Solution of ∇²T = 0 in a square domain (0 < x < 1 and 0 < y < 1) with boundary conditions:

T(x, 0) = x; T(0, y) = y; T(x, 1) = 1 + x; T(1, y) = 1 + y is

Given a function $\rm ϕ = \frac{1}{2} (x^2 + y^2 + z^2) $ in three-dimensional Cartesian space, the value of the surface integral

∯_S n̂ . ∇ϕ dS

where S is the surface of a sphere of unit radius and n̂ is the outward unit normal vector on S, is

The Fourier series expansion of x³ in the interval −1 ≤ x < 1 with periodic continuation has

If A = $\begin{bmatrix} 10 & 2k + 5 \\\ 3k - 3 & k + 5 \end{bmatrix} $ is a symmetric matrix, the value of k is _______.