EIT Mathematics & Engineering Fundamentals 3 — Questions and Answers
Question 1: The divergence theorem relates a volume integral of ∇·F to:
- The surface integral of F·n over the closed boundary (Correct answer)
- The line integral of F·dr around the boundary
- The curl of F over the volume
- The gradient of F at the centroid
Correct 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.
Question 2: A 3×3 identity matrix multiplied by any 3×3 matrix A yields:
- A (Correct answer)
- 0
- I
- A^T
Correct answer: A
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:
- μ = 0, σ = 1 (Correct answer)
- μ = 1, σ = 0
- μ = 0, σ = 0
- μ = 1, σ = 1
Correct 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.
Question 4: Using Euler's method with step h to solve dy/dx = f(x,y), the update formula is:
- y_{n+1} = y_n + h·f(x_n, y_n) (Correct answer)
- y_{n+1} = y_n - h·f(x_n, y_n)
- y_{n+1} = y_n + h²·f(x_n, y_n)
- y_{n+1} = y_n·e^(h·f)
Correct 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.
Question 5: The eigenvalues of matrix [[3, 1],[0, 3]] are:
- 3 and 3 (repeated) (Correct answer)
- 3 and 0
- 1 and 3
- 0 and 9
Correct answer: 3 and 3 (repeated)
This upper triangular matrix has eigenvalues equal to its diagonal entries, both equal to 3.
Question 6: The line integral ∮F·dr around a closed path equals zero when F is:
- Conservative (path-independent) (Correct answer)
- Divergence-free
- A constant vector
- Perpendicular to the path
Correct 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.
Question 7: The Fourier series of an odd function contains only:
- Sine terms (Correct answer)
- Cosine terms
- Both sine and cosine terms
- A constant term only
Correct answer: Sine terms
Odd functions satisfy f(-x) = -f(x), and their Fourier series contain only sine terms because cosine is even.
EIT Mathematics & Engineering Fundamentals 3
The divergence theorem relates a volume integral of ∇·F to: