formula Sheet
dz = (∂z/∂x)dx + (∂z/∂y)dy
x(∂u/∂x) + y(∂u/∂y) = nu
D = fxx*fyy - (fxy)^2
∇·F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z
∇×F = |i, j, k; ∂/∂x, ∂/∂y, ∂/∂z; Fx, Fy, Fz|
Γ(n) = (n-1)!
B(m,n) = ∫[0 to 1] x^(m-1)(1-x)^(n-1) dx
Exam Tip
Focus on the physical interpretation of Gradient, Divergence, and Curl, as many PSU questions ask for the logic behind these operators rather than just the calculation.
Common Mistakes
- Miscalculating the sign of the Hessian determinant during Maxima/Minima optimization.
- Forgetting to check the orientation of the surface normal vector in Stokes' and Gauss' theorems.
- Incorrectly identifying a function as homogeneous when determining the order for Euler's theorem.
More Revision Notes
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