EIT Mathematics & Engineering Fundamentals 3

Question 1

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

Accepted Answer

The surface integral of F·n over the closed boundary

Answer

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

Question 2

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

Accepted Answer

A

Answer

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

Question 3

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

Accepted Answer

μ = 0, σ = 1

Answer

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

Question 4

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

Accepted Answer

y_{n+1} = y_n + h·f(x_n, y_n)

Answer

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

Question 5

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

Accepted Answer

3 and 3 (repeated)

Answer

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

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