Q9Mathematics II

Question

Q.9. Find the rank of the matrix $120211302$ .

Answer 1

The classical 3D Laplace equation is mathematically written strictly as:

\frac{\partial ^{2} u}{\partial x ^{2}} + \frac{\partial ^{2} u}{\partial y ^{2}} + \frac{\partial ^{2} u}{\partial z ^{2}} = 0

In the rigorous classification of second-order linear PDEs, we strictly examine the exact coefficients of the second derivative terms. Here, the coefficients for $u_{xx}$ , $u_{y y}$ , and $u_{z z}$ are absolutely all equal to $+ 1$ , completely positive, with strictly no mixed partial derivatives or explicit time derivatives present. This mathematical structure unconditionally classifies the equation exactly as Elliptic, universally representing equilibrium or steady-state potential fields (like electrostatics or steady heat conduction).

Q9Mathematics II

Question

Answer