Engineering Mathematics Flashcards

Stokes' theorem relates a surface integral to:

Answer: A line integral around the boundary of the surface

Stokes' theorem equates the surface integral of the curl of a vector field to the line integral of the field around the boundary curve, generalizing Green's theorem to 3D.

What is the primary purpose of singular value decomposition (SVD)?

Answer: Decomposing any matrix into orthogonal and diagonal components for analysis

SVD decomposes any m×n matrix into UΣV^T where U and V are orthogonal and Σ is diagonal. It is used in data compression, pseudoinverse computation, and principal component analysis.

The error function erf(x) is commonly used in engineering for:

Answer: Calculating probabilities related to normal distributions

The error function erf(x) = (2/√π)∫₀ˣ e^(-t²)dt is directly related to the cumulative normal distribution and appears frequently in heat transfer and diffusion problems.

In regression analysis, the coefficient of determination R² represents:

Answer: The proportion of variance in the dependent variable explained by the model

R² ranges from 0 to 1 and indicates how well the regression model explains the variability of the response data. An R² of 0.85 means 85% of variance is explained by the model.

The trapezoidal rule approximation error is proportional to:

Answer: h² (the square of the step size)

The trapezoidal rule has a local error of O(h³) per interval and a global error of O(h²), making it a second-order method. Halving h reduces the global error by a factor of 4.

A homogeneous system of linear equations always has:

Answer: At least the trivial solution (all zeros)

A homogeneous system Ax = 0 always has x = 0 as a solution (the trivial solution). It may also have infinitely many solutions if the matrix A is singular (rank-deficient).