Engineering Mathematics Flashcards

The probability density function of a normal distribution is characterized by which two parameters?

Answer: Mean (μ) and standard deviation (σ)

The normal (Gaussian) distribution is fully described by its mean μ (center) and standard deviation σ (spread). This distribution is fundamental in engineering statistics and quality control.

What is the curl of a gradient field?

Answer: Always zero

The curl of any gradient field is always zero (∇ × ∇φ = 0). This is a fundamental identity in vector calculus, meaning conservative fields are irrotational.

In the z-transform, what does z⁻¹ represent?

Answer: A unit delay

In discrete-time signal processing and control, z⁻¹ represents a unit delay operator, shifting a sequence by one sample period.

What condition must be satisfied for a function to have a convergent Taylor series?

Answer: The function must be infinitely differentiable and the remainder must approach zero

A function needs to be infinitely differentiable (smooth) at the point, and the Taylor remainder term must converge to zero for the series to converge to the function value.

The method of Lagrange multipliers is used to:

Answer: Optimize a function subject to equality constraints

Lagrange multipliers introduce auxiliary variables to convert a constrained optimization problem into an unconstrained system, finding extrema on constraint surfaces.

What is the trace of a 3×3 identity matrix?

Answer: 3

The trace is the sum of diagonal elements. The 3×3 identity matrix has 1s on the diagonal, so trace = 1 + 1 + 1 = 3.