Mathematics & Engineering Fundamentals Flashcards

The divergence theorem relates a volume integral of ∇·F to:

Answer: The surface integral of F·n over the closed boundary

Gauss's divergence theorem states ∫∫∫(∇·F)dV = ∯(F·n)dS over the enclosing surface.

A 3×3 identity matrix multiplied by any 3×3 matrix A yields:

Answer: A

The identity matrix is the multiplicative identity for matrix multiplication: I·A = A·I = A.

The standard normal distribution has mean and standard deviation equal to:

Answer: μ = 0, σ = 1

By definition, standardizing a normal variable Z = (X-μ)/σ yields a distribution with mean 0 and standard deviation 1.

Using Euler's method with step h to solve dy/dx = f(x,y), the update formula is:

Answer: y_{n+1} = y_n + h·f(x_n, y_n)

Euler's method uses the first-order forward difference: the next value equals current plus step times slope.

The eigenvalues of matrix [[3, 1],[0, 3]] are:

Answer: 3 and 3 (repeated)

This upper triangular matrix has eigenvalues equal to its diagonal entries, both equal to 3.

The line integral ∮F·dr around a closed path equals zero when F is:

Answer: Conservative (path-independent)

A conservative field satisfies ∮F·dr = 0 for any closed path, which is equivalent to F = ∇φ for some scalar φ.

The Fourier series of an odd function contains only:

Answer: Sine terms

Odd functions satisfy f(-x) = -f(x), and their Fourier series contain only sine terms because cosine is even.