Mathematics & Engineering Fundamentals Flashcards

The partial derivative ∂/∂x of f(x,y) = x²y + sin(xy) is:

Answer: 2xy + y·cos(xy)

Treating y as constant: ∂/∂x(x²y) = 2xy and ∂/∂x(sin(xy)) = y·cos(xy) by the chain rule.

For the Poisson distribution with mean λ, the variance is:

Answer: λ

A unique property of the Poisson distribution is that both its mean and variance equal the parameter λ.

The rank of a matrix is defined as the:

Answer: Number of linearly independent rows (or columns)

Rank equals the maximum number of linearly independent row vectors (or equivalently column vectors) in the matrix.

What does a Bode plot display?

Answer: Magnitude and phase of a transfer function vs. log frequency

A Bode plot consists of two graphs: magnitude (in dB) and phase (in degrees) plotted against logarithmic frequency.

The inverse Laplace transform of 1/(s² + ω²) is:

Answer: sin(ωt)/ω

From standard Laplace transform tables, L{sin(ωt)} = ω/(s²+ω²), so L⁻¹{1/(s²+ω²)} = sin(ωt)/ω.

In regression analysis, the coefficient of determination R² represents:

Answer: The proportion of variance in y explained by x

R² ranges from 0 to 1 and measures how well the independent variable(s) explain the variability of the dependent variable.

Stoke's theorem relates the surface integral of the curl of F to:

Answer: The line integral of F around the bounding curve

Stokes' theorem states ∫∫(∇×F)·dS = ∮F·dr, linking the curl flux through a surface to circulation around its boundary.