Engineering Mathematics Flashcards

What is the Laplace transform of the function f(t) = e^(-at)?

Answer: 1/(s+a)

The Laplace transform of e^(-at) is 1/(s+a) for s > -a, which is a fundamental transform used extensively in control systems and circuit analysis.

A matrix A has eigenvalues 2 and 5. What is the determinant of A?

Answer: 10

The determinant of a matrix equals the product of its eigenvalues. Therefore det(A) = 2 × 5 = 10.

Which numerical method uses the formula x_{n+1} = x_n - f(x_n)/f'(x_n)?

Answer: Newton-Raphson method

The Newton-Raphson method iteratively refines a root estimate using the function value and its derivative, converging quadratically near simple roots.

What is the divergence of the vector field F = (x², xy, xz)?

Answer: 2x + x + x = 4x

Divergence is ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z = 2x + x + x = 4x. This measures the rate of flux expansion at a point.

For the differential equation dy/dx + 2y = e^(-x), what is the integrating factor?

Answer: e^(2x)

For a first-order linear ODE dy/dx + P(x)y = Q(x), the integrating factor is e^(∫P(x)dx) = e^(∫2dx) = e^(2x).

What does the Fourier series represent?

Answer: A periodic function as a sum of sines and cosines

A Fourier series decomposes a periodic function into a sum of sine and cosine terms (harmonics), essential in signal processing and vibration analysis.