1. **Problem 1:** Given $$z = x^2 \tan^{-1}\left(\frac{y}{x}\right) - y^2 \tan^{-1}\left(\frac{x}{y}\right),$$ show that $$\frac{\partial^2 z}{\partial x \partial y} = \frac{x^2 - y^2}{x^2 + y^2}.$$ Step 1: Compute $$\frac{\partial z}{\partial y}$$ and then differentiate with respect to $$x$$ carefully using product and chain rules. Step 2: Simplify the resulting expression to obtain the given formula. 2. **Problem 2:** Let $$u = (x^2 + y^2 + z^2)^{-1/2}.$$ Prove that $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0.$$ Step 1: Compute first derivatives $$u_x, u_y, u_z$$ using chain rule. Step 2: Compute second derivatives $$u_{xx}, u_{yy}, u_{zz}$$. Step 3: Sum them and simplify to show the sum equals zero, confirming $$u$$ is harmonic. 3. **Problem 3:** Given $$u = x^2 y + y^2 z + z^2 x,$$ show that $$u_x + u_y + u_z = (x + y + z)^2.$$ Step 1: Compute partial derivatives: $$u_x = 2xy + z^2,$$ $$u_y = x^2 + 2yz,$$ $$u_z = y^2 + 2zx.$$ Step 2: Sum them: $$u_x + u_y + u_z = 2xy + z^2 + x^2 + 2yz + y^2 + 2zx = x^2 + y^2 + z^2 + 2(xy + yz + zx) = (x + y + z)^2.$$ 4. **Problem 4:** Given $$z = e^{ax + by} f(ax - by),$$ show that $$b \frac{\partial z}{\partial x} + a \frac{\partial z}{\partial y} = 2abz.$$ Step 1: Compute $$\frac{\partial z}{\partial x} = a e^{ax + by} f(ax - by) + e^{ax + by} f'(ax - by) a.$$ Step 2: Compute $$\frac{\partial z}{\partial y} = b e^{ax + by} f(ax - by) + e^{ax + by} f'(ax - by)(-b).$$ Step 3: Multiply and sum: $$b \frac{\partial z}{\partial x} + a \frac{\partial z}{\partial y} = b[a e^{ax + by} f + a e^{ax + by} f'] + a[b e^{ax + by} f - b e^{ax + by} f'] = 2ab e^{ax + by} f = 2abz.$$ 5. **Problem 5:** Given $$u = \log(x^3 + y^3 + z^3 - 3xyz),$$ prove that $$\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = \frac{3}{x + y + z}.$$ Step 1: Compute $$\frac{\partial u}{\partial x} = \frac{3x^2 - 3yz}{x^3 + y^3 + z^3 - 3xyz}$$ and similarly for $$y$$ and $$z$$. Step 2: Sum the derivatives and simplify numerator and denominator using factorization of $$x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$$ to get the result. **Final answers:** 1. $$\frac{\partial^2 z}{\partial x \partial y} = \frac{x^2 - y^2}{x^2 + y^2}.$$ 2. $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0.$$ 3. $$u_x + u_y + u_z = (x + y + z)^2.$$ 4. $$b \frac{\partial z}{\partial x} + a \frac{\partial z}{\partial y} = 2abz.$$ 5. $$\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = \frac{3}{x + y + z}.$$