Engineering Mathematics Flashcards

What is the rank of a 3×3 matrix with two identical rows?

Answer: At most 2

Two identical rows are linearly dependent, reducing the maximum number of linearly independent rows to at most 2. The rank could be lower if additional dependencies exist.

Simpson's 1/3 rule requires how many equally spaced points minimum?

Answer: 3 points (2 intervals)

Simpson's 1/3 rule fits a quadratic polynomial through 3 points, requiring a minimum of 2 equal subintervals. It provides exact results for polynomials up to degree 3.

What is the gradient of the scalar field φ = x²y + yz²?

Answer: (2xy, x² + z², 2yz)

The gradient is (∂φ/∂x, ∂φ/∂y, ∂φ/∂z) = (2xy, x² + z², 2yz). The gradient points in the direction of steepest increase.

A system of linear equations Ax = b has no solution when:

Answer: rank(A) < rank([A|b])

By the Rouché–Capelli theorem, a system is inconsistent (no solution) when the rank of the coefficient matrix is less than the rank of the augmented matrix.

The second derivative test classifies a critical point as a saddle point when:

Answer: D = f_xx·f_yy - (f_xy)² < 0

When the discriminant D is negative, the critical point is a saddle point regardless of the sign of f_xx, indicating the surface curves up in one direction and down in another.

What is the order of the Runge-Kutta method commonly known as RK4?

Answer: Fourth order

RK4 is a fourth-order method, meaning its local truncation error is O(h⁵) and global error is O(h⁴), providing excellent accuracy for solving ordinary differential equations.