Dr. Brajesh Kumar Jha

Advanced Numerical Problems on Partial Differentiation Problem 1: Laplace’s Equation Problem: Verify that \( u = x^3 - 3xy^2 \) satisfies Laplace’s equation. Show / Hide Solution \[ u_x = 3x^2 - 3y^2, \quad u_y = -6xy \] \[ u_{xx} = 6x, \quad u_{yy} = -6x \] \[ u_{xx} + u_{yy} = 6x - 6x = 0 \] Hence verified. Problem 2: Total Differential Problem: If \( z = e^{xy}\sin(x+y) \), find total differential \( dz \). Show / Hide Solution \[ dz = z_x dx + z_y dy \] \[ z_x = e^{xy}[y\sin(x+y)+\cos(x+y)] \] \[ z_y = e^{xy}[x\sin(x+y)+\cos(x+y)] \] \[ dz = e^{xy}[y\sin(x+y)+\cos(x+y)]dx \] \[ + e^{xy}[x\sin(x+y)+\cos(x+y)]dy \] Problem 3: Directional Derivative Problem: Find directional derivative of \( \phi=x^2y+yz^3 \) at (1,-1,1) in direction \(2\mathbf{i}-\mathbf{j}+2\mathbf{k}\). Show / Hide Solution \[ \nabla \phi = (2xy)\mathbf{i} + (x^2+z^3)\mathbf{j} + (3yz^2)\mathbf{k} \] At (1,-1,1): \[ \nabla \phi = (-2)\mathbf{i} + 2\mathbf{j} - 3\...

Numerical Examples on Partial Differentiation Example 1: First Order Partial Derivatives Problem: If \( z = 3x^2y + 2xy^3 - 5y \), find \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \). Show / Hide Solution Differentiate w.r.t \(x\): \[ \frac{\partial z}{\partial x} = 6xy + 2y^3 \] Differentiate w.r.t \(y\): \[ \frac{\partial z}{\partial y} = 3x^2 + 6xy^2 - 5 \] Example 2: Second Order Partial Derivatives Problem: If \( z = x^3y^2 + 4xy \), find \( \frac{\partial^2 z}{\partial x^2} \). Show / Hide Solution First derivative: \[ \frac{\partial z}{\partial x} = 3x^2y^2 + 4y \] Second derivative: \[ \frac{\partial^2 z}{\partial x^2} = 6xy^2 \] Example 3: Mixed Partial Derivatives Problem: If \( z = x^2y^3 \), verify mixed partial derivatives. Show / Hide Solution \[ \frac{\partial z}{\partial x} = 2xy^3 \] \[ \frac{\partial^2 z}{\partial y \partial x} = 6xy^2 \] \[ \frac{\partial z}{\partial y} = 3x^2y^2 \] ...